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PROFESSIONAL PAPER OF THE CORPS OF ENGINEERS NO. 34

TIDAL HYDRAULICS

By

BRIGADIER GENERAL GEORGE B. PILLSBURY
U.S. A., RETIRED

NOVEMBER 1939

WAR DEPARTMENT
CORPS OF ENGINEERS, U.S. ARMY

UNITED STATES GOVERNMENT PRINTING OFFICE WASHINGTON : 1940

For sale by the Superintendent of Documents, Washington, D. Ciikeute: fa) Neve = = Price $1.00

PREFACE

The treatment of the tides, and of tidal datum planes, contained
in the first four chapters of this text, and of the reduction of measured
tidal currents, in chapter X, is drawn from the manuals issued by the
United States Coast and Geodetic Survey, and from Harris’s Manual
of Tides, published in past reports of that Survey, but now out of
print. As no engineer outside that Survey may expect the occasion
to undertake the laborious harmonic analysis of the tides at a station,
the voluminous tables required for the purpose are not included.

The cubature of a channel, described in chapter VI, is set forth in
a number of French texts. The detailed procedure explained is that
developed in the United States Engineer office at Philadelphia.

A method is developed in chapters V and VIII for computing tidal
currents from the constants commonly used for steady flow, by a
procedure somewhat analogous to that used in ordinary hydraulic
computations. Quite obviously, the varying and periodically revers-
ing flow in a tidal channel has somewhat the same relation to steady
flow that an alternating electric current has to a direct current. As
alternating currents depend upon the reactance and capacity of the
circuit as well as upon its resistance, so tidal currents depend upon
the acceleration head and the storage and release of water in the
channel as well as upon frictional resistance. When these factors
are included, computations of tidal flow should be as reliable as are
those for steady flow. The application of these principles to natural
tidal channels is taken up in chapter IX.

(IIT)

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CONTENTS

TIDES
Pages
Chapter I. General definitions—tide-producing forees—equilibrium tides_ 1
Miapterell Harmonic analysis. of tides...2-..-.._---2_---_.22.-----=. 27
eateries ©haracteristics of the tidesi...) 2! <2 5.5 2 24.5222. el ee 75
eer Micali dabumyplanes 32-2. -A2e. 22. So eek sek 87
TIDAL CURRENTS
Ghapter V. Relation of current to surface slope___=----___-___---==-=- 109
Chapter VI. Continuity of flow in long tidal channels—cubature____-____ 145
Chapter VII. Frictionless flow in a long canal of uniform dimensions;
SONCMWGS.. 5 Bee OE eae op cele en ee ca 157
Chapter VIII. Computation of tides and currents.in long canals when
Oa SBM Obst CHLOMICSS seats ae oye cr ak Sees, Boyle eS CN ST Reese De 183
Chapter IX. Tides and currents in estuaries and inlets_________-_---_-- 22%
Chapter X. Offshore tidal currents; reduction of current observations and
CRURREMUB ORE CC ULO Ils came ere tine 8g ee ae SS es Se a ee nee eolee Be 249
AGVenGixaem Mc uivalents anc constants= 25-422 2— =. oes eee 259
Appendix II. Derivation of reduction factors F(Mn) and 1.02F; and
OGRE CU OMBACE OIL plots says eee i es ee ee ee AE ee eee 260
TABLES
lee stronomicalispeeds olmmoon and) Sunes] === sss == === ae 34
Me tilrercdal-components and speeds... 2245.22 22-2 52st le ake 40, 41
IV. Mean values of equilibrium coefficients___..._--.-------+----- 70
V. Principal tidal components at selected stations___...----------- 72
UL, TPs cerirouiess 2 GLY [7 ete er eee rn en ES 93
NillemtinchimationrlonmoonssiOnbitsess= 2-9) a= 26 == ae oe eee oe 93
ORM LOL MIAO OBE Ts Sees oS ta Se ee ee to eo 99
i. Anerley ee athe So ee A ee ee 123
Rem oOnrechion saclOnN tb = eset ae ose ee oss 2 sete a 130
Miem oainucerrces: ald minutes 2222 5.522 s55-52554-~---24.-25- (259
(V)

39H

THE TIDES

CHAPTER I

GENERAL DEFINITIONS: THE TIDE-PRODUCING FORCES:
EQUILIBRIUM TIDES

Paragraphs
st OL Seen ears Meena ean te a Le ee fe ko ee —
Mian rclhallenint lame loeesmmenre pam a searchers kes es ee eon PE lee ee ee es 7-13
siGe= LOG Cl GAONCES sac ee 2 a eee Do A et BR ee ee ee 14-21
ide OROGucingspOpeMtl alas see ae yee ee Re ees De SS 22-25
Surface of equilibrium and equilibrium tide_____..______.___.__----__-.- 26-32
bitecemomunereartivs TOLAIOMN 166 28S. ae oe ee a Ske ee eae 33
Hihecmotmthe declination of the:moone=.—--22222242- 522-450-5522 05— oles 34-42
Effect of variation in the moon’s distance__________________________-_- 43
“LATE ECO WETUATOUMUITAN 7 AUG (Eppes a eae ate gy cn a 44
EG ECON OMCTIUS Meee eek ned ote Pe tiee ek as Se on Se ee 45
AN Gta CIB eS a ee ee ee ee eS ee 46-48

-

GENERAL DEFINITIONS

1. The tide is the regular periodic rise and fall of the surface of the
seas, observable along their shores. The concurrent horizontal move-
ments of the water, whether the almost imperceptible drift in the
open sea, or the strong flow through a contracted entrance to a tidal
basin, are designated, in accordance with the practise of the United
States Coast and Geodetic Survey, as tidal currents.

2. High and low water.—The maximum height reached by each rising
tide is called high water, and the maximum depression of the falling
tide is called low water. On the Atlantic coast of the United States
the tide rises and falls twice daily—or more accurately twice during
the lunar day of 24 hours and 50 minutes. The two high waters and
the two low waters are each so nearly equal that for ordinary purposes
no distinction need be made between them. On the Pacific coast the
two high waters and the two low waters occurring daily are in general
markedly different, and are designated as the higher high water, the
lower high water, the lower low water, and the higher low water. On
the Gulf of Mexico the tides are small, and toward its western end
but one tide occurs each day during a part of the month.

The heights of the high waters and of the low waters vary from day
to day. In many parts of the world, the high waters reach their
greatest height, and the low waters the least height, soon after the

(1)

2

time of full and new moon. These tides are called spring tides. The
term ‘‘spring’”’ as applied to tides has nothing to do with the season
of the year, but is the greater upspringing of the waters at intervals
of about a fortnight. Similarly the daily high waters are usually at
their least height, and the daily low waters their greatest height, soon
after the moon is in quadrature. These tides are called neap tides.
On the Atlantic coast of Europe and along the British Isles the differ-
ence between low or high water of spring tides and low or high water
of neap tides may amount to several feet, and is a matter of moment
to navigators. On the coasts of the United States the difference
between spring and neap tides is not particularly noticeable, and the
terms “spring” and “neap” tides are not in ordinary use. In this
country spring tides are commonly referred to as ‘‘tides at full (or
new) moon” or occasionally as ‘‘moon tides.”’

3. Datum planes.—The average height of all low waters at any place
over a sufficiently extended period of time is called mean low water
and is the official reference plane for the depths shown on navigation
charts, and of improved channels, in the waters of the Atlantic and
Gulf coasts of the United States. The average height of the lower of
the two daily low waters is called mean lower low water and is the
official reference datum in the waters of the Pacific coast of the
United States. In British waters the datum is usually the mean low
water of spring tides, or low-water springs. This reference plane is
also used at the Pacific entrance to the Panama Canal. The average
height of the sea, as determined usually by the average of the observed
hourly heights over an extended period of time, is called mean sea
level, and is the standard datum to which elevations on land are
referred.

4. Tidal ranges.—The difference in height between high water and
low water at a tidal station is called the tidal range. The mean range
is the average of the differences between all high waters and all low
waters; or, as is the same thing, the difference between mean high
water and mean low water at the station. The diurnal range, or great
diurnal range, is the difference between mean higher high water and
mean lower low water. The eatreme range is the maximum that has
been observed. The spring range is the difference between mean
high water and mean low water of spring tides, and the neap range
the difference between mean high water and mean low water of neap
tides.

5. Tidal currents —The tidal current setting into the bays and
estuaries along the coast is called the flood current. The return cur-
rent toward the sea is called the ebb current. The maximum velocities _
reached during each fluctuation of the current are called the strength
of the flood and the strength of the ebb, or, indifferently, the strength of
the current. Slack water is the period during which the current is

3

negligible while it is changing direction. It is specifically defined
by the United States Coast and Geodetic Survey as the period during
which the current is less than one-tenth of a knot; i.e., less than 0.169
feet per second. The slack water occurring nearest the time of high
water is called the high-water slack, and that nearest the time of low
water the low-water slack. The moment at which the current is zero
as it changes direction may be distinguished by terming it the turn
of the current.

In open waters, the direction of the current normally veers around
the compass and the current does not pass through intervals of slack
water. Such currents are called rotary, to distinguish them from the
reversing currents in a tidal channel.

6. These definitions are narrower than the common usage of the
terms. ‘Tide’ is commonly applied both to the rise and fall of the
sea and to the accompanying tidal currents. Thus the expressions
“head tide’ and ‘favoring tide” designate tidal currents that retard
or accelerate the movement of a vessel, and the term ‘“‘the ebb and
flow of the tide” is standard legal nomenclature. The term ‘ebb
tide” is often used to designate low water as well as the outflowing
tidal current. The maximum tidal stage is frequently designated as
‘high tide’ instead of “high water.’’ Its more general meaning is,
however, the higher stages of the tide. Thus it is more accurate to
say that a channel is “navigable only at high tide,” than to say that
it is “navigable only at high water.”

7. Lunitidal intervals —Casual observation shows that the tides at
any place occur a little less than 1 hour later each succeeding day.
Thus if high water is at 3 p. m. today, it will be shortly before 4
p. m. tomorrow. Closer observation shows that the high and low
waters at any place follow, by about the same time interval, the pas-
sage of the moon across the meridian of the place. Obviously, the
moon must cross the plane of the meridian twice daily—once over-
head and once underneath. These are called respectively the upper
and lower meridian transits. They mark in fact the noon and mid-
night of the lunar day. If a clock were regulated on mean lunar
time, instead of mean solar time, it would show the times the high
and low waters at a given place at about the same hour every day,
but these times would vary largely from place to place.

8. The average time interval, in solar hours and minutes, from a
lunar transit to the next succeeding high water at a given place, as
determined by an extended set of observations, is called the high-
water interval, (HWI) or the high-water lunitidal interval of the place.
Similarly the low-water interval (LW1), or the low-water lunitidal inter-
val is the average time, in solar hours and minutes, from a lunar transit
to the next succeeding low water. The high- and low-water intervals
usually are larger at the full and change of the moon, at about

4

the time of spring tides, than at other times in the month. Charts
of foreign waters sometimes give the intervals at such times, instead of
the mean intervals, designating them as HWI, F. & C., and LWI,
F. & C., respectively.

9. The charts of the United States Coast and Geodetic Survey and
other publications show the average lunitidal intervals at representa-
tive tidal stations. By computing from the Nautical Almanac the
times of upper and lower meridian transits of the moon at the place
on any day, the times of high water on that day can be approximately
determined. Although rarely of practical importance, the method of
computation is of interest.

The Nautical Almanac gives the Greenwich mean solar time of the
moon’s upper and lower transits across the meridian of Greenwich
for each day in the year. This time obviously is the interval, or hour
angle, between passage of the (mean) sun and the passage of the
moon over the Greenwich meridian. This interval increases at the
average rate of 25.2 minutes every 12 hours, or 2.1 minutes per hour.
If then the longitude of a given place is Z° west of Greenwich, the
transit of the (mean) sun over its meridian will be Z°/15 hours later
than the transit over the Greenwich meridian, and the interval between
the transits of the (mean) sun and of the moon over the meridian
of the place, or the local mean solar time of the moon’s transit, will be
the Greenwich time of transit increased by 2.1 Z°/15 minutes. For
example, the high-water interval at Sandy Hook, long. 74° W., at the
entrance to New York Harbor, is 7°.35™. For April 12, 1936, the

Almanac gives:
Upper Lower

IMI@outs Teas, (Creemyn@ll = eee eee 3555" 16521™
Correction to\Sandy Hook (74/15) 2:12 5222-2 = 10™ 10™
Local time moon’s transit, Sandy Hook__________ 4h05™ 16531™
Correction to standard time 75° meridian________. —04™ —04™
Standard time moon’s transit, Sandy Hook ___-___- 4bQim 16427™

Adding the high-water interval to the times of the moon’s transits,
the approximate times of high water at Sandy Hook are found to be
11°36™ and 24°02™; or 11:36 a. m. April 12 and 12:02 a. m. on April
13. The times given in the tide tables are 11 a. m. and 11:24 p. m.
on April 12. The time of high and low water found from lunitidal
intervals may be in error by half an hour or more.

10. The difference between the lunitidal intervals at two tidal sta-
tions, corrected if necessary for the difference in the longitudes of the
stations, gives the average difference between the times of high, or
low, water at these stations. The formula for this correction is at
once derived from the process of finding the time of high (or low)
water from the Greenwich meridian transit of the moon and lunitidal
interval, as set forth in paragraph 9. Let @ be the time, in hours, of

5

a Greenwich lunar transit, J, and J, the lunitidal intervals at the two
stations, LZ, and J, their longitudes in degrees west of Greenwich, S
the longitude of the standard time meridian of the locality, and 7,
and 7, the standard time, in hours, of high (or low) water at the two
stations. Then:

T,=G+ (2.1/60) (L,/15) + (L,—S)/15+L,
= G+ (4.14/60) LZ,—S/15+],
T,—G-+ (4.14/60) L,—S/15+1
Whence:
T,—T,=I,—I,+ (4.14/60) (L,—L2) (1)

The correction for longitude is therefore 4.14 minutes of time for
each degree of difference between the longitudes of the two stations,
due regard being had to the algebraic sign of the correction resulting
from the application of the formula. Obviously for easterly longi-
tudes the sign of the correction would be reversed.

For example, the high-water interval at Portlard, Oreg., long.
122°40’ W., is 6943™, and at Astoria, near the mouth of the Columbia
River, long. 123°46’ W., the high-water interval is 0"°41™. The dif-
ference in the time of high water between Portland and Astoria is
therefore 6'43™—0'41™+4.14 (122.56 —123.77)™=6"02™—05™=5"57™.
High water at Portland is therefore 5°57™ later, on the average, than
high water at Astoria.

11. Since the time of high water cannot be determined from
observation within a range of several minutes, the correction for the
difference in longitude between two stations may be neglected unless
it exceeds 1 minute of time. The corresponding difference in longi-
tude is about 15’ of are. No correction for longitude need be made
therefore unless the two stations are at least 10 miles apart in an east
and west direction.

12. Greenwich lunitidal intervals—A Greenwich high- (or low)
water interval at a station is the interval from a transit of the moon
over the meridian at Greenwich, as given in the Nautical Almanac,
to the Greenwich time of the following high (or low) water at the
station. For convenience, high- and low-water intervals usually are
computed by subtracting the tabulated Greenwich times of upper or
lower transits from the time of the next ensuing observed high and
low waters, as recorded on standard time at the station. The average
differences so found are then converted to Greenwich intervals by
adding the west longitude, in hours, of the standard-time meridian.
If the result exceeds the average interval of 12.42 hours between
successive lunar transits, that interval is subtracted. The local luni-
tidal intervals may then be found by subtracting the product of the
west longitude of the station, in degrees, times 0.069 hours (4.14

6

minutes), from the Greenwich intervals, increased if necessary by
12.42 hours.

Thus, the mean interval from the tabulated Greenwich transits to
standard times of observed high water at Seattle, Wash., in January
1928 was found to be 4.98 hours. As the standard time meridian at
the locality is 120° west of Greenwich, the Greenwich interval is
found by adding 120/15=8 hours. As the sum, 12.98 hours, exceeds
the interval between lunar transits, the average Greenwich interval
at the station during the month is recorded as 12.98—12.42=0.56
hours. The longitude of the station is 122°20’ W. The correction
to be subtracted from the Greenwich interval to give the local lunitidal
interval is (122%) X0.069=8.44 hours. The average local high-water
interval for the month is then 12.98—8.44=—4.54 hours.

While lunitidal intervals are conventionally given as local intervals,
the Greenwich intervals are more convenient for most purposes,
since the difference between the times of high (or low) water at any
two stations is given directly by the differences in their Greenwich
intervals, without correction for the different longitudes of the
stations.

13. Establishment of the port—The high-water interval at the full
and change of the moon is called, in England, the ‘“‘establishment of
the port,’ and the high-water interval at spring tides the “corrected
establishment.’’ These terms are not current in the United States.

While the time of full moon is commonly thought of as a day, it is
in fact an instant, duly set forth in the Nautical Almanac. The
moon’s transit nearest the moment of full or change evidently is
nearly but not quite at noon or midnight, and the mean solar time of
high water is close to the high-water interval. The establishment of
the port is also defined therefore as the local time of high water at the
full and change of the moon. The term is not further used in the
treatment of the tides herein followed.

THE TIDE-PRODUCING FORCES

14. It is an elementary principle of physics that the gravitational
attraction between two bodies varies inversely as the square of the
distance separating them; and an elementary theorem that the
attraction between two spheres, such as the moon and the earth, is
the same as though their respective masses were concentrated at their
centers. But the attraction between the moon and any individual
unit of mass in the earth depends upon the distance of this unit from
the center of the moon, which is not, in general, the same as the dis-
tance from the earth’s center to the center of the moon. The conse-
quent varying differential in the force of attraction over the earth’s
surface as compared with the average attraction per unit of mass of

ch

the earth as a whole, tegether with a similar differential with respect
to the attraction of the sun, are the tide-producing forces.

15. The tide-producing force of the moon.—In figure 1, CO is the center
of the earth, O the center of the moon, and P any point at or within
the earth’s surface, 7 the distance CP, a the radius of the earth, R
the distance between the center of the earth and the center of the
moon, D the distance from P to the center of the moon, @ (theta) the
angle between CO and CP, and P the angle between PO and CP
produced.

Let M be the mass of the moon,

uw (mu) the gravitational attraction between two units of mass
at one unit’s distance.

The attraction of the moon on a unit of mass at the point P is then
Mu/D? acting in the
direction of PO, and
its component in the
durection CP is
(Mu/D*) cos P. Sim-
larly the attraction
of the moon on a unit
of mass at the center
of the earth is My/R?
acting in the direction
CO, and its component in the direction CP is (Mu/R?) cos 6. The com-
ponent of the difference of these forces, in the direction CP, is:

FIGURE 1.

fr=(Mu/D’) cos P—(Mu/R?) cos 6 (2)

Let A be the foot of a perpendicular from the center of the moon,
O, tc the line CP produced. Then:

Deos P=PA. R cos 02=CA=PA-+r.
whence:
R cos 0=D cos P+r. cos P=(R cos 6—7r)/D.
Giving:
fr=Mu [(R cos 0@—7r)/D?—cos 6/R?|
= Mu [(cos @—r/R) (R?/D*) —cos 6]/R?. (3)

From the triangle POC:

D?=hk?-+-r?—2Rr cos 0.
Whence:
D*/R?=1—2(r/R) cos 6+ (7/R)?.

Placing, for convenience, r/R=p:

R3/D?=(1—2p cos 6+ p?) 3?
=[1—p(2 cos 6—p)]?°”.

Expanding the second member into the binomial series:

R?/D?=1+3/2 p(2 cos @—p)+15/8 p?(2 cos 6—p)?+ -:-
=1+3p cos 6@—3/2 p?(1—5 cos? 6)+ terms in the
cubes and higher powers of p.

Since the distance from the moon to the earth is approximately 60
times the earth’s radius, the cubes and higher powers of p=r/R have
values of 1/216,000 or less, and the terms containing them are too
small to be considered. Substituting, in equation (3), the expression
derived for R?/D*, reducing and again dropping the cubes of p:

fr=Mau [p83 cos? 6—1)+3/2 p?(5 cos* 6—3 cos 6)]/R?
= Muy (r/R*) (8 cos? 2—1)+3/2 Mu(r?/R*) (5 cos? 6—8 cos @). (4)

The numerical value of the coefficient of the second term of equa-
tion (4) is 37/2R times, or in the order of 1/40th or less of, the numerical
value of the coefficient of the first term. For the accuracy in general
necessary, the second term may be disregarded, giving:

fr=Mu(r/R*) (8 cos? 6—1). (5)

The distance of the moon from the earth is astronomically measured
by its parallax, which may be defined as the angle subtended by the
radius of the earth at the distance of the moon. The parallax varies
as the reciprocal of the distance, or as 1/R. Since the second term of
equation (4) contains 1/R to the fourth power, it is called the term
dependent on the fourth power of the moon’s parallax.

16. The component of the lunar differential attraction in the direc-
tion perpendicular to CP, in the plane CPO, is similarly:

fh=(Mu/D?) sin P—(Mu/R?) sin 6

from figure 1:

Disn? —OA— Ty simag
giving:

sin P=R sin 6/D,
so that:
fh=Mu(R sin 6/D®—sin 6/R?)
= Mu sin 6(R?/D®—1)/R?.

o

Substituting the expression for f?/D*® previously found, but drop-
ping the squares and higher powers of p:

fh=Mu sin 0(11+3p cos 6—1)/R?
=3Mu/(r/R’) sin 6 cos 0=3/2 Mu(r/R?) sin 20. (6)

The terms containing “‘the fourth power of the moon’s parallax”’
being omitted.
17. When P is at the surface of the earth, r becomes a, the earth’s
-radius. The line CP is evidently the vertical at P. Therefore the
vertical component of the lunar tide-producing force is:

fr=Mu(a/R’) (8 cos? @—1) (7)
and the horizontal component, in the direction of the moon, is:
fh=3/2 Mu(a/R*) sin 20. (8)

Since the vertical line CP is directed toward the zenith of the place
P, it is also clear that the angle @ is the zenith distance of the moon,
or the complement of the moon’s altitude above the horizon.

18. Characteristics of the lunar tide-producing force.—It is evident
from equation (7) that the vertical component of the tide-producing

Moon

FIGURE 2.—Directions of tide-producing force.

force is a maximum when §6=—0 and 180° and is then 2Mua/R*. It
is zero when cos @=4/1/3; i.-e., when 6 is 54°44’, 125°16’, 234°44’,
and 305°16’. It reaches a maximum negative value of —Muya/R*
when 6=—90° and 270°. Similarly the horizontal component increases
from zero, when §6=0, to a maximum of 3/2 Mua/R* when 6=45°, and
then decreases to zero when §=90°, repeating this variation with
appropriate changes in sign in each quadrant. The resultants of
the horizontal and vertical components of the tide producing force,
for various values of 6, are shown graphically in figure 2.

The attraction of the moon tends to pull the water of the oceans
toward it on the side of the earth nearest the moon, and to pull the

10

earth away from the water on the other side. The resultant tide-
producing forces on the side of the earth away from the moon must
balance the tide-producing forces on the side toward the moon, for
otherwise the total attraction between the earth and the moon would
not be the same as though the respective masses of these two bodies
were concentrated at their centers. Because, however, of the some-
what greater attraction by the moon on the nearer area of the earth,
the tide-producing forces on the two sides of the earth are not exactly
symmetrical. This variation in the tide-producing force is expressed
by the term containing the fourth power of the moon’s parallax.
It tends to mould the surfaces of the ocean into a very slightly pear-
shaped variation from a perfect oval (fig. 8, par. 32).

19. The solar tide-producing force——Designating the mass of the
sun by S, and its distance from the earth by R,, and its zenith distance
at the point P by 6;, the vertical component of the solar tide-producing
force at the earth’s surface is evidently, from equation (7):

Fri=Sp (a/R) (8 cos? 6,—1) (9)
and the horizontal component, from equation (8):
fh, =3/2Sy(a/R,3) sin 26; (10)

The maximum value of the vertical component is 2Sua/R,’. Its
ratio to the maximum value of the vertical lunar component is:

(2:Spa/R,*)/(2Mya/R*) = (S/M) (B?/h,’)

The mass of the sun, S, is 27,000,000 times the mass of the moon,
M,; but the distance of the sun from the earth, A,, is about 389 times
the distance, R, of the moon from the earth. Substituting these
values, the ratio of the maximum values of the solar to the lunar
tide-producing force becomes 27,000,000/58,863,869=0.46. Despite
its enormously greater mass, the tide-producing force of the sun is
less than half that of the moon, because of its greater distance.

20. A consideration of figure 3 shows that when the moon is full,
M’’, or at change, M’, the solar tide-producing force will tend to
increase the lunar tide-producing force, while when the moon is at
quadrature, at MM’ ’’ and M* the solar tide-producing force will tend
to decrease the lunar tide-producing force. At the full and change
of the moon, therefore, high waters tend to be higher and low waters
lower, than at other phases of the moon, thus producing the spring
tides at full and change, and neap tides at quadrature (par. 2).

21. The tide-producing forces are minute.—The force of gravity at
every point on the earth’s surface is Hy/a?, EK being the mass of the
earth, a its radius, and y» the gravitational attraction between two

ita

units of mass at one unit of distance. The ratio of the vertical com-
ponent of the lunar tide-producing force to the force of gravity is,
from equation (7):

(Mua/R?)(3 cos? @—1)/(Eu/a?) = (M/F) (a*/R?)(3 cos? 6—1)

This ratio reaches a maximum of 2(//E)(a?/R*) when 6=0.

Since the mass of the earth is approximately 80 times the mass of
the moon, and its distance from the moon approximately 60 times the
earth’s radius:

M/E=1/80 a/R=1/60
a
a ee
, \
po \
“9 QD ow
Woe Aethe Te:
N Y
tA

SS
= O.-
M

FIGURE 3.—Phases of moon.

The substitution of these values shows that the maximum value of
the vertical component of the lunar tide-producing force is about
1/8,640,000 of the force of gravity. The maximum value of the hori-
zontal component is similarly found to be about 1/17,280,000 of the
force of gravity. The maximum values of the components of the
solar tide-producing force are less than half of those of the lunar com-
ponents. Such small forces evidently are not directly measurable
by the most delicate instruments, nor can they sensibly affect the
levels of limited bodies of water even as large as the Great Lakes.
The accumulated effect of these small forces over the vast areas of
the oceans is however sufficient to produce the tides.

THE TIDE-PRODUCING POTENTIAL

22. The effect of the tide-producing forces upon the waters of the
oceans is indicated by the potentials of these forces. The potential
of a force at any point is defined as the work required to move a
unit of mass against the force to a position where the force is zero.
Since the tide producing force is zero at the earth’s center, the tide-
producing potential at P, distant 7 from the center C (fig. 4) is the

192750—40——2

12

work required to move a unit of mass against the force, from P to C.
If the mass be moved along the radius PC, the radial component is
the only part of the force against which work is done. The radial

a aN

FIGURE 4.

component of the lunar tide-producing force is as shown in equation

(5):
Mu(r/R*) (3 cos? @—1)

As derived, this force is positive in the direction CP.
The lunar tide-producing potential at P is therefore:

‘ 0
v=—| Mu(r/R*) (3 cos? 6@—1)dr= — (Muy/R?*) (3. cos? oa) J om
= 4Mu(r"/R*)@ cos’ @—1) -

23. Relation of potential to force—It follows from the definition of
the potential of a force, that its rate of change, in any direction, is
the component of the force acting in that direction. Thus the rate
of change of the lunar tide-producing potential in a direction per-
pendicular to the radius (in the plane of the moon, the point, and
the earth’s center) is:

dV ,/d(ré) =dV,/rd6=%Mu(r/R?)d(3 cos? 6—1)/de
=—3Mu(r/R*) cos 6 sin 6=—3/2 Mu(r/R?) sin 26

as found in equation (6). The negative sign results from the fact
that the direction of the force is opposite to the direction in which
§ is increasing, as will be apparent from a reference to figure 1.

24. It is evident from the preceding paragraph that when the po-
tential varies from point to point over a water surface, such as the
surface of the oceans, the water tends to move from areas of low
potential toward areas of high potential, just as it would tend to
move from areas having a higher elevation toward the areas having a
lower elevation. When a water surface is in equilibrium, the total
potential of all forces acting upon it evidently must be the same at
all points on the surface.

25. The lunar tide-producing potential at any point P on the sur-
face of the earth is found at once by substituting the earth’s radius
a for r in equation (11), and is:

Vi=%Mu (a/R) (3 cos? 6— 1) (12)

13

This potential is evidently a maximum at P, and P,, figure 5,
where 6=0 and 180°, respectively, and a minimum on the great
circle P3;P,, where 0=90°. The difference in the potential will
therefore tend to cause the water of the oceans to pile up toward P,
and P; as was shown from the analysis of the tide-producing forces
in paragraph 18. To an observer at any point on the great circle.

ie

3

cae Sos

nU

%

FIGURE 5.

P,P, the moon is on the horizon; at P, directly overhead. The tide-
producing potential at any point is therefore a minimum when the
moon is on the horizon, and a maximum when it attains its greatest
altitude above (or below) the horizon.

THE SURFACE OF EQUILIBRIUM AND THE EQUILIBRIUM TIDE

26. Lunar equilibrium tide—If the earth, instead of rotating daily
about its axis, rotated once in a lunar month, so that the same side
of the earth was al-
ways presented to the
moon, the lunar tide-
producing force evi-
dently would create
two permanent bulges
or distortions in the
surface of the oceans,
which would be di-
rected toward the
moon on one side
of the earth, and in
the opposite direction
on the other. The surface of the oceans would then conform
to the surface of equilibrium resulting from the joint action of
the force of gravity and the lunar tide-producing force. If
the oceans entirely covered the earth, this surface of equilibrium
evidently would take the form of a prolate spheroid of revolution,
with its axis pointing toward the moon, as shown in figure 6. The
displacement of this theoretical tidal surface of equilibrium from the

FIGURE 6.—Tidal surface of equilibrium.

14

spherical equilibrium surface produced by the action of gravity alone,
affords a yardstick for measuring the effect of the tide-producing
force of the moon, and is called the lunar equilibrium tide.

27. Equation of the tidal surface of equilibrium.—Let r be the dis-
tance OP (fig. 6) from the center of the earth to any point P on the
equilibrium surface, 6 the angle between CP and the axis of the sur-
face, and as before H and M the masses of the earth and moon, respec-
tively, R the distance between their centers, a the radius of the earth,
and u the coefficient of gravitational attraction. Let V, and V, be,
respectively, the lunar tide-producing potential and potential due to
eravity at P.

The force of gravity becomes zero when 7 is infinite. The gravity
potential is then, from the definition in paragraph 22:

Wee { (Bal dr= Bul. (13)

Since, as shown in paragraph 24, the total potential at all points on
the surface of equilibrium is constant:

Vea — Ce

Substituting the expression for V, found in equation (11), and for
V, in equation (13), the equation of the surface of equilibrium be-
comes:

YMu(r?/R*) (8 cos? 6—1)-+ Ep/r=C. (14)

28. An indefinite number of surfaces are given by this equation as
various values are assigned to C. If the oceans were continuous, the
particular surface to be chosen would have a volume equal to that of
the sphere with radius a, since the volume cannot be altered by the
tidal disturbance. It will be shown that this condition is fulfilled by
the surface whose radius vector is equal to the earth’s radius where the
tide-producing potential is zero, 1. e., where cos? 6=1/3. Such asurface
will intersect the sphere in the small circles P,P:, and P3P, in figure 6. -
The resulting value of the constant is found by placing r=a and cos? 6=
1/3 in equation (14), giving:

HH}
and the equation of the surface of equilibrium is therefore:
¥Mu(r?/R) (3 cos? 0—1) 4+ Hy/r= Ep/a
which reduces to:
¥ (Ma?/ER?) (3 cos? 6—1) =a (r—a) /r°

Representing the height of the equilibrium tide by w, it follows from
the definition in paragraph 26:

T—G=—U.

15
Substituting this expression in the preceding equation,
¥, (Ma?/ ER?) (3 cos? 6—1)=a7u/(a+u)?. (15)

Since wu is very small in comparison with a, the value of (a+)? is
always very close to that of a’. Equation (15) then becomes:

u=\(Ma'/ER?®)a(3 cos? 6—1) (16)

‘This is the equation of the lunar equilibrium tide.

29. The volume of the tidal surface of equilibrium evidently is the
same as the volume of the undisturbed sphere if the total positive tidal
volume over the zones P,P2P; and P3P.P, in figure 6 is equal to the
negative tidal volume over the zone P,P;P,P;; or, what is the same
thing, if the positive and negative tidal volumes in the hemisphere
P2P’P™ are equal.

In figure 7, P is any point on the tidal-equilibrium surface; CP its
radius vector, 7; P,P the equilibrium tide, wu, at that point; CP, the

FIGURE 7.

radius, a, of the undisturbed sphere; 6 its angle with the line CO
directed toward the moon; and CP’,P’ the position of CP,P when 6
is increased by the differential angle dé. The equilibrium surface is,
as has been seen, a surface of revolution whose axis is CO. <A well
known theorem establishes the volume of a solid formed by rotating
a plane figure about an axis in the same plane as the product of the
area of the plane figure by the length of the circumference of the circle
‘described by its center of gravity.

The area of the differential triangle PP’C is 4 r? dé and the radius of
the circle described by its center of gravity is 2/37 sin 6. The differ-
ential volume resulting from the rotation of the triangle about CO
is therefore:

2/3 ar? sin 6d0=2/3 r(a+u)? sin 6dé

The corresponding differential volume of the sphere is
2/3 ma® sin 6 dé

The difference between these volumes is the elementary tidal
volume, dq, generated by the rotation of PP,P,;’P’ and is:
dq=2/3 z[(a+u)?—a’] sin 6 dé
=2/3 ra3(3u/a+3u?/a?+u?/a?) sin 6 dé

16

Since the ratio u/a is extremely small, its squares and higher powers.
may be dropped, giving:

dq=2ra’u sin 6dé
Substituting the expression for uw in equation (16) and integrating:

q= S 1(Ma'‘/ER*) (3 cos? @—1) sin 6dé0
=1(Ma‘/ER*)(f3 cos? 6 sin 6dd— f sin 6d@)
= 7 (Ma‘/ER*) (—cos? @+cos 0) +C (17):

Taking gq=0 when @=0, the constant of integration becomes zero,
and the expression for the tidal volume in the zone measured by the:
angle 6 is:

q=7 (Ma'‘/ ER?) (cos 6—cos? 6) (18)
This volume reaches a maximum when 3 cos? 6—1=0 and is then
a (Ma®/ ER?) (¥/1/3—1/3-+/1/3) =2/9 «(Ma*/ER®)/3

which is the volume of the positive tide over the zone P;P,P, in figure
6. The volume of the negative tide is the same, as q reduces again to.
zero When 06=90°. The condition of continuity is therefore fulfilled
by the expression for the equilibrium tide given in equation (16).

30. Magnitude of the lunar equilibrium tide.—Assigning to the con-
stants in equation (16) their numerical values, the ratio, M/E, of the
mass of the earth to the mass of the moon is 1/81.45; a, the mean
radius of the earth, 3,959 statute miles; R, the mean distance to the
moon, 238,857 statute miles. The coefficient %(Ma?/ER*)a then is.
0.584 feet. The corresponding height of the lunar equilibrium tide
in feet is therefore:

u=0.584(3 cos’ 6—1)

The factor 3 cos? @—1 has a maximum value of 2 when 6=0, and
a minimum value of —1 when 6=90°. The maximum range of the
lunar equilibrium tide is then 3X0.584=1.752 feet. This distortion
of the water surface of the earth is very small in comparison with the
distortion due to the earth’s rotation, since the latter, as measured by
the difference between the equatorial and polar radii, is 13.35 miles.
The tidal distortion is however superimposed upon and not measur-
ably affected by the distortion due to the earth’s rotation.

31. Solar equilibrium tide.-—The solar equilibrium tide is, by trans-
posing in equation (16);

u,= %(Sa?/ER,’)a(38 cos? 6,—1)

bi?

The numerical value of S/H is 333,432; and the mean distance, Ry,
from the earth to the sun, is 92,897,416 statute miles. The substitu-
tion of these values gives:

U,=0.270(8 cos? 6,—1)

The maximum range of the solar equilibrium tide is 3 X0.270=0.810
feet.

32. Equilibrium tide dependent on the fourth power of the moon’s
parallax.—lf the second term of equation (4), paragraph 15, is included
in the derivation of the lunar tide-producing potential, paragraph 22,
- and of the lunar equilibrium tide, paragraph 28, the equation of the
latter (equation 16) becomes:

u='4(Ma?/ER*)a(3 cos? 6—1)+%(Ma*/ER*)a(5 cos? @—3 cos 6) = (20)

The second term of this equation, 4(Ma‘*/ER*)a(5 cos? 6@—3 cos 8), is
the “lunar equilibrium tide dependent on the fourth power of the
moon’s parallax.”

Substituting the numerical values for the constants in the coefficient
of this term, this part of the tide has the value, in feet, of

0.007(5 cos? @—3 cos 84)

The factor (5 cos? @—3 cos 6) has a maximum value of 2 when @=0,
decreases to — 0.894 when §6=63°26’,
increases to 0.894 when 6=116°34’
and again decreases to a minimum
of —2 when 6=180°, repeating this
variation in the third and fourth
quadrants. This part of the equili-
brium tide is shown, on a greatly
exaggerated scale, in figure 8.

It will be noted that the equili-
brium tide dependent upon the
fourth power of the moon’s parallax
goes through three fluctuations from
maxima to minima as 6 goes through
a cycle from 0 to 360°; but that its
maximum range is but one-quarter
of aninch. It is superimposed upon and produces but an immaterial
distortion of the principal equilibrium tide due to the third power of
the moon’s parallax, previously developed.

Since the ratio of the radius of the earth to its distance to the sun
is but 1/389th of its ratio to the distance to the moon, the equilibrium
tide dependent upon the fourth power of the sun’s parallax is too
small to be considered.

FIGURE 8.—Surface of equilibrium dependent on
fourth power of moon’s parallax.

18

EFFECT OF ROTATION OF THE EARTH AND THE MOVEMENT OF THE MOON
AND THE EARTH IN THEIR ORBITS

33. The effect of the rotation of the earth.—As shown in paragraph 25,
the lunar and solar tide-producing forces each create two areas of high
potential on the surface of the earth, one facing the moon, or sun,
and the other opposite. As the earth spins around its axis, these
areas make the circuit of the earth and set up the slight oscillations of
the oceans which make the tides. The rise and fall of the actual tide
at any locality, and the times of high water and low water, depend on
the conformation of the ocean shores and beds and on the momentum
of the water masses as well as on the tide-producing potential. The
equilibrium tide affords a measure of the effect of tide-producing
potentials alone. The variations of equilibrium tides resulting from
the movements of the moon and earth in their orbits indicate the
variations to be expected in the actual tides because of these move-
ments.

34. Effect of the declination of the moon and sun.—The surface of
equilibrium of the oceans due to the lunar tide-producing potential

S

FIGURE 9. FIGURE 10.
Effect of moon’s declination on tides.

has been shown to be a prolate spheroid, with its axis pointing to the
moon (par. 26). When the moon is in the plane of the earth’s equator,
as shown in figure 9, the equilibrium tide at any point P on the earth’s
surface quite evidently goes through two equal fluctuations during
one rotation of the earth around its axis NS, as measured from the
position of the moon; i. e., two equal lunar equilibrium tides then
occur each lunar day. At the earth’s equator the range of these two
tides is 1.75 feet when the moon is at its mean distance from the earth
(par. 30), this range decreasing with the latitude of the tidal station.
When, on the other hand, the moon is above or below the plane of the
earth’s equator (fig. 10) the two daily fluctuations of the lunar equi-
librium tide quite obviously are unequal, except on the earth’s equator,
the inequality depending on the latitude of the tidal station, and the

19

angular distance of the moon from the equator. This angle is the
declination of the moon. The solar equilibrium tides vary similarly
with the declination of the sun. It may be noted that the solar
equilibrium tides are equal when the days and nights have the same
length, and that the inequality of the two daily tides when the moon
or sun are off the equator is for a cause analogous to that of the
inequality of the days and nights.

35. Periodic variations in the declinations of the sun and moon.—The
changes in the form and range of the equilibrium tide at a given station
produced by the changing declinations of the moon and sun, and the
corresponding changes in the actual tides, obviously run through
cycles whose respective periods are the periods of the declinations.
It will be recalled that as the sun moves along the ecliptic, its apparent
path on the celestial sphere, it crosses the celestial circle of the earth’s
equator, and has therefore a zero declination, at the vernal equinoz,
passing this point yearly in the latter part of March. It then ascends
north of the equator and its declination reaches a maximum angle of
23°.452 at the summer solstice, late in June.. This angle is the in-
clination of ecliptic to the equator, and may be considered as constant
so far as tidal computations are concerned. At the summer solstice
the sun is directly overhead at noon on the tropic which separates the
torrid from the temperate zone in the northern hemisphere. The sun
again crosses the equator at the autumnal equinox in late September,
and reaches its maximum negative (south) declination of —23°.452 at
the winter solstice in late December. The period of its travel from
vernal equinox to vernal equinox is the tropical year of 365 days,
5.813 hours.

The moon, in its movement along the celestial circle marking its
orbit, similarly crosses the earth’s celestial equator monthly at the
ascending intersection, reaches a maximum north (positive) declination
in about a week, again crosses the equator at the descending intersec-
tion in another week, to reach its maximum south (negative) declina-
tion. The period of its travel, from ascending intersection to ascend-
ing intersection, is the tropical month of 27 days, 7.718 hours.

The points at which the moon’s celestial orbit crosses the ecliptic
are called its ascending and descending nodes, respectively. The plane
of the moon’s orbit has a constant inclination of 5°.145 to the plane
of the ecliptic, but because of a slow retrograde movement of the
moon’s nodes along the ecliptic, the inclination of the moon’s orbit to
the equator, and hence the maximum monthly declination of the
moon, slowly varies. The moon’s node makes the circuit of the eclip-
tic in approximately 19 years.

20

In figure 11:
U is the vernal equinox.
O the moon’s ascending node.
Yi I the intersection of the moon’s
orbit with the equator.
Vee a UO the ecliptic.

eee” UT the celestial equator.

IO the moon’s orbit on the celestial
sphere.

When the moon’s ascending node coin-
cides with the vernal equinox the inclina-
tion of the moon’s orbit to the equator has
its maximum value of 23°.452+5°.145=28°.597. When 9% years
later it coincides with the autumnal equinox the inclination of the orbit
has a minimum value of 23°.452—5°.145=18°.307. The maximum
monthly declinations of the moon, both positive and negative, range
between the same limits.

36. Longitude of the moon’s node.—The angular distance on the
ecliptic from the vernal equinox U to the moon’s ascending node 0,
figure 11, is the longitude of the moon’s node, and is designated by the
letter N. It determines the inclination, J, of the moon’s orbit to the
equator. The value of J may be found from N by the solution of the
spherical triangle ZUO, since in this triangle the angle JUO is the
known inclination of the ecliptic to the equator and JOU is the known
inclination of the orbit to the ecliptic. The values of J in terms of N
are tabulated in manuals on tidal analysis.

37. The lunar equilibrium tide in terms of the latitude of the tidal
station and the moon’s declination.—In figure 12, CN is the axis of the
earth, Nits north pole,
P,M, the equator, the
angle P,CP the lati-
tude, \ (lambda), of a
tidal station P, NPP,
the meridian through
P, the angle ,CM
the declination, 6
(delta), of the moon,
NM’'M, the hour cir-
cle through the line OM joining the centers of the earth and
the moon, and the spherical angle PNM, the hour angle, H, of the
moon with respect to the meridian through P. The angle PCM’ is
then 6, the zenith distance of the moon.

From the spherical triangle PNM’:

FIGURE 11.—Moon’s orbit and the
ecliptic.

FIGURE 12.

cos 6=cos (90°—X) cos (90°—6)-+sin (90°—X) sin (90°—6) cos H
=sin \ sin 6-++cos \ cos 6 cos H (21)

21

The expression for the equilibrium tide, uw, in terms of cos @ is given
in equation (16):

u=4(Ma?/ER?*) (3 cos? @—1)a=3a(Ma*/ER*) (cos? 6— 3)
Substituting the value of cos 6 given in equation (21):

=3 a(Ma?/ER®) (cos” d cos? 6 cos? H+2 cos \ cos 6 sin 2X sin 6 cos H
+sin? d sin? 6— 4) (2)

Since cos? H= (1-+ cos 2H), sin \ cos A= sin 2X etc., this equation
reduces to:

u=%*%, a(Ma?/ER?) (cos? d cos? 6 cos 2H+sin 2X sin 2 6 cos H
+ cos? \ cos? 6+2 sin? \ sin? 6—% (23)

Substituting for cos? \ and cos? 6 in the third term their equivalents,
1—sin? \ and 1—sin? 6 respectively, and reducing, the equation for w
becomes:

u=%*% a (Ma3/ER?*) cos? ) cos? 6 cos 2H+ % a(Ma?/ER?*) sin 2) sin 26 cosH
4 a(Ma?/ER*) (1—3 sin? \) (1—8 sin’ 6) (24)

38. Semidwurnal and diurnal parts of the lunar tide —Equation (24)
shows that the lunar equilibrium tide at any tidal station is composed
of the following parts:

(a) That represented by the term * a(Ma?/ER?*) cos? \ cos? 6 cos 2H.
Since the angle 2H obviously goes through two complete cycles from 0
to 360° while H is making one cycle in a lunar day, this part goes
through two cycles every lunar day and i is therefore called the sem-
diurnal part of the tide.

(6) That represented by the term % a(Ma?/ER?*) sin 2 \ sin 6 cos H.
This part goes through one cycle each lunar day and is the diurnal part
of the tide.

(c) That represented by the term:

\ a(Ma3/ER?*) (1—3 sin?A) (1—3 sin? 6).

Since this term is independent of the angle H, it undergoes no change
because of the rotation of the earth. It is therefore the height of the
daily mean sea level above that of a sea undisturbed by tidal forces.
Its variation due to the changing declination of the moon will be
later discussed (par. 42).

39. A typical example of the diurnal and semidiurnal fluctuations
of the lunar equilibrium tide, and of the total tidal fluctuation re-
sulting therefrom (disregarding the variation due to the changing
distance between the moon and the earth) is illustrated in figure 13,
which shows these fluctuations at a station at 40° north latitude during
a lunar day in which the declination of the moon increases from 7°
to 13°.

22

Upper Transit
Lower Transit
Upper Transit

Total Tide

Diurnal Tide

Semidiurnal Tide

Q 90° 180° 270° fe)
Lunar hour angle

FIGURE 13.—Diurnal, semidiurnal, and resu!tant tides.

It will be noted that the semidiurnal part of the lunar equilibrium
tide follows the sinusoidal curve of the cosine function, with a very
slight skew because of the change in the moon’s declination during the
day. The skew of the diurnal part of the tide because of this change
is more pronounced. The combination of these two parts produces
the inequalities of the daily tides inferred in paragraph 34. The
ranges of the two parts of the equilibrium tide are affected differently
by the latitude of the tidal station. At high latitudes the diurnal
part may mask the semidiurnal fluctuations and produce an equili-
brium tide which is predominantly or wholly diurnal.

40. Variations in the amplitude of the diurnal and semidiurnal parts
of the lunar equilibrium tides with the moon’s declination —The ampli-
tude of the fluctuations of the semidiurnal part of the lunar equilib-
rium tide evidently varies with the coefficient of cos 2H in the first
term of equation (24). At a tidal station whose latitude is 4, it con-
sequently varies with cos? 6. It is therefore a maximum when the:

23

declination of the moon is zero, as the moon crosses the ascending
intersection of the moon’s orbit with the celestial Equator. The
amplitude then decreases to a minimum when, a week later, the moon
has its maximum north declination; again increases to a maximum
when, in another week, the moon is at its descending intersection;
and decreases to a second minimum when the moon reaches its maxi-
mum south declination. The amplitude of the fluctuations of the
semidiurnal part of the lunar equilibrium tide therefore varies between
a maximum and a minimum twice during a tropical month.

The amplitude of the fluctuations of the diurnal part of the lunar
equilibrium tide at any tidal station (except those on the earth’s
Equator) varies with sin 26. It therefore increases twice during the
tropical month from zero, when the moon has a zero declination, to a
maximum when the moon has its maximum declination north or south
of the Equator.

The amount of the variation in the amplitudes of the fluctuations
both of the semidiurnal and the diurnal parts of the lunar equilibrium
tide slowly changes with the inclination of the moon’s orbit to the
Equator (par. 35) and hence with the longitude of the moon’s node
(par. 36).

41. Variations in the range of the actual tide with the:moon’s declina-
tion.—Since the equilibrium tides are a measure of the astronomical
causes of the actual tides, 1t may be expected that the part of the actual
tide due to the moon is made up of diurnal and semidiurnal elements,
each varying with the declination of the moon in the same manner
manner as the equilibrium tides; the amount of the variation slowly
changing with the longitude of the moon’s node. It does not follow
however that the diurnal and semidiurnal parts of the actual tides
change with the latitude in the same manner as the parts of the
equilibrium tide; for the latter, while affording a measure of the
astronomical causes of the variation in the tide at a particular station,
afford no indication of the relationship between
the tides at two different stations.

42. Lunar fortnightly tide.—In figure 14, JM,
is the celestial Equator, // the celestial circle
of the moon’s orbit, J the intersection, MW the ae ee
position of the moon at any time, N/M, the
hour circle through M@. M,M is then the moon’s
declination, 6. Let IM, the angular distance of

the moon from the intersection, be represented uae) |
by/. Then in the right spherical triangle J1/,M: ;
sin 6=sin / sin J (25)

where J is the inclination of the moon’s orbit to the Equator.
Substituting this expression for sin 6 in the last term of equation

(24), % a (Ma*/ER*) (1—3 sin? dA) (1—3 sin? 6), this becomes

24

¥, a (Ma?/ER*) (1—3 sin? A) (1—3 sin?/sin? I). Again substituting for
sin? 1 its equivalent, 4 (1—cos 2/) the term reduces to the two terms:

34 a (Ma?/ER?) (1—8 sin? d) sin? I cos 21
V a (Ma?/ER*) (1—3 sin? \) (1—3/2 sin? J)

The first of these terms evidently goes through two cycles from 0
to 360° while J is making one cycle in the tropical month. It is called
the lunar fortnightly component of the tide.

The last term remains constant as the earth revolves about its
axis and the moon changes its declination. It represents therefore a
permanent distortion of the ocean surface, so far as these movements
are concerned.

The substitution of the numerical of the constants in the expression
for the lunar fortnightly equilibrium tide shows that, at the earth’s
Equator, its range varies from 0.086 foot when the inclination of the
moon’s orbit is a minimum, to 0.20 foot when the inclination is a
maximum. The range decreases to zero at a latitude of 35°16’ north
or south of the Equator, increasing again toward the poles. The
actual fortnightly tide is correspondingly small. At most tidal sta-
tions this component produces, however, a fortnightly fluctuation of
an inch or more in the daily mean height of the sea.

43. Effect of eccentricity of the moon’s orbit—As the moon travels its
orbit, its distance, R, from the earth varies. The point at which it is
nearest the earth is its perigee, and the point at which it is the most
distant is its apogee. Because of the disturbing effect of the attraction
of the sun, the moon’s orbit varies somewhat, but its distance from the
earth at apogee ordinarily exceeds by more than 10 percent the dis-
tance at perigee. The moon makes the circuit from perigee to perigee
in the anomalistic month of 27 days, 13.309 hours. Since the coeffici-
ents of the terms in equation (24) each contain the factor 1/R®, the
amplitudes of the diurnal and semidiurnal parts of the lunar equilib-
rium tides, as derived from these terms, tend to vary from a maximum
to a minimum once during the anomalistic month, as well as varying
twice during the tropical month because of the changing declination
of the moon. A corresponding variation may be expected in the actual
tides.

To gage the amount of the variation in the tides because of the
elliptical form of the moon’s orbit, let P be the distance of the moon
at perigee and A—P-+d its distance at apogee. The ratio of the ampli-
tude of the equilibrium tides at perigee to those at apogee is then

(4Mat/EP®)/(4Mat/EA®) = A3/P*= (P+ d)/P®
=1-+3 d/P+3 (d/P)?+ @/P) (26)

25

Neglecting the squares and cubes of the ratio d/P, it is apparent
that the variation in the equilibrium tide is three times the variation in
the distance to the moon. As the ratio d/P usually exceeds 10 per-
cent, the amplitudes of the diurnal and semidiurnal lunar equilibrium
tides are subject to a variation of 30 percent or more during the
anomalistic month. This variation is sometimes called the parallax
umequality, since the distance to the moon is measured by its parallax
(par. 15).

The lunar fortnightly component of the tide is so small that its
parallax inequality is not taken into consideration. The variation in
R produces, however, a variation in the fixed term:

4, a (Ma?/ER?)/(1—3 sin? d) (1—% sin? 1)

developed in paragraph 42, giving rise to a small monthly tidal com-
ponent with the period of an anomalistic month.

44, The solar equilibrium tides —The equilibrium tide due to the
sun is similarly made up of a semidurnal part, which goes through
two complete cycles in a mean solar day of 24 hours; a diurnal part
which goes through one cyle per day; both of which vary with the
declination of the sun; together with a semiannual component of rela-
tively small range. Since the eccentricity of the earth’s orbit around
the sun is much less than the eccentricity of the moon’s orbit, the
parallax inequalities of the solar equilibrium tides are small in compar-
ison with those of the lunar equilibrium tides.

45. As will be shown in the following chapter, the varying semi-
diurnal and diurnal fluctuations of the tide, because of the changing
declinations of the sun and the moon, and the varying distances
of the earth from these bodies, may be resolved into components of
fixed amplitudes with periods not far from 12 hours and 24 hours
respectively.

THE ACTUAL TIDES

46. The actual fluctuations of the surfaces of the oceans because of
the tide-producing forces are somewhat akin to the slopping around
of the water in a basin on a moving train. The momentum of the
moving masses in the deep seas tends to pile up the water in the
shallow depths along the coasts, producing tides whose ranges may
greatly exceed the range of the equlibrium tide. The tidal ranges at
different points along the shores, and the times of high and low water,
depend upon the contour of the ocean beds and the conformation of
the coasts and cannot be determined by abstract calculation. The
relation of the times of the actual to the equilibrium tides may possibly
be more apparent if the varying pull upon the waters of the oceans
due to the attraction of the moon and sun be conceived to be replaced

26

by the varying push that would come from a heaving of the beds of
the oceans of the magnitude and sequence of the equilibrium tides.
Since the equilibrium tides move across the oceans at the same speed
as the moon and the sun, or upward of 660 miles per hour in the
range of latitudes of the United States, the actual tides generally lag
behind the equilibrium tides by a substantial interval of time. But
while the range and the times of high and low water vary widely
from those of the equilibrium tide, the periods of the fluctuation of the
components of the actual tide must clearly be exactly the same as
the periods of the forces which cause them, and consequently conform
exactly to the periods of the equilibrium tides. Since however the
oceans may be expected to respond differently to the diurnal fluctua-
tions of the tide producing forces than to the semidiurnal fluctuations,
the diurnal and semidurnal elements of the actual tides are generally
displaced with respect to the respective timing of these elements of the
equilibrium tides. Similar displacements may be expected in the
various components into which the diurnal and semidurnal tides may
be resolved. Last of all the ranges of the various components of the
actual tides bear a definite relationship to the range of the like com-
ponents equilibrium tides, in that small components of the latter will
produce small fluctuations in the actual tide; and the variations of the
various components at any tidal station, due to the longitude of the
moon’s node (par. 36) correspond to the variations in the equilibrium
tide.

47. Meteorological tides —Besides the systematic fluctuations due to
the tide-producing forces, irregular fluctuations of the oceans are
caused by winds and the varying barometric pressure over their sur-
face. These accidental fluctuations are called meteorological tides.

48. Hramples of tides —The recorded tides at a few representative
places are shown in figure 15. It is apparent that these tides vary
widely as to type. The effect of the diurnal tidal variations in the
tide at San Francisco and Galveston may be especially noted.

i Se a eos

L\ fe
SS Stan eae
mai)
a

Feet on aaqe

Recorded Tides - eae of San Francisco - June 25 -July to , 1935:

@ Recorded TWdes - ee Gara Zone - December 1-16 1937.

Apogee
D 1B qarter Perigee
O Full Moon North Tropic

€ 3c quarter Sovth Tropic
Equatorial

FIGURE 15.—T ypical tide curves. 192750—40 (Face p. 26)

*'b e
a
® *
wh

te

a

Cuapter II

HARMONIC ANALYSIS OF THE TIDES

Paragraphs
HeEumnicicomponents: defined... 2.2 ..22 2-2. ee kee 49-50
Conmpmerionusoficomponents= =. 5-5) 5.2 22 Soe eee ee eee 51-59
ASTRO MDT CAN OST YC ls see se es a a re ee ee ee ee ee ee 60-63
PRE maiatica nCOMPONeMts. S.-42 2 = 22 22.2. 22255 22s eens ce 64-76
Computation of hourly component heights____-__--___.-.______+_______ 77-90
Computation of amplitude and initial phase_______________________=__-_ 91-96
ee EAEN IOS ACL OLS as see eS hee eet Ps oe ge ee ye 97-98
Setarearme cicero ek rary SE SN oy ee oe ee er A le 100
Pa ero OMECOMNPOMCM tS! = mes ee oe 5 ha ee ee ol ew 101
Mie tmenalies and iepochs, defined. 2. _2---=_.. 22 _2 2 22. ee 102-103
PPemIonA Ot eQUUMOrIUIM tdes =. 2-2 -.t 222222202) 24-8 ene es 104-111
Ete grareenrU Ina eyr TUITE Mt ean SF A pt ee = a Ra LE ee 112-114
Hip ganimimMeECOIMPONECMUS. eos 2 Se ee Se 115-116
ermine OnCDOCNS oo! tm oe oe ee a ee 117-122
Ae SEE SACS i, ae za a ee ae ee ey 123-128
pies ume ecrOMmCOCINCICN Ga yo. ee 2 a ee ee 129
Mai CEO CONSiAlisem ns yee aye heat ol Se Dene a ae 130-131
RRMEE TOBE COGS! 2 sete Oe) Be NS a Se 132-133
ERE MEMULONVESCOMSUAMUS = — Sato oo oe 5s Le ee ps sot nl 134
sPLEUEC GT OUP UGS 2 kee a ee ee ep oes ne pe Sg 135-138

49. Harmonic components of the tide—It is reasonable to assume,
and will later be shown, that, except as affected by irregular meteoro-
logical disturbances, the tide curve at any station is the resultant of
a limited number of sinusoidal (cosine or sine) curves, whose periods
are determined by the periods of the tide-producing forces. This
relation is expressed by the equation:

y=H)+ A; cos (at-+-a,) + Az cos (dat a) + Az cos (agt-+a3)+ * * * (27)

in which y is the height, at the time ¢, of the tide above an arbitrarily
chosen datum, H) is the height of mean sea level above this datum,
and the subsequent terms in the form A cos (at-+a) are the component
tides. Of each component the coefficient A is the amplitude or
semirange, a is the speed, the angle at-++a is the phase at the time ¢, and
a (alpha) is the initial phase. Placing:

a=—3607/f oral=360° (28)
| it follows that at increases from zero to 360° as ¢ increases from zero

to 7; and again as ¢t increases from 7 to 27, and soon. Tis therefore
the period in which the component goes through its cycle of fluctua-

192750—40——3 _ (27)

28

tion. From the discussion in paragraph 46 it is clear that the speed,
a, of each component is determined by the astronomical movements
of the moon or sun, or both. The amplitude A, and the initial phase
a may be determined from the recorded tides at the place, by a
method hereinafter described. In the ensuing discussion, the ampl-
tude A will be expressed in feet; the time ¢ and the period 7’, in mean
solar hours; the speed, a, in degrees per hour (unless otherwise indi-
cated) and the initial phase a in degrees.

Each component is graphically represented by the projection on the
Y axis (fig. 16), of a generating radius CP, of length equal to the

FIGURE 15.— Generating radius and tide curve of a component.

amplitude of the component, rotating at the constant speed of the
component around the origin, C, its initial angle with the Y axis being
the initial phase of the component.

The direction of the rotation of the generating radius is taken as
positive in a counterclockwise direction, in accordance with the usual
trigonometric convention.

The graph of the component is the sinusoidal cosine curve shown
on the right in figure 16, in which the abscissas represent time and the
ordinates the height of the component above mean sea level.

It is sometimes more convenient to write equation (27) in the form:

y=Ho+ A, cos (ayt— $1) + Az cos (d2t— 2) + As cos (ast—f3) + +--+ (29)

in which the angles designated as ¢ (zeta) are numerically equal to the
respective initial phases, a, of the several components but opposite
in sign. Since each component reaches its maximum when at —¢=0,
and hence when t=¢/a, it is evident that ¢/a is the time of high water
of the component next after the origin of time.

50. ‘“‘Astres Fictifs.’’—The tide represented by a single component
would have high waters and low waters of constant heights occurring
at equal intervals of time. Such a tide would be generated by a
moon traveling at a constant angular speed along a circular orbit in
the plane of the earth’s Equator. Hach component of the tide is
therefore sometimes treated as the tide due to a fictitious moon, or
‘“ostre fictif,’’ moving at a uniform speed along the earth’s celestial

29

Equator. This conception appears, however, to complicate rather
than to simplify a consideration of the tidal components and is not
herein pursued.

51. Combination of components.—A consideration of the manner in
which two or more components combine with each other will afford
a basis for the selection of the speeds of the particular components
which reproduce the tide at any station as it varies with the changing
positions and distances of the moon and sun.

Taking the two components:

Yi=A, cos (at+ ay) Y2—= A, cos (dgt+ a») (30)
let CP, and CP, (fig. 17) be the positions of their generating radii at

FIGURE 17.—Resultant of com ponents of unequal speeds.

any instant of time. Completing the parallelogram CP,P;P, it is
at once apparent that the projection on the Y axis of the resultant
vector CP; is the algebraic sum of the ordinates of the two compo-
nents y; and y, at that instant. This resultant vector CP, will there-
fore generate the tide curve of the resultant of the
two components, as shown on the right of the
figure.

If CP,, CP2, and CP; (fig. 18), are the generat-
ing radii of three components at any instant,
the length and position of the resultant vector
is given by the line CP, found by drawing P,P;
parallel and equal to CP:, and P; P parallel sige agi
and equal to CP;. The resultant vector of any
number of components may be drawn in a similar manner.

52. Two components of the same speed—-If two components have
the same speed, the angle between the generating radii CP; and CP;
(fig. 17), remains constant, the parallelogram CP,P3P, does not change
its shape as the radii rotate around C, and the resultant vector CP;
therefore remains of constant length and rotates at the constant
speed of the two components. It will therefore generate a sinusoidal
curve as shown in figure 19.

30

FIGURE 19.—Resuitant of two components of the same speed.

Designating the common speed of the two components by a, the
constant length of the resultant radius by <A3, and its initial angle
with CY (when t=0) by az, it follows that

A, cos (at-+-a,)-+ A, cos (at+a,) =A; cos (at+ az) (81)

It will be seen therefore that any two components having the same
speed unite into a component of the identical speed.

53. Two components of different speeds ——If two components have
different speeds, the faster of the two generating radii CP, or CP; is
continuously gaining on the slower, the angle between these radii
progressively changes, the parallelogram CP,P;P, steadily changes
its form, and the length of the resultant vector CP;, together with
its speed, varies with the time. The curve generated by the result-
ant vector takes various forms, depending on the amplitudes and
speeds of the components; but the periodic variations in the length
and speed of the resultant vector only need be here considered.

54. Variation in the length of the resultant vector—When the ampli-
tudes of the two components differ, A; may be taken as the amplitude
of the major component, a, its speed; and A, the amplitude of the
minor component. Let 6 be the algebraic difference between the
speeds of the components, so that a,=a,+6. The faster of the gener-
ating radii of the two components evidently will overtake and pass
periodically the slower. Taking for convenience the origin of time
at a moment when the generating radu coincide, the two components
then have the form

Y= A, Cos At Yo= Az cos (a,-+b)t (32)

In figure 20, CP, and CP, are the positions of the generating radii
of the two components at any time ¢t, and CP; the resultant. Since
the angle YOP, represents a,t and the angle YCP, represents (a,+))t,
the angle between the radii, P,;CP,, is (a,+6)t—at=0t.

ol

FIGURE 20. FIGURE 20 A;
Variation in length of resultant vector.

If then in figure 20A, Op, is laid off equal to A,, the angle Wp,p;
equal to bt, and p,p; equal to A,, the triangle Op,p; in figure 20A
reproduces the triangle CP,P; in figure 20, and Op, is the length of
the resultant vector CP;. As the time increases the point p3 evi-
dently describes a circle of radius A, around p, as a center. The
length of the resultant vector fluctuates between O/=<A,-+ A, and
ON=A,—A,. The period in which the point p; completes the circuit
around p; as a center obviously is 360°/b. This interval is called the
synodic period of the two components, and, without regard to the
algebraic sign of 6, is the interval between the successive times at
which the resultant vector reaches its maximum length, and also
between the times at which this vector has its minimum length.

55. Variation in the speed of the resultant vector —A consideration
of figure 20A makes it apparent that the resultant vector alternately
leads and lags behind the radius vector of the major component; and
that its mean speed is the speed of the major component. A study
of the figure shows further that when 6 is positive, the point p; rotates
around p, in the same direction that CP, rotates around C, and the
speed of the resultant is a maximum at the point M, when the length
of the resultant isa maximum. When 6 is negative, p; rotates in the
opposite direction and the speed of the resultant is a maximum at
the point N, when the length of the resultant is a minimum.

56. Speed of the resultant of two components of equal amplitudes.—It
the amplitudes of the two components are equal, the resultant CP;
(fig. 20), evidently bisects the angle P,CP2:, and the angle YCP; is
therefore equal to (a,+%b)t. The speed of the resultant therefore
has the constant value of a,+ 4b, the average of the speeds of the two
components.

57. Form of resultant of two components whose speeds are nearly
equal.—lIf the difference between the speeds of two components is
relatively small, so that the length of their resultant vector changes

32

little during one revolution of the component radi, the curve repre-
senting the resultant of the two components evidently takes the general
form of a sinusoidal curve, with an amplitude slowly fluctuating be-
tween the sum and difference of the amplitudes of the components,
as shown in figure 21.

FIGURE 21.—Tide curve of two components whose speeds are nearly equal.

If A, is the amplitude of the major component and 6 the numerical
difference between the speeds of the two components, the equation of
the resultant has the form:

y=A, cos (a;t-+ a) +A, cos [(ai+b)t+ a] (33)
when the speed of the minor component is the greater, and:
y—Ay cos (at-+ a) + Ae cos [(a,—b)t+ ars] (34)

when the speed of the minor component is less than that of the major.
From the discussion in paragraph 55 it is apparent that in the first
case, (equation 33), the speed of the resultant is greater than that of
the major component when the amplitude of the resultant is large,
and less when it is small. The high and low waters of the resultant .
shown in figure 21 will then progressively lead those of the major com-
ponent when the amplitude of the resultant is large, and progressively
drop back again when its amplitude is small. In the second case
(equation 34), the high and low waters of the resultant will progres-
sively lag behind those of the major component when the amplitude
of the resultant is large, and progressively catch up with them when
the amplitude is small. By taking the sum of the three components:

y= A, cos (a,t+ a) + Ap cos [(a,-+b) t+ a2] + As cos [(a; —b) t+ a] (35)

the timing of the high and low waters of the resultant may be made,
by the selection of the relative values of A, and A;, to conform to a
systematic variation from the timing of the high and low waters of the
principal component. If A, is equal to A;, the timing of the high
and low waters of the resultant will conform exactly to those of the
simple harmonic component A; cos (at-+ a), since the speed of the
resultant of the last two terms in equation (35) is, as shown in para-
graph 56:

% [(a, +6) + (—6)]=ay. (36)

B35)

58. If the amplitudes of two components are equal, A,—A,—0
and if their speeds are nearly equal, the curve of the resultant takes
the form shown in figure 22.

FIGURE 22.—Tide curve of two components of equal amplitudes.

Since the speed of the resultant in this case is the average of the
speeds of the two components, the equation of the resultant con-
veniently may be written:

y=A cos [(a+b)t+a:]+A cos [(a—b) t+ ay] (37)

where a is the speed of the resultant and 6 the difference in the
speeds of the two components.

59. The form of the curves shown in figures 21 and 22 indicates the
manner by which the periodical variations in the semidiurnal and
diurnal parts of the tidal fluctuations at any station, due to the
changing declinations and distances of the moon and sun (pars.
40-44), may be represented by a combination of components of prop-
erly chosen speeds, fixed by the accurately established periods and
speeds of the movements of the moon and sun.

ASTRONOMICAL PERIODS DETERMINING THE SPEEDS OF
TIDAL COMPONENTS

60. Mean solar and lunar days.—The intervals between the suc-
cessive transits of the true sun across the meridian of a place vary
slightly with its declination and distance, increasing as its declination
increases either north or south of the equator, and increasing also as
its distance decreases. The mean interval is the mean solar day of 24
mean solar hours. The speed at which the hour angle of the true sun
increases obviously must be the least when the interval between its
transits is the greatest. The speed of the hour angle of the true sun,
in degrees per mean solar hour, as influenced by the declination, is
therefore a maximum when the declination is zero, and a minimum
when the declination is at its maximum. As influenced by the
distance, the speed is a minimum at perihelion, when the distance to
the sun is the least, and a maximum at aphelion, when the distance
is the greatest. The angular speed of the hour angle of the moon goes
through similar but more rapid changes, with further disturbances
because of the attraction exerted by the sun on the moon. The mean

34

interval between the upper, or visible, transits of the moon across the
meridian of a place is the mean lunar day of 24.841,202,4 mean solar
hours.

61. It may be noted that the slow tilting to and fro of the plane of
the moon’s orbit with respect to the plane of the earth’s Equator
because of the changing longitude of the moon’s node (par. 36) does
not affect the length of the mean lunar day, the mean speed of the
lunar hour angle, or the mean period of the travel of the moon from
intersection to intersection. The intersection moves to and fro along
the Equator in a small are on either side of the equinox, its mean posi-
tion being the equinox itself.

62. The mean periods and the speeds, or mean angular changes per
mean solar hour, of the movements of the sun and moon pertinent to
the development of the speeds of the tidal components are as follows:

TABLE I
Period in Speed in
Cycle mean solar degrees per
hours solar hour
Migantsolanidayi=s-- bees aoe Sot eS Ce ee ee ee eee 24 15
VT sara rir saree Cl hye oe ee ne ee ee ea 24.841,202,4 | 14. 492, 052, 1
Tropical month—moon’s travel from intersection to corresponding intersec-

BLOTS ect ee eek Sea en ae ge ne Se tere ce a gals REE) ee ee 655. 717, 96 . 549, 016, 5
Anomalistic month—moon’s travel from perigee to perigee (par. 43)__________ 661. 309, 20 . 544, 374, 7
Synodichmonth— tual sare orto wir) Lira 0 ora eee ee 708. 734, 1 . 507, 947, 9
Tropical year—sun’s travel, equinox to corresponding equinox_-_--_--_- 5 8765. 812, 7 . 041, 068, 6
Anomalistic year—sun’s travel, perihelion to perihelion_________-__-_-_-_____ 8766. 230, 9 - 041, 066, 7

63. The “speeds” in the preceding tabulation are the quotients of
360° divided by the corresponding period, in accordance with the de-
finition expressed in equation (28). The speed of the mean lunar
day is the mean hourly change in the angle between the hour circle
through the moon and the meridian of the place. The speed of the
tropical month is the mean hourly change in the angle between the
hour circle through the moon and that through the intersection of the
moon’s orbit with the Equator. The sum of these two speeds is
therefore the mean hourly change between the meridian and the inter-
section. Similarly the sum of the speeds of the solar day and the
tropical year is the mean hourly change between the meridian of the
place and the equinox. Since the mean position of the intersection
is at the equinox, the two sums are the same, i. e.,

15.000, 000, 0 14.492, 052, 1
.041, 068, 6 .549, 016, 5

15.041, 068, 6 15.041, 068, 6

35
SPEEDS OF THE TiDAL COMPONENTS

64. Semidiurnal lunar components.—It has been shown in para-
eraphs 40 and 41 that the amplitude of the semidiurnal part of the
lunar tide increases and decreases twice during the tropical month
because of the changing declination of the moon, and in paragraph 43
that it increases and decreases once during the anomalistic month
because of its changing distance. Since the effect of one of these
variations on the other is small, these variations may be reproduced
by a combination of a major component and two pairs of minor com-
ponents in the form indicated in paragraph 57; i. e.,

y=Mb cos (msf+a,)+K,y cos [(m.+b)t+a]+K5 cos[(m:,—b)t+as]
+L, cos [(m2+¢)t+ as]+ No cos [(m,—c)t-+ as] (38)

In equation (38), Mz, is the amplitude of the major component.
Its speed m,; is the mean speed of the semidiurnal component of the
tide, and is therefore twice the speed of a lunar day, or 28.°984,104,2
per mean solar hour (par. 55).

Either or both of the next two terms will produce a variation in
the amplitude of the resultant, of the same period as the varia-
tion in the amplitude of the actual tide due to the changing decli-
nation of the moon, if 360°/b is made equal to that period (par. 54).
Since the period of these fluctuations is one half of a tropical month,
b is twice the ‘‘speed”’ of the tropical month, or 1.°098,033,0 per
mean solar hour. The relative amplitudes of this pair of minor
components should produce a variation in the speed of the resultant
conforming to the variation in the speed of the semidiurnal tide, and
hence in the true speed of the hour angle of the moon, due to the
changing declination of the moon. Since, when the effect of the vary-
ing declination is alone considered, the hour angle reaches its maxi-
mum speed when the declination is zero (par. 60) and the ampli-
tude of the semidiurnal part of the lunar equilibrium tide is then
a maximum (par. 40), the component with the greater speed,
K, cos [(m,+6)t+ a] must be the dominant one of the pair (par. 57).
A mathematical derivation of the tidal components, later outlined,
shows that this component correctly reproduces the entire variation
in the amplitude of the semidiurnal part of the lunar equilibrium tide,
and hence of the actual tide as well, because of the changing declina-
tion of the moon. The third term in equation (38) therefore dis-
appears.

Similarly either or both of the fourth and fifth terms of equation
(38) will produce a variation in the amplitude of the resultant of the
same period as that of the actual tide due to the changing distance
of the moon if 360°/c is made equal to the anomalistic month, or if ¢
is the speed of the anomalistic month, 0.°544,374,7 per mean solar hour.

36

Since the speed of the hour angle of the moon tends to become a
minimum at perigee, when the increase in the amplitude of the semi-
diurnal part of the lunar equilibrium tide because of the decreased
distance of the moon is a maximum, the component with the lesser
speed, Nz cos [(m,—c)t+-as], is the larger. Both of this pair of com-
ponents are found necessary, however, to represent the variations in
the tide because of the changing distance of the moon.
65. Writing then equation (38) in the form:

y= Mz cos (Msf-+a;) + Ky cos (ket-+ a2) +L» cos (t+ a3) Nz cos (nett ax)
(39)
M,—28°.984,104,2 per mean solar hour.
k,»=m,+6=30°.082,137,2 per mean solar hour.
],=m,)+c=29°.528,478,7 per mean solar hour.
Ny=M,—C= 28°.439,729,5 per mean solar hour.

The detailed mathematical analysis shows that components having
these speeds are sufficient to reproduce with substantial accuracy
the lunar semidiurnal part of the tide. Certain other small compo-
nents are developed by that analysis, principally to account for the
variations in the tides due to the irregularities in the movement of
the moon because of the sun’s attraction, but these are of no sub-
stantial importance.

66. Designation of components.—The capital letters designating the
amplitudes of the components whose speeds are identified in the pre-
ceding paragraphs are those conventionally assigned to these com-
ponents. The subscript 2 indicates that the component is a semi-
diurnal one; i. e., that its speed is in the vicinity of 30° per mean
solar hour. Diurnal components, with a speed not far from 15° per
hour, are given the subscript 1. The speed of the component is
conventionally designated by the corresponding small letter of the
alphabet, and is given the same subscript. Components are custom-
arily referred to by the letter and subscript designating the amplitude;
1. e., as the ““M. component,” and “‘K, component,” etc. The com-
ponents previously identified are named as follows:

Mp, the principal lunar component,

No, the larger lunar elliptic, semidiurnal,

Ly, the smaller lunar elliptic, semidiurnal,

Ks, the discussion in paragraph 63 indicates that a solar com-
ponent of the same speed is to be anticipated. This
component is therefore called the lunar portion of the
lunisolar semidiurnal.

67. Solar semidiurnal components.—The speeds of the solar semi-
diurnal components corresponding to the lunar semidiurnal compo-
nents already identified may be written at once by substituting the

37

solar for the lunar day and the tropical year for the tropical month.
These components are:

S., the principal solar. Since its speed is twice that of a mean
solar day, s:=30° per hour.

Ts, the larger solar elliptic, semidiurnal. Its speed is the differ-
ence between that of the principal solar and that of the
anomalistic year and is therefore tz=30—0°.041,066,7=
29°.958,933,3 per hour.

Ro, the smaller solar elliptic, semidiurnal. Its speed is the sum
of that of the principal solar and of the anomalistic year,
and is therefore r,.=30+0°.041,066,7=30°.041,066,7
per hour.

K;, the solar portion of the lunisolar semidiurnal. Its speed is
twice the sum of the speeds of the solar day and the
tropical year, and is therefore 30°.082,137,2 per hour.

Since the eccentricity of the earth’s orbit around the sun is rela-
tively small, the T, and R, components are small in comparison with
the N» and L, components respectively. Since the lunar and solar
parts of the lunisolar semidiurnal components have the same speed,
they unite into a single component of that speed (par. 52), designated
as the lunisolar semidiurnal component Kg.

68. Lunar diurnal components.—The amplitude of the lunar diurnal
part of the tide has been shown to increase from zero to a maximum
and back again to zero twice during a tropical month because of the
changing declination of the moon. This part of the tide follows
therefore a curve of the characteristic form shown in figure 22. Such
a curve is represented by the sum of the two components:

y=A< cos [(a+ 4b)t+a|+A cos [(a— 4b) t+ ay] (40)

in which a is the speed of the resultant of the two components, and
360°/6 is the period of the fluctuation of the resultant (par. 58). The
speed of the resultant lunar diurnal tide is the speed of the lunar day,
designated as m,. If Tis the period of the tropical month

360°/b=%
whence 360°/4%6=T

It follows from the definition in paragraph 63 that 'd is the ‘“‘speed”’
of the tropical month.

The data given in paragraph 62 show that the numerical values of
the speeds of the two components in equation (40) are respectively:

m,-+ 4b=14°.492,052,1-+-0°.549,016,5=15°.041,068,6
— b= 14°.492,052,1—-0°.549,016,5=13°.943,035,6

38

The first of these speeds is one half of the speed of the lunisolar semi-
diurnal component K;. It is therefore designated as k;. The second
is conventionally designated as 0}.

Since the amplitude of each of the lunar diurnal components just
developed also fluctuates during the anomalistic month because of the
varying distance between the earth and the moon, each should be
made up of a principal and two minor components, giving the six
components in the form

y=K, cos (k,t-+-a)+K’, cos [(k,; +e) ¢+a]+K’’; cos [(ki—c)t+ a]
+O, cos (o;¢+-a)-+-O’, cos [(0;+¢)t+a]+0’’; cos [(0;—¢)t+a] (41)

where c is the speed of the anomalistic month, and the value of a in
the various terms is not generally the same.
The speeds of the minor components then have the values:

k, te=15°.041,068,6-+0°.544,374,7 =15°.585,443,3
kj —c=15°.041,068,6—0°.544,374,7 = 14°.496,693,9
0, te=13°.943,035,6-+0°.544,374,7=14°.487,410,3
0;—¢=13°.943,035,6—0°.544,374,7 = 13°.398,660,9

The component having the speed k,-++c is designated as the J; com-
ponent and that having the speed of 0,—c¢ the Q, component. The
other two have speeds so close to that of the lunar day, besides being
intrinsically small, that they are replaced by a component designated
as M,, with the speed of the lunar day.

69. The components identified in the preceding paragraph are
named as follows:

K,, lunar portion of the lunisolar diurnal, whose speed is the
sum of those of the lunar day and tropical month.

O,, principal lunar diurnal, whose speed is the difference
between those of the lunar day and tropical month.

J,, small lunar elliptic, whose speed is the sum of those of the
lunisolar diurnal and the anomalistic month.

Q., larger lunar elliptic, whose speed is the difference between
those of the principal lunar diurnal and of the anomal-
istic month.

M,, smaller lunar elliptic, whose speed is that of the lunar day.

70. The corresponding solar diurnal components are:

K,, solar portion of the lunisolar diurnal, whose speed is the
sum of those of the solar day and the tropical year, or
15°.041,068,6. This unites with the lunar tide of the
same speed to form the lunisolar diurnal component.

og

P,, principal solar diurnal, whose speed is the difference
between those of the solar day and tropical year,
15—0°.041,068,6=14°.958,931,4.

The solar elliptic diurnal components corresponding to J, and Q,
are too small to be recognized. Daily land and sea beeezes and daily
variations in atmospheric pressure may however give rise to small
fluctuations having the period of the mean solar day. A meteoro-
logical component, S,, with a speed of 15° per hour is therefore
recognized.

71. Long period components.—The following long period compo-
nents have already been developed in the discussion of the solar and
lunar equilibrium tides:

The lunar fortnightly (par. 42).—This component is conventionally
represented by the symbol Mf. Its speed is twice that of the tropical
month, or 1°.098,033,0 per mean solar hour.

The lunar monthly (par. 43).—Conventionally represented as Mm.
Its speed is that of the anomalistic month, or 0°.544,374,7 per mean
solar hour.

The solar semiannual (par. 44).—Conventionally represented as Ssa.
Its speed is twice that of the tropical year, or 0°.082,137,2 per mean
solar hour.

In addition, recurring seasonal meteorological effects produce a
solar annual component, designated Sa, whose speed is that of the
tropical year, or 0°.041,068,6 per solar hour.

72. Overtides.—A distortion of the tides is produced in the compara-
tively shallow waters of estuaries and other coastal areas. This dis-
tortion gives rise to tidal components whose speeds are multiples of
the speeds of the astronomical components heretofore developed.
They are called overtides because of their analogy to overtones in the
theory of musical tones. The only overtides of sufficient magnitude
to be of importance are those of the principal lunar and solar com-
ponents M, and S,. They are designated M., Ms, Ms, and S, and
S,, the subscripts denoting the ratio of their speeds to that of the mean
lunar or solar day.

73. Compound tides—Besides producing overtides, the distortion
of the tides in shallow water gives rise to components whose speeds
are the sums or differences of the speeds of the elementary components.
The recognized compound tides are MS, with a speed of m;-++s,; MN
with a speed of m.+ns, MK with a speed of m.+k,, 2MK with a
speed of my—k,, and 2 SM with a speed of s;—m,. These components
are generally quite small. Certain other compound tides have the
same speed as some of the primary components and therefore unite
with them.

74. Tide depending on the fourth power of the moon’s parallax.—tit
has been shown in paragraph 32 that the lunar equilibrium tide due to

40

the fourth power of the moon’s parallax is very small, and that it goes
through three maxima and minima as the zenith distance of the moon
makes‘a cycle from 0° to 360°; while the principal part of the tide,
due to the third power of the moon’s parallax, goes through two
maxima and minima during this cycle. It is not difficult to see,
therefore, that the fluctuations of both the equilibrium and the actual
tides due to the fourth power of the moon’s parallax go through three
fluctuations each lunar day. This tide is therefore approximated by
the component:

y=Ms cos (m3t+a) (42)

where m; is three times the speed of the lunar day, or 43°.476,156,3 per
mean solar hour. Additional components to account for the variations
in this part of the tide due to the changing declination and distance of
the moon, and other variations from this primary component, are too
small to be recognized.

75. Résumé of components identified —For the purpose of harmonic
analysis, it is convenient to group together the components whose
speeds are multiples of another. The components identified in the
preceding paragraphs fall into groups and individual components as
follows:

TaBLeE II
Speed in
Symbol Name See
hour

M Groupe
Mi Smailleninnariellipiicraiinn a) ees ee ee ee ee 14. 492, 052, 1
Me Lenebovenjoye) [itbuarene GspoaIOhIDE OC eo aes eh ee esate Sees Sesoses =e 28. 984, 104, 2
M3 IVconaa Zura NODE weaoVN ide TmNE Me — = ee Le ne BaP S/T De cle ® 43. 476, 156, 3
My LunllarOvertide 23225 = ae ee ee ee ee eee 57. 968, 208, 4
Ms © Ge eae ES et TAR Dias tsa ce SOF TUE es Oa) ee 86. 952, 312, 6
Mighe HL Sse8 Oe ee a es Se es ee ae ee ee 115. 936, 416, 8

S Group
S: Meteorological: Sat.- ee SES ee Sa ae a SE ee 15. 000, 000, 0
So 1Brhavor oul Cole Genooviobiby moe ne ee eee 30. 000, 000, 0
S4 Solar overtide 2.22.2 2see2 Ses Se a aes eo es ee ae 60. 000, 000, 0
Se. lessee (6 Uo pen em me mc eS ee Ee Oe oe Re 90. 000, 000, 0

K GROUP
Ki Ljnisolar:- diurnal. == oe 15. 041, 068, 6
Ko Lunisolar'semidiurnal. - .22_-2 12 ea ee ee ee eee 30. 082, 137, 2

INDIVIDUAL DIURNAL COMPONENTS
O; Pprincipaldunaridiurial= 32 et ae ee ee ee oe ee 13. 948, 035, 6
P, Prineipalisolaridivrn alee Se Re ee a ee ee 14. 958, 931, 4
Q; Larger lunarellipticidtirimal 2 Se ees Se ee eee eee 13. 398, 660, 9
Jy Smaildunarielliptici@ivutmal = === saa ee ee 15. 585, 443, 3
INDIVIDUAL SEMIDIURNAT, COMPONENTS

Na NGA TOTS LUA TET Np GL CSE TINT CU UL Te a a a 28.439, 729, 5
Le Smallerlunanellipticisem diurnal ee sss ae nee 29. 528, 478, 7
T. IGS ool eMlbhoa Ve Ceraenolibin Mls 2 eee eet See ee ee 29. 958, 933, 3
Ro ShaeylDecoleie cllbyoynte cecan Gera ON. a ee eee ee eS 30. 041, 066, 7

4]

Taste I]—Continued

Speed in
r. degrees per
Symbol Name mean SAG
hour
a LONG PERIOD COMPONENTS
Mf Une O Gen oh Ghyjaeeaa eae oo eee oe ee ee ee SUR ae oa eee 1. 098, 033, 0
Mm UTE AT ATA OM GL Veaes pean Same ee a ae ea, ee a ee eS . 544,374, 7
Ssa Solbirgaeaey aa elle ees eS oe eee Se SRS Se ee ae ee . 082, 137, 2
Sa Solariannuale -22--25 = ee ee ee ee ee ee ee - 041, 068, 6
COMPOUND TIDES

TRUS) 1) oa oe me pe og eRe am RR ep ET 58. 984, 104, 2
WONT 2228 5 2 Lies ee ees ee er ee ee ee ee ee ee ee ee 57. 423, 833, 7
FY 1G en | Pn yt Seen Mire em oe as A ev dee ee we Se 44.025, 172,9
DINK fas Sete 5 Sees wee RR aa a ee aie Ree ee ee ee eee 42. 927, 139,8
GIN lla sa ace ae SUR PRR CS SRE AE ee is ae i ee ei a 31. 015, 895, 8

76. Other components.—Additional small components disclosed by
the mathematical analysis later outlined are here appended for
convenient reference.

TaBLeE II]

Symbol Name Speed
2N Lunar elliptic, 2d order, semidiurnal______- x 27. 895, 354, 8
y2(nu) Larger lunar evectional, semidiurnal ______ 28. 512, 583, 1
Ax(ambda) | Smaller lunar evectional, semidiurnal 29. 455, 625, 3
wo(mu) Wariationales san as: See nae s oe aes eset ao 27. 968, 208, 4
OO Munarciirnalpodiordersa: = =a. 2eee ee a ee eee ee 16. 139,101, 7
2Q unavellipticn2drorderidiuria lea e222 ee aes ee 12. 854, 286, 2
pi(rho) ILanaar loa Oyacitoroe |, Chupa a = ee ee ee a 13.471, 514,5
MSf MIMisolaTsyNo die MOntnl Sh thy = see aa = ee ee 1.015, 895, 8

HARMONIC ANALYSIS OF TIDES

77. The amplitude and initial phase of each component of the tide
at any tidal station may be computed from the observed hourly tidal
heights for a sufficient period of days, and the predetermined speed
of the component, by the process of harmonic analysis, which will now
be explained. The observed heights used for this computation are
taken at (mean solar) hourly intervals, beginning at midnight (0 hour)
each day, giving 24 observations per day. The observations ordi-
narily are on standard time; but early records may be on local time.

78. Separation of S group of components.—The repeating form of
the sinusoidal curve representing any component (fig. 16) shows at
once that the value of the component at any instant is repeated at the
intervals of time given by the period of the component, and by any
multiple of that period. Thus since the period of the S, component
is 360°/30°—12 hours, this component has exactly the same value at
say 3 p.m. on any day as at 3 a. m.; and has the same value at 3 a. m.
on every succeeding day. The solar overtides S, and S,, with periods
of 6 hours and 4 hours respectively, each have similarly the same value
at the same hour each succeeding day, as has the small S; component.
All other components have values which progressively vary at the

42

same solar hour each succeeding day. The progressive variation of
the M, component, whose period is 12.420 hours, is illustrated for
example, in figure 23, in which P; P2, and Ps, etc., show the relative
values of the component at 3 a. m. on 3 successive days; the amplitude
and initial phase of the component being taken at random. In this

FIGURE 23.—Variation in Ms component at given solar hour.

progressive variation, these components run through their entire
range of values, both positive and negative.

While, therefore, the sum of the successive values of the S group of
components at a given hour over a period of days increases directly
with the number of days in the period, the average value remaining
constant, the sum of all other components at that hour does not so
increase. On the contrary, at the intervals at which the positive
and negative values of a component offset each other, the sum of the
values of that component nearly disappears. If the number of days
in the period is suitably chosen, the average value of the resultant of
all of the components at a given hour reduces therefore to nearly the
value of the S group at that hour.

79. The observed hourly tidal heights are ordinarily scaled from
the record of a recording tide gage, and give these heights above an
arbitrarily chosen datum, usually set low enough to make all of the
readings positive. Each of the recorded heights is then the algebraic
sum of the height of mean sea level above datum, plus the resultant
of all the tidal components at the hour, plus the accidental variations
due to meteorological disturbances, as well as to inaccuracies of
observation. The average of the heights at a given hour of the day
over a suitably chosen number of days, then closely approximates
the height of the resultant of the S group of components at that hour
above the datum plane, the other components, together with the
accidental variations, being averaged out by the process.

80. Example.—The period of the diurnal component K, is so close to
that of the S,; component that 6 months of observations are necessary to
segregate the S group of components as a whole. The principal
solar component, S, (with the overtides) may however be approxi-
mately determined from a set of observations extending over 15 days,

43

by averaging the hourly tidal heights at the 12-hour period of this
component. For example, the hourly tidal heights at Sitka, Alaska,
for the first 3 days, and the average of both the morning and after-
noon hourly tides for a 15-day period beginning July 1, 1893, are shown
below, the heights during the other days of the period being omitted
for brevity.

Hourly height of tide, in feet

Total Total
DD nyeeeeenen -AC Le | -- 1 2 3 3 days 15 days Average
Hour a.m p.m. a.m p.m a.m. | p.m.

(eee ot 13.9 9.5 13.1 8.4 11.6 Tell 63. 6 309. 2 10. 31
ios Jao ee es 14.5 11.4 14.1 10.5 13.0 9.2 PRT 317.9 10. 60
2. 14.2 12.5 14.4 alzyal 13.8 T1s2 78. 2 319.5 10. 65
ena em = 13.0 12.8 13.8 12.9 13.8 12.4 78.7 313.5 10. 45
epennmerrar eee 10.9 12.3 12.2 12.8 12.9 12.8 73.9 301.4 ° 10.05
Geter 8.5 Tals 1 10.1 12.0 2, 12.4 65.3 287.0 9. 57
Gane eer Ls 6.0 9.8 aD) 10.7 8.9 11.3 54. 2 272.9 9.09
eee te Fs 4.3 8.8 5.5 9.4 6.8 10.0 44.8 264. 6 8. 82
Cp ies) 3b) 8.4 4.1 8.5 4.9 8.8 38. 2 262. 0 8.73
O63. 3 3.9 8.8 3.9 8.3 4.1 7.9 36.9 268. 0 8. 93
il) =. 2 eee 5.3 10.0 4.7 8.9 4.2 8.0 41.1 280.5 9.35
ii. =. ee abt) Tile 6.5 10. 2 5.83 8.7 49.7 296. 2 9. 87

AQ 0s se sees Eesessee | Stee see |e Sess ces aaa elle nee I oer ee eee S| Pe ae ete 9. 70

The average heights derived in the last column are then the ap-
proximate hourly heights of the S, component, and its overtides,

ky

Po
©
o
o
Gs Complete Tide
Re So component ———-—
‘S
bo
“eo
A)
oq
oO

Mean Solar Hours

FIGURE 24.—S: component at Sitka, Alaska.

above datum. The recorded hourly heights for the first day, and of
the S, component as thus computed, are plotted in figure 24.

81. The amplitude of the S, component at Sitka shown by the plotted
curve conforms well with value of 1.137 feet found from a year’s
observations. The symmetry of the curve indicates that this com-

192750—40——-4

44

ponent is not affected by overtides of any substantial amplitude.
A complete analysis for a year confirms this fact. This symmetry
shows also that a summation for even this short period nearly elimi-
nates the other components.

82. The computation of the S, component in paragraph 80 is ar-
ranged to illustrate the process most directly. In the conventional
form of computation, the observed hourly heights, numbered from 0
to 23 each day, are entered in daily vertical columns on standard
sheets. The S group of components is computed by summing the
observed heights horizontally, the aggregates of these sums being
checked against the aggregate of the sums of the daily columns.
The amplitude and the initial phase of each component of the S
group are then computed from the average hourly heights of the whole
set of observations by a method to be explained later.

83. Separation of other components—The observed tidal heights at
each mean /unar hour similarly could be taken off and tabulated by
lunar days, and the M group of components segregated by averaging
the tidal heights at each hour of the lunar day. In like manner all
of the components could be determined by averaging the tidal heights
at their component hours, tabulated by component days. The process
of taking off new observed tidal heights for each of the several com-
ponents would be a laborious one. Instead, the height at each lunar
or other component hour is taken as the observed height at the nearest
mean solar hour. These heights are sometimes a little greater and
sometimes a little less than the true heights at the component hour,
but their average over a sufficient number of days closely approximates
the average at the given component hour. The same process of averag-
ing which separates the component sought from the others, reduces
the observed heights as taken at the nearest mean solar hour to the
heights at the component hour. The correction for a small systematic
error resulting from the process is developed in paragraph 97.

84. Component days and hours.—To select the observed mean solar
hourly heights which are to be taken as the component hourly heights,
a tabulation is prepared showing the component hour nearest to each
solar hour on each successive calendar day. For diurnal components,
the component day is the period of the component, and the component
hour, one twenty-fourth part of that period. For semidiurnal com-
ponents, the component day is twice the period of the component,
and the component hour is one-twelfth of the period. For components
of shorter periods, the component day is that multiple of the period
nearest to a mean solar day. It will be observed that the com-
ponents M,, Mz, M;, My, Me, and Msg all have the same component
hour; and that the components K, and Kg, similarly have the same

45

component hour. Since the period, in mean solar hours, of a com-
ponent whose speed is a, is 360°/a, the length of the component hour
of a diurnal component is 360/24a=15/a solar hours, and 1 solar
hour is a/15 component hours. For semidiurnal components, 1 solar
hour is a@/30 component hours.

85. Tabulation of component hours.—A tabulation of the mean
lunar hours corresponding to mean solar hours will illustrate the
process of preparing such a tabulation for any component. Since
the speed of the lunar day (and of the M; component) is 14°.492,052,1
per hour, 1 solar hour is equal to 14.492,052,1/15=0.966,136,8 lunar
hours. In the tabulation at the end of this paragraph, the left-hand
column lists the mean solar hours each day. The next column gives,
in parentheses, the corresponding lunar hours, to three places of
decimals, for the first 15 hours of the first calendar day, and the third
column, the corresponding nearest whole lunar hour. Succeeding
columns give these whole lunar hours on the following calendar days.
It will be observed that the values of the corresponding lunar hours
diminish hourly by 1.—0.966,136,8=0.033,863,2. Between the four-
teenth and fifteenth mean solar hours the cumulative diminution
passes half a unit, so that in whole numbers the fourteenth lunar hour
corresponds to both the fourteenth and fifteenth mean solar hours.
Obviously from the fifteenth solar hour on, until the progressive
diminution passes 1.5, the corresponding lunar hours are 1 hour less
than the solar hours. After 1.5/0.033,863,2=44.296 hours, i. e.,
beginning with the twenty-first hour of the second day, the corre-
sponding lunar hours drop back another unit, and may be found by
subtracting 2 from the solar hour (or adding 22 if the remainder
would be negative), and so on. ‘To fill out the tabulation it is neces-
sary only to find the solar hours at which the lunar hours drop back a
unit. These occur at intervals of 1/0.033,863,2 =29.53058 hours begin-
ning with 0.5/0.033,863,2=14.76529 hours. They are, therefore, the
integers following:

14.76529, or 1st day—15 hours. (—1)

44.29586, or 2d day—21 hours. (—2)

73.82644, or 4th day— 2 hours. (—3)
and so on.

The hours beginning at which successive numbers are to be added or
subtracted to give the nearest component hour of each of the har-
monic components are tabulated in Manuals of Harmonic Analysis
of Tides under the heading “Tables for the Construction of Primary
Stencils; ”’ from which the data for the M group of components, up
to the twenty-ninth day, is extracted.

46

M components

Differences | Day | Hour Differences | Day | Hour

! 1 0 +16 —8 10 6
+23 —1 1 15 +15 -—9 11 12
+22 —2 2 21 +14 —10 12 17
+21 —3 4 2 +13 —11 13 23
+20 —4 5 8 +12 = 114 15 4
+19 —5 6 13 +11 —13 16 10
+18 —6 7 19 +10 —14 17 15
+17 —7 9 0 +9 =15 18 21

Differences | Day | Hour
+8 —16 20 2
+7 —17 21 8
+6 —18 22 13
+5 —19 23 19
aL || =) | 25 0
+3) —21 26 6
+2 —22 27 iy
+1 —23 28 17

1 29 22

It is for the preparation of such tables for 369 or more days that
the speeds of the components are extended to seven places of decimals.
The application of these tables is illustrated in the extension of the
subjoined tabulation to include the seventh day.

Calendar days Calendar days
Mean Mean
solar Mea lunar 1 2 3 4 5 6 7 solar Mea lunar 2 3 4 ay. |) 8 ef
inaiiie our hour our
Nearest whole lunar hour Nearest whole lunar hour
© || Meeese-=- ON 23225522) e 21s 20, 19 nln (G01 593) ee 1) Wal |) 1) 9 8 8 7
ie (0966) =e 1 (@) |) PS¥ || Fas Be |) Za 20 Tle} |) (IP ae) ese) fp lP2 j) ANIL |). 0) 9 8 8
2a e932) eae 2 1 ON e23ele2an ieee 21 145) (32525) es 14 LS |e ee 9 9
3, | (22898) 222 == 3 2 1 0 One23 22 15 | (14.491)_____ 14) | 4) SS ese LD 10
4 | (3.864) _____ 4 3 2 1 1 0 23 LOpiee eae ees ly | Palsy pach) ai) We) sal inl
i) GeeBn)) 5 4 3 2 2 1 0 BY ig |e ea NG aKey |) ales aes |p alee |). tp 12
Gl) Gian) s2-. 6 5 4 3 3 2 1 ARS} eset Ee eS V7) LZ |) 16 Vom ts 13
7 |) (@7G8)) = 5= TEM GN ee) Gb eet] 83 2 OR Reeeaese =. se TEM alee 1K} |) 1G) |) EL |) 713
Sein(@Ai29) Se By wp 4b 4 3 Pty || See eae 19 | 19 | 18| 17] 16] 15} 14
9 | (8.693) -_-_- 9 8 7 6 5 5 4 Zils Petes OL ees 20 | 19 | 19 | 18 |] 17 | 16 15
10 | (9.661) _---- 10 9 8 ili 6 6 5 22} eis = ee 21} 20) ||) 20) |) LSS Ss a7, 16
Tit |) (Oz). |) Wl |} 30) 9 8 qf i 6 Pia el oe 5 eee 2a 2 PPL PAW Nera |) 720) || ik) || ike} 17

The height of the M group of components at zero component hour
is found by summing and averaging the observed heights at the hours
marked 0 on the tabulation; the height at the first component hour
by averaging those marked 1, and so on for the 24 component hours.

86. Example —The hourly heights of the M components found from
the application of this process to the observed heights at Sitka, Alaska,
over a period of 29 days beginning July 1, 1893, are as follows:

Component
raeriie 0 1 2 3 4 5 6 i 8 9 10 11
Sumses -e2 == 334.9 |370.4 |369.9 [371.8 |345.4 |277.7 |239.9 |203.8 |187.3 |197.0 |233.6 |272.3
Divisors.—----- 29 29 28 29 30 28 29 29 29 29 29 28
Component
height__.____] 11.55 | 12.77 | 13.21 | 12.82) 11.51) 9.92] 8.27) 7. 03 | 6.46] 6.79 | 8.06) 9.72
Component : 4
ote 12 13 14 15 16 17 18 19 20 21 22 23
SumsSes3---==—— 334.7 |882.9 |386.2 |373.8 |341.3 |289.6 |241.8 |213.6 |182.2.|197.6 |244.4 |288.8
Divisors- - ----- 29 30 29 29 29 29 29 30 28 29 30 29
Component
hehtaes==—= 11. 54 | 12.76 | 13.32 | 12.89 | 11.77 9.99 8. 34 7.12 6. 51 6. 81 8.15 9. 96

47

The heights of the M components thus computed, and for compari-
son the recorded ‘heights for the first day’s observations are shown
in figure 25.

Foy

> 10

o

o

oH

&
ae

a Complete Tide

sey M Components —--——
n

Lunar Hours

24

Mean Solar Hours

FIGURE 25.—M components at Sitkr, Alrska.

87. Stencils —The summation of the hourly heights is facilitated by
the ingenious device of cutting stencils which, when laid over the
tabulated observed heights on a sheet in standard form, show through
the openings the heights to be added to give the sums for each com-
ponent hour. Two stencils are prepared for each successive 7-day
period shown on a standard sheet, one for the even component hours
and one for the odd, lines being drawn on the stencil to connect the
observations to be taken for each component hour.

88. The stencils ordinarily used are prepared for the program of
computation illustrated for the M component, in which each observed
height enters once and only once in the summations of the 24 com-
ponent hours. On the stencils for those components whose component
hour, like the lunar hour, is longer than the mean solar hour, a com-
ponent hour is repeated at the intervals at which its tabulation must
shift back a unit to most nearly correspond to the mean solar hour;
and both of the observed values so indicated are included in the sum-
mation for the component hour. If the component hour is shorter
than a mean solar hour, a component hour is omitted from the stencil at
the corresponding intervals at which the tabular values must shift
forward a unit. The aggregate of the sums for the 24 component
hours taken from stencils in this form may be checked against the
sum of all of the observations for the period, but the divisors for com-
puting the averages may not be the same as the number of component

48

days in the period. Stencils may be prepared in which the summations.
are made for 1 and only 1 component hour in each component day.
In such stencils 1 of the observed heights is omitted at the shifts of the
component hour when it is longer than the mean solar hour, and 2 com-
ponent hours are assigned to the same observed height at the shifts
when the component hour is shorter than the mean solar. The latter
system is not generally favored.

89. Secondary stencils —The speed of the K, component is so nearly
that of the solar day that its component hours shift with respect to
the solar hours but once in about 8 days. To facilitate the computa-
tion of this component from a year’s observations, it is permissible to
assemble on standard sheets the 7-day sums of the observed hourly
heights, and to use these sums, instead of the daily observations, in
computing the hourly heights of this component. A number of com-
ponents similarly have speeds so close to those of others that they may
be computed from the 7-day hourly sums of their primary component.
The stencils prepared for such computations are called secondary
stencils.

90. Number of days of observations required—To eliminate a com-
ponent by harmonic analysis, the tidal observations should extend
over a period, or multiple thereof, in which the successive values of
the component to be eliminated, at the component hours of the com-
ponent sought, run through their entire range of values, both positive
and negative (par. 78). If ais the speed of the component A, which is
to be segregated, and 6 the speed of a component B which is to be
eliminated, component B gains 6—a degrees on component A each
solar hour. Its successive daily values at any component hour of A
will then run through their whole range of values in 360°/24 (b—a) =
15/(b—a) days, or 360/(b—a) hours, the synodic period of the two
components (par. 54). A consideration of the relative speeds of the
tidal components establishes a minimum period of 14 days for diurnal
and 15 days for semidiurnal components. The periods adopted by the
United States Coast and Geodetic Survey are 14-15, 29, 58, 87, 105,
134, 163, 192, 221, 250, 279, 297, 326, 355, and 369 days. The stand-
ard period for a complete analysis for tide predicting purposes is 369
days. A period of 29 days affords, however, fair determinations if
corrections are applied to eliminate the residual effect of interfering
components.

91. Computation of amplitude and initial phase-—The determination
of the heights of the various components at their component hours has:
been described in the preceding paragraphs. The amplitude and initial

49

phase of each component are computed from its hourly heights by a
process based on the arithmetical integration of the coefficients of
Fourier’s series. Components having the same component hour are
separated by the process.

92. Using the form indicated in equation (29), the height, at any
time ¢, of the resultant of a group of components having the same
component hour is given by the equation:

y=H,+A, cos (at—£1) +A, cos (2at—§) +A; cos (Bat—f3) + . . .(43)

where H) is the height of mean sea level above datum, A, the ampli-
tude of the diurnal component, a its speed, and ¢/a the time of its
high water (par. 49); A, is the amplitude of the semidiurnal compo-
nent and ¢/2a the time of its high water; and the other terms similarly
represent minor components and overtides. But a few terms are
needed in any group of components having the same component hour,
and for single components the equation reduces to one variable term
in addition to the constant term HH.

The expansion of the cosines in equation (43) gives the equation:

y=H),+ <A, cos at cos £,+ A; sin at sin £,+ A, cos 2at cos f
-+- A, sin 2at sin ¢,+ A; cos 3at cos (3+ A; sin dat sin f3... (44)

Placing:
PPCOStG =e pA Siti =——65, As COS Ca—Cz, Ay SIN (o>—G>, etc. (45)
equation 44 becomes:

y=H)+c, cos at+s, sin at+c: cos 2at+s2 sin 2at-+-c; cos 3at
+s, sin 3af+ ... > *46)

The values of the angles and coefficients in equation (43) may be
found readily from the coefficients ¢;, 81, C2, S2, ¢3, 83, etc., of equation
(46), since, from equations (45):

tan C/G tan 62=So/C2 tan 63=S83/C¢s, etc. (47)

and:
i=¢,/Cos G—s;/sin Az =C/cos f2=S,/sin fo, ete. (48)

It may be noted that by expressing equation (43) in the form indi-
cated by equation (29), rather than equation (27), negative signs are
avoided in equations (46), (47), and (48).

50

93. The demonstration of Fourier’s series shows that if y is any
function of x, its values between the limits of r=0 and r= are given
by the series:

y=B,+B, cos all) +(Q, sin (r2/l) +B, cos (27x/l) -+C, sin (272/L)
+B; cos (87rx/1) +0; sin (87z/l)+ ... (49)

in which:
= (UN) | ude,
B,=2/d) | y cos (ra/l)dz, C= 2/) |, y sin (rax/l)dz,
B= (2/1) ii ee CAD ee OO) ie y sin Omen

i l
B= ein |. y cos (872/l)dx, C3= @/d |. y sin (bra/l)dz, (50)

and so on. The angles in these equations are expressed in radians.
94. If 7'is the length (in mean solar hours) of the component day,
then since 27 radians=360°, 27/7'=a, the speed of the diurnal com-
ponent.
Placing x=t and /=T, equation (49) becomes:

y=B,+ B, cos %at+C, sin %at+B, cos at+C, sin at+-B; cos 3/2 at
+0; sin 3/2 at+B, cos 2at+C, sin 2at+ ... (51)

Equation (51) is the development of any function of ft. It becomes
the development of the particular function of t expressed by equation
(46) if it is identical with the latter, 1. e., if the coefficients of the identi-
cal terms in the two equations are equal, the coefficients of the terms
in equation (51) not appearing in equation (46) becoming zero. It
follows therefore that

T
Hy=B)=(1/T) |, at
T T
C=) — ein) |. y cos at dt, b= || y sin at dt,
T T
o=B=Q/D) |) y cos 2 at dt, ow y sin 2 at dt, ete. (52)

An examination of the form of the integrals in equations (52) dis-

closes that (1 i) |, "yat is the mean value of y between the limits of

51

ii
0 and T; @/) [ y cos at dt is twice the mean value of y cos at
0

between the same limits; and that similarly all of the remaining values
of the coefficients are twice the mean values of the expressions inte-
grated. If then the values of y, or the determined heights of the
component at its successive component hours, are designated as ho,
hy, hz *** hoz, it immediately follows that:

Ay=Vea(ho thy t+he+ cee + hy3) (53)

ie)

The angle at has the value of == 15° at the end of the first com-
ponent hour, 30° at the end of the second component hour, etc.
Twice the average values of y cos at; y sin at; y cos 2at; y sin 2at
are then:

C1=Mo2(hy cos O+h; cos 15°+ he cos 80°+.+hy3 cos 345°) (54)

S:= Mo(ho sin O+/, sin 15°+hAy sin 80°-+.+A,3 sin 845°) ~— (55)

C2= Ye(ho cos O+h, cos 380°+h,z cos 60°-+.+h23 cos 330°) (56)

So= Vo(hy sin O+h, sin 30°+-f2 sin 60°-+.-+-Az3 sin 330°) (57)
and so on.

Equation (53) merely expresses the evident fact that the elevation
of mean sea level is the mean of the heights at the component hours.
The amplitude and initial phase of the diurnal component are de-
termined by computing c, and s, from equations (54) and (55) and
applying to them the relations expressed by equations (47) and (48).
The amplitude and initial phase of the semidiurnal component are
similarly derived from the computed values of cz; and s,; and those of
other components from the corresponding coefficients, the equations
for determining which may be written by analogy to equations (54)
to (57).

The computations of the coefficients from equations (54) to (57)
may be greatly abbreviated by combining the terms whose sine or
cosine factors have the same numerical value. For example, in find-
ing the values of c. and s, from equations (56) and (57), it is apparent
that these factors for hi. are respectively cos 360° and sin 360°, those
for hy; are cos (360°+30°) and sin (360°+30°) and so on. The suc-
cessive factors for the last 12 terms are consequently the same as for
the first 12 terms. Furthermore, the factors for hg are cos 180° and
sin 180°, respectively. Since cos (180-+¢)=—cos¢ and sin (180°+¢)
——sin ¢, the successive factors for the second 6 terms are equal but
opposite in sign to those of the first 6 terms.

95. Erample.—Taking the heights of the M group of components
at their component hours at Sitka, computed in paragraph 86, the
computation of the amplitude and initial phase of the M, component

at this station and at the

as follows:

52

(1) Hourly heights ho—hy, 11.55 12.77 13.21 12.82 11.51 9.9? 8.27 7.03 6.46
(2) Hourly heights hyz—ho3 11.54 12.76 13.32 12.89 11.77 9.99 8.34 7.12 6.51
KO) ENS UTS As oe te ee 23.09 25.53 26.53 25.71 23.28 19.91 16.61 14.15 12.97
(4) Sums he to hyy------__ 16.61 14.15 12.97 13.60 16.21 19.68
(5) Differences. -________- GxAS Ie SSimlos Onell olemn ye ON 23
1 2 3 4 5 6 |
Com-
bined | Angles| Cosines | Products (1X3) Sines | Products
: (1X5)
heights
6. 48 0 1 6. 48 0 0
11. 38 30° . 866 9. 855 ao 5. 690
1183, Bie) 60° a5) 6. 780 . 866 11. 748
12.11 90° 0 1 12.110
7.07 120° —.5 —3. 535 . 866 6. 123
573) 150° —.866 —.199 ae) eras
23.115 —3. 734 35. 781
—3. 734
12c2=19. 381 12s9=35. 781
c= 1.6151 s= 2.9818 o=

The expression for the M, component is therefore:

y=3.391 cos (m,t—61.6°)

period of the observations may be arranged

6.79 8.06 9.72
6.81 8.15 9.96

13.60 16.21 19.68

log s2 = 0. 47448
- log c2 = .20820
log tané = .26628
=61°.6
log 82 = .4744%
log sin ¢ = 9.94414
log Mz = .53034
Mz = 3.391
237. 27
oA =9. 886

Figure 26 shows the graph of the M, component and the plotted
heights (referred to mean sea level) of the resultant of all of the M
It is apparent that at this station the other

components as derived.

Lunar Hours.

FIGURE 26.—M:2 component and hourly values of M group, Sitka, Alaska.

components of the group are small and that the summation has
nearly effected the elimination of components outside of the M group.

It is of interest to note that in the computation as set forth, the
summation of lines (1) and (2) automatically elimmates the M,; and M;
components, and the subtraction of line (5) from line (4) eliminates the
constant height of mean sea level and the M, and M, components.

96. A reference to the list of components in paragraph 75 shows

53

that the complete analysis of the M group of components at a station
includes the separation of six components. Computation programs
for this purpose are shown on standard forms of the United States
Coast and Geodetic Survey. The analysis of the S group of com-
ponents includes the separation of four components; and of the K
group 2. All other components require but a single analysis.

97. Augmenting factors ——In the computations of the components
(except those of the S group) the tidal height at each component
hour is taken as the average of the observed heights at the nearest
mean solar hour (par. 83). When the computation is made from
stencils in the form ordinarily used (par. 88), these observed heights
are scattered quite uniformly over an interval extending from one-
half a component hour before to one-half a component hour after
the exact component hour.

It is graphically apparent from
figure 27 that on a sinusoidal curve
the average of these heights is
always somewhat less, numeri- |
cally, than the height at the |
middle of the period, and that a | at
small systematic error is intro- |
duced by using the average value. | | |
This error is readily corrected,
- since on a cosine curve the mid
height has a constant ratio to the mean height over an are of given
length. This ratio is called the augmenting factor.

98. To determine the augmenting factor for a component whose
equation is y= cos (at+a), let 7 be the length of the componen thour.
The average value of y, between the limits of t)—%r and t)-+ ris then:

FIGURE 27.

(1/r) ay tot A A cos (at+a)dt

=(A/ar)[sin (at) + kar+a)—sin (at)—%ar+a)]
=(A/ar)[sin (at)+ a+ Kar) —sin (atf)+a—ar)|
=2(A/ar) cos (atp+a) sin %ar (58)

The ratio of the mid value to the mean value is then:
A cos (at)+a)/(2A/ar) cos (at) +a) sin Yar=ar/2 sin Yar (59)
in which ar is an angle expressed in radians, whose equivalent, in
degrees, is war/180°. The expression for the augmenting factor is

therefore:
mwar/360° sin kar

54

For diurnal components, the length of the component hour, 7, is
15°/a, for semidiurnal components 30°/a, and so on. The values of
the augmenting factor are then:

Diurnal components, a4 = aan = 1.00286
Semidiurnal components, D4 at 15° = 1:01 152
Ms, 24 Sap 102br4
M,, ST gg = 1.04720
M,, si st ago He
Mg, om =1.20920

24 sin 60°

99. A review of the process by which the amplitude and initial
phase of each component are found (par. 94), shows that the appli-
cation of the augmenting factor to the hourly component heights.
(above mean sea level) will increase the amplitude in the same ratio,
but will not affect the initial phase. The augmenting factor is there-
fore applied directly to the computed amplitude. The application
ef this correction to the amplitude of the M, component at Sitka,
for example, gives a corrected value of 3.391 1.01152=3.430 feet.
Evidently, no augmenting factor should be appled to the S com-
ponents. The more complicated factors for computations from
secondary stencils are given in the Manual of Harmonic Analysis
of Tides.

100. Elimination.—The hourly component heights derived from
the process of averaging that has been described will contain the
residuals of components other than that sought. After a first deter-
mination has been made of the amplitudes and initial phases of the
several components, corrections may be computed from them to
eliminate from each the effect of the other components. The process
is explained in the Manual of Tides, but is not of sufficient general
interest to be here included.

101. Long period components.—The components listed in para-
graphs 75 and 76 include 2 having a fortnightly, 1 a monthly, 1 a
semiannual, and 1 an annual period. The first 3 of these are too small
to be of much importance, but periodic meteorological causes may
produce substantial annual and semiannual variations in the sea
level. Since a long period component does not change appreciably
during a calendar day, the daily averages of the observed tidal heights,
instead of the hourly heights, may be used for its determination, or
the daily sums may as well be used, the final result being divided by

15)

24. The computation of the amplitude and initial phase follows the
general method heretofore described for the diurnal, semidiurnal,
and short-period components. For the fortnightly and monthly
components, a component month replaces the component day. It is
divided into 24 parts, corresponding to the component hours. A pre-
pared tabulation designates the daily sums to be taken as the height
at each component ‘“‘hour.’”? These heights are then summed and
averaged, and the amplitude and phase of the component computed
therefrom. For the annual and semiannual components, the com-
ponent year similarly replaces the component day. Since the average
of the observed heights during a calendar day contains residuals of
the short-period components (other than the S components) the
amplitudes and phases of the long-period components are corrected
by the process of elimination heretofore referred to.

MEAN VALUES AND EPOCHS OF COMPONENTS

102. Mean values —The amplitude of each component of the actual
tide increases and decreases with the changing inclination of the
moon’s orbit to the plane of the earth’s Equator, and the amplitudes
computed from a particular set of observations must therefore be
corrected before they may be used at another period. The correction
is based on the logical assumption that the change in the actual
components is proportional to the change in the corresponding equilib-
rium components. For convenience, the amplitude of each com-
ponent of the actual tide is reduced to its mean value, which is obvi-
ously independent of the period from which it was derived.

103. Hpochs.—The computed initial phase of each component is
that at the beginning of the particular set of observations from which
the component was derived. The phase of a component of the actual
tide depends upon the accidental configuration of the sea bed, while
that of the corresponding component of the equilibrium tide is depend-
ent upon astronomical causes alone. Since both components have
the same speed, the difference in their phases, at any tidal station,
is constant at all times. This difference is called the epoch of the
component and is conventionally represented by the general symbol x
(kappa). It may readily be found by taking the difference between
the initial phase of the equilibrium component, at the zero hour of the
observations (as determined from astronomical data) and the initial
phase of the actual component, as determined from the observations.
The phase of the actual component at any other origin of time can
then be found by applying its epoch to the phase of the equilibrium
tide at that time. The epoch of a particular component is designated
by the symbol for its amplitude with a degree mark. Thus the epoch
of the M, component is designated as M,°.

56

104. Mathematical derivation of lunar equilibrium tide.—To derive
the formulae for reducing the components of the actual tide to their
mean values, for converting these mean values to the amplitudes
applicable at any given period, and for determining the epochs of
the components, it becomes necessary to develop the mathematical
expressions for each component of the equilibrium tide in terms of
astronomical constants. The expressions for the lunar and solar
equilibrium tides in terms of the hour angle and declination of the
moon and sun, derived in equation (24), afforded the means for
developing the characteristics of the tide, and for inferring therefrom
the speeds of most of the components. The much more elaborate
expression necessary to develop the coefficients and phase relations
of the components of the lunar and solar equilibrium tides will now
be developed in outline.

105. In figure 28, Nis the north pole of the earth’s axis on the celes-
tial sphere, UIS,M/,P, the
celestial Equator, UOS
the ecliptic (the path of
the sun), U the vernal
equinox and S the posi-
tion of the mean sun
at a given instant; JOM
the moon’s path (orbit),
I its intersection with
the Equator, O the
moon’s node, M the
position of the moon,
P the zenith of a tidal
station, NPP, its celes-
tial meridian, NMM, the hour circle of the moon, NSS, the hour
circle of the mean sun, and U, the foot of the great circle drawn
through the vernal equinox perpendicular to the moon’s orbit.

Then:

§ (theta), the are PM, is the zenith distance of the moon.
5 (delta), the arc M,M, the declination of the moon.
(lamda), the are P,P, the latitude of the station.

H, the angle PNM, the hour angle of the moon.

N, the are UO, the longitude of the moon’s node.

T, the angle M4,IM, the inclination of the moon’s orbit.

The symbols conventionally assigned to other arcs and angles,
and to pertinent astronomical constants are:

T, the are P,S;, the hour angle of the mean sun.

s+k, the arc U,M, the true longitude of the moon.

s, the mean longitude of the moon; i. e., the longitude which it
would have it if travelled at the average rate along its orbit.

FIGURE 28.

57

k, the correction to be added to the mean longitude of the
moon to give its true longitude.

h, the are US;, the mean longitude of the sun.

p, the mean longitude of lunar perigee, the arc measured from
U, to the position of lunar perigee, if the latter moved at its
mean rate.

£ (xi), the are U;J, the longitude, in the moon’s orbit, of the
Intersection.

vy (nu), the are UJ, the right ascension of the Intersection.

€=0.05490, the eccentricity of the moon’s orbit.

m=0.074804, the ratio of the mean motion of the sun to that
of the moon.

R, the true distance from the center of the earth to the center
of the moon at a given moment.

€=238,857 statute miles, the mean distance, earth to moon.

a=3,958.89 statute miles, the mean radius of the earth.

The values of e and m given are those for January 1, 1900, but
they change but little with the time.
106. The height of the lunar equilibrium tide is, from equation (16):

u=\(Ma?/ER?*)a(3 cos? 6—1) (16)

In which JM is the mass of the moon and £ the mass of the earth.
The ratio M/E has a value of 1/81.45.
As shown in equation (21):

cos 6=sin ) sin 6+ cos \ cos 6 cos H (21)

From the right spherical triangle JM4,M:

sin 6=sin J sin JM (60)

From the right spherical triangle 1/M/,P,

cos 6 cos H=cos P.M (61)

And from the spherical triangle 7P,M:

cos P,;M=cos IM cos JP;+sin IM sin IP, cos I (62)

From the figure:

IM=U,M— U,I=s+k—£ (63)

IP,\=US,+8,P,— UI=h+T—» (64)

58

107. By substituting, in equation (21), the expressions for sin 6, and
cos 6 cos H, derived from equations (60) to (64), an expression for
cos 6 may be derived in terms of s, k, h, T, £, and ».

The astronomical formula for the correction k (in radians) is:

k=2e sin (s—p)+5/4 e sin 2(s—p)+15/4 me sin (s—2h+p)
+11/8 m? sin 2 (s—h) (65)

and since k is small, its angle, in radians, may be substituted for its
sine.
The astronomical formula for 7/R, in equation (16) is

1/R=1/e+e cos (s—p)/c—e?) +e? cos 2(s—p)/cU—e’)
+15/8 me cos (s—2h+-p)/c(1—e?) +2? cos 2(s—h)/e(1—e’) (66)

108. The expression for the lunar equilibrium tide in terms of the
angles \, 7, s, h, £, and v, and astronomical constants, is then derived
by substituting these expressions for cos @ and 1/f in equation (16)
and successively converting the products of the sines and cosines of
the angles 7, s, h, &, and v, into sines and cosines of their sums and
differences, by the application of the elementary trigonometric for-
mulas:

cos x cos y= cos (x—y) + cos (x+y)

sin x sin y= cos (x—y)— 8 cos (x+y)

sin z cos y= sin (x+y)+% sin (xy)

cos z sin y= sin (x7+y)—% sin (e#—y) (67)
cos? z= (1+ cos 22)
sin? r= %(1—cos 22)

sin x cos c= sin 2x

109. The result is an equation for w which contains 63 terms, and
which would cover more than a printed page. It is not here repeated.
But 21 of the variable terms have coefficients of sufficient numerical
value to require consideration. These give the following working
equation for the lunar equilibrium tide, now designated as y:

V—=3)2) (Masi yay.

{cos? cos* KI [(4—5/4 e?) cos (27+ 2h—2s+2&—27) M,
4+7/4 e cos (2T+2h—3s-+p+2é—2y) Np
+1/4 e cos 27T+2h—s—p+2é—2y+180°) [L,]
+17/4 e cos (27T+2h—4s+2p+2&— 2p) 2N
+105/32 me cos (27+4h—3s—p- 2é—2yp) V2

4115/32 me cos (2T—s+p+2é—27+180°) ne

59

+23/16 m? cos (27+ 4h—4s+2£—27)| Le

+cos? \ sin? J [(1/4+3/8 e?) cos (27+2h—27) [K,]

+3/8 e cos (27+ 2h—s+ p—2y7)] [Lo]

+sin 2) sin I cos? J [(1/2—5/4 e?) cos (T-+h—2s+2&—7+90°) O,
+7/4 e cos (+h—3s+p+2é—7+ 90°) Q,

+1/4 e cos (T+h—s—p+2é—v—90°) [M,]

+17/4 e cos (T+h—4s+2p+2&—»+90°) 2Q
+105/32 me cos (T+3h—3s—p+2E—7+90°)] 01
+sin 2d sin J sin? I (1/2—5/4 e?) cos (T+h+2s—2t—vy—90°) OO

+sin 2 sin 27 [(1/4+3/8 e?) cos (T+h—v—90°) [KK]

+3/8 e cos (T+h+s—p—v—90°) Ji

+3/8 e cos (T+h—s+p—v—90°)] [M,]

+ (1/2—3/2 sin? \) sin? [(1/2—5/4 e?) cos (2s—2¢) Mf
+ (1/2—3/2 sin? \) (1—3/2 sin? J) [e cos (s—p) Mm
+m? cos (2s—2h)]}. [Msf]

(68)

110. Solar equilibrium tide—The corresponding equation for the
solar equilibrium tide may be written at once from equation (68) by
substituting:

S, the mass of the sun, for / the mass of the moon.
¢;, the mean distance of the sun, for c.

é,, the eccentricity of the sun’s orbit, for e.

w (omega), the obliquity of the ecliptic, for J.

pi, the longitude of the sun’s perigee, for p.

The angle s, the mean longitude of the moon, becomes identical
with h, the mean longitude of the sun. The angles ¢ and », the longi-
tude and right ascension of the intersection, become zero, as does m,
the relative motion of the moon and the sun; and e, is so small that a
number of terms dependent upon this constant may be dropped.
The equation of the solar equilibrium tide then becomes:

y=3/2 (Sa?/Ec3)a{cos? \ cost ’w [(1/2—5/4 e,) cos 27 Se

+7/4 e, cos (2T—h+ 9) ahs

+1/4 e, cos 27-+h—p,+180°)] Rs

+ cos? \ sin? w(1/4+3/8 e,7) cos (27+ 2h) [K,|
+ sin 2X sin cos? 4w(1/2—5/4 e,7) cos (T—h+90°) P,
+ sin 2X sin 2w(1/4+3/8 e) cos (T+h—90°) [Ky]
-- (1/2—3/2 sin? \) sin? w(1/2—5/4 e,) cos 2h}. Ssa
(69)

111. Tide depending on fourth power of moon’s parallaz.—This may
be derived by substituting in the second term of equation (20) an

192750—40——_5

60

expression for cos @ derived as explained in paragraph 107 and reducing,
by the general method pursued in determining the tides due to the third
power of the moon’s parallax. All of the resulting terms are very
small, the only one recognized being:

y=3/2 (Ma'/Ec*)a[5/12 cos*)d cos® 4I cos (3874+ 3h—38 +3£—3r)]. M3 (70)

112. Hquilibrium argument.—Each term of equation (68) contains
the general factor 3/2 (Ma*/Ec*)a, which has a constant value of 1.7527
feet; a factor composed of a function of the latitude of the tidal sta-
tion, which is constant at a given station; a factor composed of a
function of J, which changes very slowly, a constant factor containing
e and m, and the cosine of an angle formed by the algebraic sum of
simple multiples of the angles 7h, s, p, , andy. This angle is called
the equilibrium argument. The term in equation (70) is in the same
form but with a different general factor.

Similarly each term of equation (69) contains the general factor
3/2 (Sa*/Ec,*?)a, which has the constant value of 0.8091 feet; a factor
composed of a function of the latitude of the tidal station; a factor
composed of a function of w, which does not change; a constant factor
containing ¢, and the cosine of an equilibrium argument containing 7’,
h, and 7, only.

113. Since 7 is the hour angle of the mean sun at the tidal station,
it is zero at noon, mean local time, at the station, and increases at the
rate of 15° per mean solar hour.

The values of the angles h, s, p, and p,, the longitudes of the mean
sun, moon, and lunar and solar perigee, respectively, at the beginning
of each calendar year at Greenwich are given in Manuals on Harmonic
Analysis of Tides. Their rates of change remain practically constant
for a century of time, and are as follows:

Angular change in degrees per
mean solar hour
Angle
Symbol Value

4h 6 (theta) 145)

h n (eta) . 041, 068, 64.
8 go (sigma) . 549, 016, 53.
D w (omega) . 004, 641, 83.
pi w1 - 000, 001, 96.

That part of the equilibrium argument made up of the angles T, h,
s, p, and p; which change at a constant rate, together with any con-
stant term formed by the introduction of 90° or 180°, is convention-
ally represented by the symbol V. This part then has the form:

=mT+nh+ngs-+nsgp+nspi+n90° (71)

where 7;, 72, 73, etc., are small positive or negative integers, or zero.

61

114. The remaining part of the equilibrium argument is made up
of simple multiples of the angles » and é, the longitude and right
ascension of the intersection, represented by the arcs UJ and U,J,
figure 28. The are UO is the longitude of the moon’s node, conven-
tionally represented as N; the angle WOS=JOU is the constant angle
~ between the moon’s orbit and the ecliptic, and the angle JUO is the
constant angle w between the equator and the ecliptic. The value of
y in terms of N and these known angles may therefore be determined
by the solution of the spherical triangle JOU and the value of ¢ then
found from the right spherical triangle JUU,. As the moon’s node,
O, makes the circuit of the ecliptic in its period of 19 years, the ascend-
ing intersection, 7, moves to and fro over a comparatively small arc
on either side of the vernal equinox, U, the angle » increasing slowly
from 0 to 13°.02, then decreasing to —13°.02 and increasing again to
zero. The angle é similarly fluctuates between the limits of 11°.98
and —11°.98. The maximum change in these angles during a year is
about 5°. The slowly fluctuating part of the equilibrium argument
formed by these two angles is conventionally designated by the
symbol uw. The total equilibrium argument is then represented by
V+u.

The value of N at the beginning of each calendar year at Greenwich
is tabulated with those of h, s, p, and p,. Its rate of change is
—19°.326,19 per calendar year, or —0°.002,206,41 per mean solar
hour. Its value at any instant is therefore readily found. The
values of vy and é at that instant can then be found from a table giving
these angles for each degree of N.

115. Components of the equilibrium tide —If Vo is the value of V at
any given instant, taken as the origin of time, then at any time ¢ there-
after,

V=V,-+at,
in which a is a constant whose value is:
A=7N04+ Noy tnzgotmotnsar (72)
Each term of equations (68), (69), and (70) then has the form:
y=A cos (_V+u)=A cos [at+(Vo)+4u)] (73)

The form of this expression shows at once that each term represents
a component of the equilibrium tide. For lunar components the
values of A and wu change slowly with the longitudes of the moon’s
node, but may be considered as constant during a limited period of
time such as a month or even a year. For solar components, A is
constant and w is zero.

116. The numerical value of the speed of each component of the
equilibrium tide may be readily computed from the speeds of the

62

constituents of V given in paragraph 113. Thus the speed of the
component represented by the first term of equation (68), viz,

3/2(Ma?/Ec*)a cos? cos* ¥I(1/2—5/4 e?) cos (27 +2h—2s+2é+4+ 2p)
is
a=20+2n—2c0=30°+0°.082,137,28— 1°.098,035,06=28°.984,104,22

This is then the M, component of the equilibrium tide, its speed
being identical with that previously identified for that component
(par. 75). All of the other terms in equations (68), (69), and
(70) may be similarly identified as the equilibrium components corre-
sponding to the components listed in paragraphs 75 and 76. Their
conventional symbols, conforming to those previously listed, are shown
opposite each term. It will be noted that the semidiurnal components
are those whose arguments contain the term 27; the diurnal compo-
nents are those whose arguments contain the term 7, and the long-
period components are those in whose arguments 7’ does not enter.
The lunar and solar components designated as K, and K,, each have
the speed of 26+2n and 6+ 7, respectively. As previously pointed
out, these pairs each unite into a single component. Their symbols
are therefore enclosed in brackets to indicate that they are parts of a
combined component. Two other lunar components, L, and Mb,
appear twice in the list in brackets. The speed of the L, component
represented by the third term in equation (68) is 20+27—c—6=
29°.528,478,92 and that of the ninth term is 20+2y—o+6=
29°.537,762,58. The difference in these speedsis evidently 24 =0.00928366
and the synodic period of the two components (par. 90) is
15/0.009 283,66 days=1,720 days. They therefore cannot be separated
by analysing observations over a period of even a year, and conse-
quently are treated as a single component. The evaluation of the
coefficients of the two terms shows that the first is the larger, and its
speed is therefore assigned to both. The speed of the M, component
represented by the twelfth term is similarly 0+-y—o—6 while that of
the eighteenth term is 6+y—c+6. Since the difference in these
speeds is also 26 they also cannot be separated by a year’s observa-
tions. For convenience they are treated as a single component having
a speed of 14°.492,052,1 whose component hour is the same as that
of the principal lunar diurnal component M,. The speed of the lunar
fortnightly component MSf is exactly the same as that of a compound
tide whose speed is the difference of the speeds of the M, and S,
components, and this component is therefore also bracketed.

117. Determination of the epoch of a component of the actual tide.—
As shown in paragraph 103, the phase of a component of the actual

63

tide differs from that of the corresponding equilibrium component by
a fixed angle, which is designated as its epoch, and is conventionally
represented by the symbol x. If then the equation of the equilibrium
component is (equation (73)) :

y=A, cos [at+ (Vo+u)]
the equation of the component of the actual tide is:
y=A cos [at+ (Vo. +u—x)] (74)

Comparing this equation with the equation for a component of the
actual tide in the form given in equation (29):

y=A cos (at—f), (75)
it is evident that:
Qa U=K==6, (76)
whence:
K—Vo-U-6 (77)

The computation of ¢ from a series of tidal observations was shown
in paragraphs 94 and 95, the origin of time being taken at the beginning
of the series. To determine the value of «, the value of V, at the same
origin of time must be computed. Since w is regarded as constant
during the period of the observations, its value is taken as that at the
middle of the period.

118. Computation of V +u.—For simplicity, the hourly tidal
heights from which the components of the actual tide are computed
begin at 0 hour (midnight). The time is usually the standard time
at a time meridian, whose longitude, S, differs from the longitude, LZ,
of the tidal station. Taking longitude west of Greenwich as positive,
and east as negative, the Greenwich time of the beginning of the
observations is then the S$/15th hour of the initial date. The expres-
sion for V)+w for each component is in the form:

Vo tu=mTo+ noho+Ng8o+ NsPot Ns (P1)0-+-7690° +néiit+ngy, (78)

in which 7), ho, So, Po, and (p;)o are the values of the respective angles
at 0 hour on the initial date of the observations, and é,, and »; are the
values of £ and v at the middle of the period.

Since 7, the hour angle of the mean sun, is zero at noon, mean
local time of the tidal station, it is +180° at midnight (0 hour) mean
local time, and (S—L)+180° at midnight, standard time. For
dirunal components, n,;=1, and n,7)>=S—L+180°; for semidiurnal
components n,=2, and n,Ty=2(S—L) +360°=2(S—L).

64

As has been stated, the values of h, s, p, p1, and N, at 0 hour Green-
wich mean civil time on January 1 of each calendar year are given in
tables contained in manuals for the harmonic analysis of tides, to-
gether with the differences to be added successively to give these values
on the first day of each month, on each day of the month, and at
each hour of the day, Greenwich time. The values of ho, 89, 9, and
(p1)o are the values of h, s, p, and p;, at Greenwich time of 0 hour on
the initial date of the observations. The value of N, the longitude of
the moon’s node, is similarly taken off for the middle of the period of
observations, and from it the values of £, and v, taken from the table
showing the value of these angles for each degree of N. Entering these
values in equation (78), the value of V)+-u is immediately determined.
This value, added (algebraically) to the value of ¢ found from the
observations, gives the value of «. The value of « so derived is, it
may be observed, independent of the meridian on which the times of
the observations are based.

If the observations are made on local time, instead of standard
time, S=Z, and the angle S—LZ becomes zero.

119. Example——The computation in paragraph 95 shows that for
the M, component of Sitka, Alaska, long. 135°20’ W., for the 29-day
period beginning at 0 hour, mean local time, July 1, 1893, ¢=61° .6.
The Greenwich time of the beginning of the period is then 9.02h,
July 1. The equilibrium argument for the M; component is from
equation (68):

Vtu=2T+2h—2s+2&—2p.

Since the observations are on mean local time,27=0. The values
of h and s, at 9.02h, July 1, 1893, Greenwich time, found from the
tables, are:

hy=99°.64 SUS OS

and the value of N for 9.02A July 15, 1893, is 24°.17. The corre-
sponding tabular values of »; and &, are:

Tyseea ALS esa Oil,
Then

Vo tu=0+199°.28—256°.06+8°.02—8°.90= — 57°.66=302°.34
K=61°.6-+302°.3—360°=3°.9.

The value of « derived from observations for a year is 3°. (Table
V, par. 134.)

120. Greenwich epochs —The Greenwich epoch of a component at a
station is the difference between the phases of the equilibrium com-
ponent at Greenwich and of the actual component at the station.

65

At any given origin of Greenwich time, the initial phase of the equilib-
rium component at Greenwich is, since S and L are both zero:

Votu=m(+180°) + nho+ns8o4 NsPotNs(Ps)o+N90°+u (79)

while the initial phase of the actual tide at a station whose latitude
is J is:

Votu-—c=n(—L+ 180°) + Ngho+ N38 + NsPo+ Ns (Di )o+n90°-+u . (80)

The difference is:

_ It may be observed that 7, is the same as the subscript of the com-
ponent. The formula for Greenwich epoch is usually written:

Gales. 2 re)

In which G' is the Greenwich epoch, p the subscript of the component,
L the west longitude of the station and «x the local epoch.

The difference between the Greenwich epochs of a component of
the tide at any two stations is the constant difference between the
phases of the component at the two stations.

121. Hquilibrium arguments of overtides and compound tides.—The
equilibrium argument of an overtide is taken as the indicated mul-
tiple of that of the primary tide. The equilibrium arguments of com-
pound tides are the sums or differences of those of the components com-
pounded.

122. Expression for u of the Ky, Ke, Le, and M, components.—A
reference to equations (68) and (69) shows that the K, component is
the resultant of a lunar component whose
equilibrium argument is 7’+h—v—90° and a
solar component whose argument is T-+h—90°.

The relation of the resultant to the components
is graphically shown in figure 29, in which CP,
is the amplitude of the lunar component, CP;
the amplitude of the solar component, and CP;
the amplitude of the resultant. The angle
YCP, is T+h—90°—», the angle YCP, is
T+h—90°, and the angle P,CP, is v. Placing
the angle P;CP,=v’ the equilibrium argument
of the resultant is 7+A—90°—»’. If Ais the foot of the perpendicular
drawn from P; to CP, produced, then

FIGURE 29,

sin v’ = AP;/CP;— P,P, sin v [CP;=CP, sin v [CP; (83)
cos 7 — (CP, + P,A)/CP;= (CP,+ CP, cos v) (OP. (84)

66
whence
tan v’=CP, sin »/(OP2+ CP, cos vy) =sin y/(cos y+ CP,/OP,) (85)

The amplitude CP, is, from equation (69), after substituting the
numerical value of the general factor of this equation (par. 112):

0.8091 (1/4+3/8 e?,) sin 2w sin 2X,
and the amplitude CP, is, from equation (68):
1.7527 (1/4+3/8 e?) sin 27 sin 2d

The substitution of these values in equation (85) gives, after
applying the numerical values of e, e; and w

sin vy sin 27 (86) |
cos vy sin 2/+0.3357

py bane
The equilibrium argument for the K, component may similarly be
shown to be: V+-u=27-+2h—2’’, where

sin 2ysin? J (87)
cos 2y sin? [+0.0728

iy oa

Since »y and J are both functions N, the longitude of the moon’s
node, v’ and 2y’’ are also functions of N. The values of v’ and 27’’
for each degree of N are included in the tables showing the values of v
and & (par. 114).

The equilibrium arguments for the L, and M, components are taken
from special tables, contained in Manuals for the Harmonic Analysis
of Tides. These components are not important, and the derivation
and application of these tables is here omitted.

MEAN VALUES

123. Equilibrium components—Kach component of the lunar.
equilibrium tide developed in equation (68) is in the form:

y=ZJ cos (V+u) (88)

in which J is made up of factors formed by astronomical constants; a
factor formed of a trigonometric function of \, the latitude of the
tidal station, which is constant at any given station; and a factor
formed of a trigonometric function of J, the inclination of the moon’s
orbit to the Equator, which slowly changes with the longitude of the
moon’s node. If this last factor is represented by ¢ (J), then

J=C 91) (89)

67

In the first term of equation (78), for example:
C=3/2 (Ma3/Ec*)a(1/2—5/4 e?) cos? dX, and ¢(1)=cos* %4/.

The amplitude, J, of the equilibrium component fluctuates slowly
between fixed limits with the changing values of J. A rigid analysis,
which need not be here repeated, shows that the mean value of J
during the circuit of the moon’s node around the ecliptic, is

Jv=C [o(L)o X [eos u]>=CM (90)

where [¢(J)|, is the mean value of ¢ (J), [cos uw] is the mean value of
cos uw, and M is the numerical value of their product.
From equations (89) and (90)

Jo/J=M/9(L) (91)

124. Mean value of amplitudes of the actual components.—As a basic
assumption, the fluctuation of the amplitude of a component of the
actual tide with the changing values of J is proportional to the con-
current fluctuation of the corresponding equilibrium component
(par. 102). If then R is the value of the amplitude of a component
of the actual tide as determined from a particular set of observations,
H the mean value of the amplitude, and J the amplitude of the
corresponding component of the equilibrium tide when J has the value
prevailing during the period of the observations:

A/R=J)/J=M/¢() (92)

The factor M/¢(J) is conventionally designated as F. Its recipro-
cal, ¢(1)/M is designated as f. Hence

H=FR ee (93)

125. Expressions for F.—A reference to equation (68) shows that
the expressions for ¢ (J) and for uw in the terms representing the
various components are as follows:

Components (1) u
M,, Np, 2N, Vo, Noy Mey cos* 1 ) 2E—2p
O;, Q:, 2Q, pi, sin I cos? J, Qi—y
OO, sin J sin? \J, — (2é+ p)
Ji, sim 2) I, —y
Mf, sin? I, —2¢
Mn, 1—3/2 sin? J, 0

The mean values of these functions of J, and of the corresponding
expressions for cos u are found by deriving the expressions for J, &,
and » in terms of N from the spherical triangles OUJ and JUV, figure

68

28; transforming the functions of J, and the expressions of u above
listed into functions of N; and finding the mean value of these
expressions as N varies from 0 to 360°. This somewhat lengthy
derivation, which need not be here repeated, shows that the prod-
ucts of these mean values are as follows:

c M, numer-

Function [¢(Z) Jo[cos wo “ell walbre
Costaeie eo cos! 14 w cos! Yi__-____________ 0. 9154
sin J cos? 4I______- sin » cos? Y4w cost Wi__________ . 3800
sin J sin? 47______- sin w sin? lgw cos! Yi__________ . 0164
sin at Bi os 2 ee gin Pi) (I= BYP Stee 7) . 7214
Sl 2b Sai Sin2i@iCOSeto a ee . 1578
l= 3/2 Sinceeeeee (1—3/2 sin? w) (1—3/2 sin? 7) __ - 1532

The numerical values in the last column are found by substituting
the values of w=23°.452; and 7=—5°.145 in the expressions in the
second column. It may be noted that as the moon’s orbit tilts to
and fro, the median value of its inclination J to the Equator is the
inclination w of the ecliptic to the Equator. Since the values of
cos! 4 and of (1—3/2 sin? 7) are very close to unity, the mean values
differ but little from the value of the function when J~wo.

The expressions for the reduction factor F' are then:

For Mz, No, 2N, v2, Ae, and we, F=0.9154/cos* %

For O,, Q;, 2Q, and pi, F=0.3800/sin I cos? 4I

For OO, F=0.0164/sin I sin?

For J,, Hy 2 WAY Sime 2h

For Mf, HO. 1578/sme yh

For Mm, F=.0.7532/(=3/2, sina) (94)

126. The reduction factors for the lunisolar components K, and K,
are more lengthy functions of J, and those for the L, and M, compo-
nents are still more complicated. The derivation of these factors is
explained in full in Special Publication No. 98; United States Coast
and Geodetic Survey, and is not here described.

127. Application of reduction factors.—The logarithms of the reduc-
tion factors for the several lunar and lunisolar components, corre-
sponding to each tenth of a degree of J, are tabulated in Manuals on
the Harmonic Analysis of Tides, special tables being included from
which the factors for the L, and M, may be found. To find the mean
value of the amplitude of a component from the value determined .
from a particular set of observations, the value of N at the middle of
the period is taken off as described in paragraph 118, the corresponding
value of J taken from a table, and from it the logarithm of F ascer-
tained. Thus the amplitude of the M, component at Sitka, Alaska,
for the 29-day period beginning July 1, 1893, corrected by the aug-
menting factor, was found in paragraph 99 to be, R=3.430 feet.

69

The value of N at the middle of the period was found in paragraph
119 to be 24°.17. The corresponding tabular value of I is 28°.22;
and for this value the tabular value of log F' for the 14, component is
0.0148.

Then log R=0.5353
log F= .0148

lost 5501

H=3.549

The value of H derived from a year’s observation is 3.591 (table V,
par. 134).

Since solar components do not vary with J, no reduction factor is
to be applied to them.

128. Reduction factors for other components.—It has been seen that
for the M, component

(1) =cos* ¥I Uu=2—E—2p

The corresponding expressions for the M; component are, from
equation (80):
o(1)=cos® kl U=3&—3p

Since cos® 4J= (cost J) */?, it may be presumed that the reduction
factor for the M; component is

i Cok Vip) 2

and this relation is established by a detailed analysis.

Similarly the reduction factor for the lunar overtides are taken as
the squares, cubes, etc., of the fundamental tide. These factors are
then:

For M,, F=(f of M.)?; Me, F=(F of M.)’, and so on.

No reduction factors are to be applied to the solar overtides.

The factors for compound tides are taken as the products of the
factors of the components compounded, the factor for any solar
component entering into the compound tide being unity.

129. “Mean values of coefficients.’”—An examination of equations
(68) and (69) shows that the amplitude of each semidiurnal com-
ponent of the equilibrium tide is the product of a coefficient, whose
numerical value may be determined from astronomical data, times
cos?\; the amplitude of each diurnal component is a coefficient times
sin 2\; and the amplitude of each long period component a coefficient
times (1/2—3/2 sin?\). The mean values of these coefficients there-
fore show the relative magnitudes of the mean values of the ampli-
tudes of the semidiurnal, diurnal, and long-period equilibrium com-
ponents, respectively, at a given station. The “mean values of the

70

coefficients’ conventionally used, and shown in table IV at the end
of this paragraph, are the complete coefficients divided by 1.7527,
the numerical value of the general factor 3/2 (a?/Hc?)a, in equation
(68); but these afford an equally good measure of the relative magni-
tudes of the mean amplitudes of the equilibrium components in the
three classes. Thus the mean value of the coefficient of the M, com-
ponent is (1/2—5/4 e?)M, in which M has the numerical value of 0.9154
derived in paragraph 125. The mean value of the coefficient of the
S; component is G(1/2—5/4 e?,) cos? kw, in which G is the ratio of the
general factor in equation (69) to the general factor in equation (68),
this ratio being 0.46164. .

TasBLE 1V.—Mean value of coefficients

Semidiurnal Diurnal Long period
Moe 0. 4543 Ky 0. 2655

Se 2120 O; . 1886 Mf 0. 0763
Noe 0880 Py . 0880 Mm 0414
Ko 0576 Qi . 0365 Msf 0042
Lo 0126 M, . 0149 Ssa 0365
To 0124 Ji 0149

V2 0123 (exe) 0081
2N 0117 P1 0051

jt) 0074 2Q 0049

. 0018
do . 0018

INFERENCE OF AMPLITUDES AND EPOCHS

130. It has been found that the mean values of the amplitudes of
the semidiurnal components of the actual tides at any station are
generally proportional to the mean values of the amplitudes of their
corresponding equilibrium components, as are the amplitudes of the
diurnal components. The amplitudes of the minor components may
therefore be approximately determined from those of the larger com-
ponents of the same type by applying this proportion, without gomg
through the laborious process of determining them by harmonic
analysis. The ratio of the amplitudes of the equilibrium compo-
nents is given by the ratio of the ‘‘mean values of the coefficients”
listed in table IV. It has also been found that the difference in the
epochs of components of the same type is proportional to the differ-
ence in their speeds. Thus if x, «2, x; are the epochs of three com-
ponents, and a’, a’’, and a’’ their speeds

(k3—k1) | (kg) = (@’’’ —a") /(@” —’)
whence
kg=K1+ (ky—K1) (a —’)/(a" —a’) (95)

If then the epochs x; and x, have been determined by analysis, the
epach x; can be determined by inference.

131. In the harmonic analysis of small components, accidental
variations in the tidal heights may conceal, to a relatively large meas-

fal

ure, the systematic variation sought for, and the amplitudes and
epochs derived by inference may be preferable to those determined
by direct analysis, particularly if the period of observations is short.
A considerable number of these small components are customarily
determined by inference.

SUMMARY OF THE METHOD OF HARMONIC ANALYSIS

132. The harmonic analysis of the tide at a station comprises:

(a) Some six or more separate summations of the observed hourly
tidal heights for a period generally of 369 days to obtain the hourly
component heights of the S, M, and K group of components, and the
larger individual components (pars. 78 to 89).

(b) The computation of V, for each component at the initial hour
of the observations and of wu at the middle of the period (par. 118).

(c) The preliminary determination of the epochs, x, and of the
amplitudes, R (corrected by the augmenting factor) of the compo-
nents of each group, and the larger individual components, from their
computed hourly component tidal heights (pars. 91-99, and 118),
and the preliminary inference of the remainder for use in elimination
(par. 130).

(d) The elimination of the effect of one component on another
(par. 100).

(e) The reduction of the corrected amplitudes to their mean values,
HT (par. 127) and the final inference of the constants of the compo-
nents not analyzed.

133. Standard forms to systematize these computations, and tables
giving the requisite data are published in the Manual of Harmonic
Analysis and Prediction of Tides of the United States Coast and Geo-
detic Survey. The labor entailed in the analysis of the tide at a
station is apparent. The dependability of the results of a tidal
analysis is illustrated by a comparison between the separate deter-
minations of the harmonic constants at Fort Hamilton, New York
Harbor, for three periods of 369 days beginning January 1, 1900, 1904,
and 1928, respectively. Omitting the constants derived by inference,
the determinations are as follows:

Amplitude H, in feet Epoch x, in degrees
Com- Pid au
ponent |
1900 1904 1928 1900 1904 1928
M2 25212 2. 208 2. 256 PPC es 220°. 7 220°. 7
No 459 . 496 473 203°. 9 204°. 5 202°. 1
S2 440 . 450 461 248°: 8 247°.0 248°. 6
Ki 320 . 324 316 103°. 1 104°. 0 102255
O; 178 . 167 171 98°. 1 98°. 9 100°. 5
P; 102 . 095 096 102207 109°. 1 108°. 8
Ko 148 . 132 120 244°. 1 235°. 8 2o2ae
Mi 008 . 007 012 86°.8 123°.0 123°. 0
M! 028 . 030 055 333°. 4 345°. 3 S13e56
Ms 051 053 063 35°.8 34°.9 35°. 8
S1 044 036 050 68°. 5 58°. 6 60°. 1
Sy 035 042 030 75°.8 63°.9 59°.8

72

134. Principal tidal components at representative stations—The
harmonic components of the tide have been determined at a large
number of tidal stations throughout the world. The amplitudes, in
feet, and the epochs, in degrees, of the five principal components at
stations used in the ensuing chapters to illustrate the characteristics
of the tide and the determination of tidal datums, are abstracted as
follows from the extensive data given in Special Publication, No. 98,
United States Coast and Geodetic Survey (1924).

TasBLe V.—Principal tidal components

Station Mz: M,.° Se S2° Ne N2®° Ky Keo O; 0;°
|

iMastport, Maine: == === 2 8. 576 326 1. 399 6 25) 298 0. 480 129 0. 377 ill
Pulpit Harbor, Maine_-_-__-__- 4.899 | 320 777 | 355 | 1.049 | 288 -457 | 129 . 365 108
IPo;ilandaeViainea==- = 4.372 324 . 699 0 . 949 292 - 462 132 2000 111
BOSOM IVIASS = ase aes 4. 371 330 . 699 5 . 995 300 - 449 134 . 348 117
Fort Hamilton, New York

Lar hOnecs seer See wee ane as 2. 210 221 «445 248 - 478 204 . 322 104 iP 98
Hemandina, Was =) == 2. 854 228 - 509 258 - 585 213 - 345 127 a 7ay? 129
Galveston auexe eae ee . 308 108 . 094 111 . 074 91 . 386 315 . 358 309
MristobalG@.. Zee seree ee ee . 268 6 .044 | 197 -087 | 329 cen ||. Gil . 196 160
Balboa, C. Li Bin PINE) ye Za eat 6. 000 89 1.616 146 1. 260 58 - 443 343 . 128 355
Presidio, San NREIMGISIED) Califa|) La 330 -404 | 335 .376 | 304 1.208 | 106 . 756 89
Seattle; Wash. --2=-222-2==2 3.494 128 . 846 154 - 686 97 2. 697 156 1. 502 133
Ketchikan, ALASK AS oe aoe 6. 138 if 2.014 28 1. 241 342 1. 648 129 1.014 114
Sitka Alaskan 5: = 22s eee 3. 591 3 1.145 34 . 758 335 1. 504 125 - 905 110
Sheerness, England_____---__| 6.297 1 1. 750 56 1.046 337 .377 14 - 451 193
Do Son, Indochina__--------- SIBaL |) 1g} .098 | 140 - 025 99 | 2.362 OT 252970 35

HARMONIC PREDICTION OF THE TIDES

135. The chief use made of the harmonic constants is in the predic-
tion of the heights and times of high and low waters at a tidal station.

The height of the tide at any time during a particular year is given
by the equation:

y=H,+fd: cos (t+ Vetu—di°) +fKy cos (t+ Vo+u—Kya°)+ -
+-fK2 cos (katt Votu—K,°)+ - - - (96)

in which H, is the height of the mean sea level above any standard
reference plane,

J,, K,, Ky, ete. are the mean values of the amplitudes of the
several tidal components,

f is the mean value during the year (taken as the value at the
middle of the year) of the factor to reduce the mean value of
the amplitude to its value for the year (par. 124),

t is the time in hours after the beginning of the year,

j1, ki, ks, ete., are the speeds of the several components,

V, is the value of V for the component at the beginning of the
year,

u is the value of wu for the component at the middle of the year,

J1°, Ky°, K2°, etc., are the epochs of the components.

73

The times of high water and of low water are those at which y is a
maximum or minimum respectively, and therefore at which dy/dt=0.
They are then given by the equation:
dy/dt= —jifJ, sin (j,¢-+ Vot+u—Ji°) —kif Ky sin (kit+ Vo +u—K,°)

ea Rte tar ave ee 0... (97)

136. Tide-predicting machine.—The arithmetic evaluation of y for
successive values of ¢ from equation (96) would be too laborious to
be practicable. The solution of equation (97) for the successive
values of ¢ at high and low water could be accomplished mathematically
only by an even more laborious process of successive approximations,
An elaborate machine, called the tide-predicting machine, has been
devised and constructed, by which the values of y and of dy/dt in
these equations are mechanically summed for values of ¢ measured
by the angular travel of the mechanism. The height of mean sea
level, Hy, the values, fH, of the amplitudes of the several components
for the year of prediction, and the values of the initial phases, V,+u—x«x
of the several components at the beginning of the year, are all set
on the dials of the machine. The machine is then put in motion.
When the pointer indicating the value of dy/dt (the sine summation)
crosses its zero mark, the machine is stopped, the height of the tide is
read off the dial which indicates the summation of y, and the time of
the tide is read off a dial which mdicates the time corresponding to
the angular travel of the mechanism. These are the height and time
of the first high or low water of the year. The machine is again set
in motion, the height and time of the next low or high water read off,
and the process continued until the predictions for the year are
completed.

137. Tide Tables published annually by the Coast and Geodetic
Survey give the predicted heights and times of the tides at some one
hundred reference stations throughout the world, with data showing
the corrections to be applied to give these heights and times at nu-
merous secondary stations.

138. Accuracy of tide predictions—The predicted times of high and
low water which are published in the tide tables, obviously must be
those which would occur without the accidental disturbances due to
winds and other meteorological causes. A comparison between the
actual and the predicted tides at Portland, Maine, and at Seattle,
Wash., in May and November 1919, shows that the maximum de-
parture in the time of the actual from the predicted high and low
waters was 24 minutes; but that the times were generally in much
closer agreement. The height of one of the tides at Portland differed
by 1.9 feet from the predicted height; and it was not unusual for the

74

tides at both stations to differ by more than half a foot from those
predicted; but most of the observed heights were within half a foot
of the predictions (The Tide, Marmer, p. 205, et seq.).

AM PM
June 25,1935 Tuly 7, 193s

FIGURE 30.—Recorded and predicted tides, San Francisco (predicted tides enclosed in circles).

The plot in figure 30 of the recorded tides and the predicted high
and low waters at San Francisco, Calif., on 2 days chosen at random,
indicates the correspondence ordinarily to be expected.

Cuapter III

CHARACTERISTICS OF THE TIDES

Paragraphs
Tr SPURS) DAR LAOS A a ek eS 139-140
Mercamponent, semidiurnal tides_=-:.---==+--------.----.-2.-24L.- 141-142
SAECOMPOMeMG SpRINembIdes — 9 588. See 2 ee eee ee eee 143
IANO AGO. 22 See esas ee Se ene ee ae ee eee 144
NeEIcoMponent= pemseannideS= 2 5.— 5 -=-5= s=—- SS e ee Skee ee see 145
Pella Q. S252. OAS Sees eee eS ee a ae reese 146
Compimed effect of S; and No components. —--_....-.----.-1----==-- 147-148
K, and O; components, tropic tides, and diurnal inequalities ___-_-_-_-___- 149-152
anne ea cc meee sete ee ae Fe ea Se ee Se ee 153
(SD UORIMELG SL 2c 5 See OS = EE es 155

139. Types of tides—The tides throughout the world are of three
general types, which are determined by the relative magnitude of the
semidiurnal part of the tide (as indicated by the amplitudes of the
principal semidiurnal components, My, S2, and N,) and of the diurnal
part of the tide (as indicated by the principal diurnal components,
Kk, and O,). These types are:

(a) The semidiurnal or semidaily tides —Tides of this type have two
nearly equal high waters and two nearly equal low waters each lunar
day of 24 hours 50 minutes. They occur when the amplitudes of the
diurnal components are small in comparison with those of the semi-
diurnal components. This type is found along both coasts of North
and South Atlantic Oceans, and at other places as well.

(b) Mixed tides —This type is characterized by two markedly un-
equal high waters, or two markedly unequal low waters, or both, on
each lunar day, during most of the month. Tides are of this type
when the amplitudes of the diurnal components are considerable in
comparison with those of the semidiurnal components, but do not
greatly exceed the latter. Such tides are common, but not universal,
along the coasts of the Pacific Ocean.

(c) Diurnal tides —Tides of this type have but one high water and
but one low water each day during a substantial part of or all of the
month. Such tides are common along the coasts of large enclosed
seas with restricted entrances, such as the Gulf of Mexico, the Carib-
bean, the waters of the East Indies, and the Mediterranean; and some-
times at oceanic islands. Diurnal tides are usually quite small and
irregular.

140. Tides of the semidiurnal type usually have some diurnal in-
equality during the two periods in each month when the moon is

192750—40——6 (75)

76

farthest from the Equator and the diurnal tidal impulses are conse-
quently a maximum (paragraph 40). In tides of the mixed type, the
two daily high waters and the two daily low waters become nearly
equal when the moon is near the Equator, and the diurnal tidal im-
pulses are a minimum. When the moon is at its maximum declina-
tion, near the celestial tropics, one of the two daily low waters or high
waters of tides of the mixed type occasionally may disappear, pro-
ducing a diurnal tide. On the other hand, tides of the diurnal type
usually break down into two daily tides during a part of the month,
although in the Gulf of Tongking in Indo-china, the tide remains
diurnal throughout the month.

Obviously the types of tide merge into each other. The accepted
criterion distinguishing the types is the ratio (K,+0O,)/(M:+S,),
derived from the harmonic components at the station. If this ratio
is less than 0.25, the tide is classed a semidirunal; if between 0.25 and
1.25 as mixed; and if over 1.25 the tide is classed as diurnal.

THE EFFECT OF THE PRINCIPAL SEMIDIURNAL COMPONENTS
ON THE TIDES

141. The M, component, semidiurnal tides—When the tide is of
the semidiurnal type, the M, component, with rare exceptions, is the
dominant one, with an amplitude nearly but not quite one-half of the
mean tidal range. Generally, its amplitude may be taken as 0.47
times the mean range.

142. Relation of epoch of M2 component to lunitidal intervals —-As has
been seen (paragraph 117), the expression for the M, component of the
actual tide may be written in the form:

y=M;, cos (m.t+ Vo tu—M.°) (98)
in which

Mm, is the speed of the component, and has the numerical value
of 28.984° per solar hour.

V,+w is the value of the equilibrium argument at any arbi-
trarily chosen origin of time.

M.° is the epoch of the component.

The expression for the M, equilibrium component is
y=M, cos (m2t+ Vo+4) (99)
At the high water of the actual tide
m¢+ V,tu—M.°=0

whence
t=[M.°— V,>—u]/m,z (100)

77

Similarly, at each high water of the equilibrium tide

The nature of the equilibrium tide is such that the high waters of
its M, component must occur at the moon’s transits across the meri-
dian of the tidal station. When the origin of time is taken at a lunar
transit, equation (101) shows that V,+~ must be zero. The time of
high water of the M, component of the actual tide is then, from equa-
tion (100), M.°/m, hours after a lunar transit.

The M, component is the dominant one when the tide is of the semi-
diurnal type, and largely determines the time of high water of the
entire tide. The other semidiurnal components, alternately advance
and retard the time of high water. The diurnal components and lunar
overtides may produce a systematic difference in the time of high
water. Denoting this systematic difference by At, the average inter-
val between a lunar transit and the time of high water at a station is
then (M;°/m,)+At. This average interval is the high-water interval at
the station (paragraph 8) and is designated as HWI. It follows
therefore that:

M,.°=m,(HWI— At?) (102)

The low water of the Mz component similarly occurs when:

m.t—M,.°= == 180°
or when
t= (M,.°+180°)/m, (103)

Since the diurnal components and lunar overtides retard (or
advance) the time of low water by the same amount that they advance
(or retard) the time of high water

M,°=m,(LWI-+ Af) F 180° (104)

where LW] is the low-water interval at the station.
Combining equations (102) and (104) to eliminate Af, and sub-
stituting for m, its numerical value:

M.°=14°.492 (HWI+LWI) + 90° (105)

The negative sign is applied to the last term when the HW1 is less
than the LWI.

For example, at Fort Hamilton, New York Harbor, the high-water
interval is 7.67 hours and the low-water interval is 1.64 hours. The
epoch of the M:; component, from formula (105), then is:

14°.492(7.67-+1.64) +90°=225°

Its values from harmonic analysis is 221°.

78

At Philadelphia, the high-water interval is 1.49 hours, and the
low-water interval 8.97 hours. The epoch of the Mz component
from formula (105) then is

14°.492(1.49-+-8.97) —90°=62°

Its value from harmonic analysis is 49°.

Formula (105) gives evidently only an approximate value of the
epoch of the M2 component.

143. The S. component—spring and neap tides.—At stations having
a tide of the semidiurnal type, the amplitude of the S, component is
generally from one-sixth to one-half of that of the M2 component.
Since the difference in the speeds of these two components is relatively
small, the resultant of the two fluctuates slowly from a maximum
of M.+S., when the generating radu of these components coincide,
to aminimum ot M,;—S, when they are 180° apart (par. 54), the period
of the fluctuation, from maximum to minimum, being the synodic
period of the two components, or 360°/(s;—m,:)=354.367 hours.
This period is one-half of the lunar synodic month, the average
interval from full moon to full moon.

Other tidal influences disregarded, the high and low waters occur-
ring nearest the time at which the resultant of the M, and S, com-
ponents is at a maximum are respectively higher and lower than at
other times, and the tidal range is the greatest. These are the spring
tides, and their range is the spring range (pars. 2 and 20). The tides
nearest the time at which the resultant is a minimum are similarly
the neap tides. The times at which the generating radii of the M;
and S, components are in coincidence, and their resultant a maximum,
may be called the time of spring tides, although this time is not
generally the exact time of either spring high water or spring low water.
Similarly, the time at which these generating radii are opposed may
be called the time of neap tides. Because of the effect of the other
components, the average spring range somewhat exceeds 2 (M2++S2)
and the average neap range somewhat exceeds 2 (M:—S)).

144. Phase age.—The interval between the instant of full or new
moon, and the time of spring tides is called the phase age. At the
instant of full or new moon, the S; and M, components of the equilib-
rium tides quite evidently are in conjunction, and the difference in
their phases is zero. Since the phases of the corresponding com-
ponents of the actual tides differ from those of the equilibrium com-
ponents by their respective epochs, S.° and M,°, the difference in the
phases of these components of the actual tides at the instant of full
or new moon is S.°—M.,°; and since the S, component gains on the
M; component at the rate of s;—m,° per hour, they are in conjunction

79

(S:°— M,°)/(ss—mz) hours after the time of full or new moon. There-
fore:
Phase age (in hours)

= (S,°—M,°)/(30° —28°.984) =—0.984 (S,°— M,°) (106)

It is easily shown that this expression gives also the time of neap tides
after the instants at which the moon is in quadrature.

For example, at Fort Hamilton, New York Harbor, S,°=248° and
M.°=221°. The phase age is therefore 0.984 (248—221) hours=26.5
hours. At this station therefore spring tides occur a little more than
one day after the moon is at full or change (new), and neap tides at
the same interval after the moon is at quadrature.

The phase age at tidal stations throughout the world ranges up
to 3 days. It rarely is negative.

145. The Nz component—perigean and apogean tides——The ampli-
tude of the Nz component generally is between one-sixth and one-
third of that of the M, component. At stations on the Atlantic
coast of the United States, the N. component usually has a larger
amplitude than the S, component, but at stations on the Atlantic
coast of Europe, and along the British Isles, the amplitude of the
Ns component is materially less than that of Sy.

It is evident from the preceding paragraphs that the resultant of
the M, and N; components fluctuates between a maximum of M,+Ne
and a minimum of M;—Nb, the period from maximum to maximum
being

360° (m.— nz) =360°/ (28.9841 — 28.4397) =360°/0.5444=661 hours.

This is the length of the lunar anomalistic month (par. 62).

The maximum amplitude of the resultant obviously is due to the
maximum attraction of the moon at perigee, and is called the perigean
tide. Its minimum amplitude results from the minimum attraction
of the moon at apogee, and is called the apogean tide. The average
perigean range of the entire tide slightly exceeds 2 (M.+N2) and the
average apogean range 2 (M.—N,).

146. Parallax age—The interval between lunar perigee and the
time of perigean tides is called the parallax age. Since the M» and
N. components of the equilibrium tides are in conjunction at lunar
perigee, the phases of these components of the actual tides then differ
by the difference in their epochs, M.°—N2°; and these components
of the actual tides come into conjunction (M2°—N2°)/(m:—n,) hours
later. The expression for the parallax age is then:

Parallax age (in hours)= (M.°— N,°)/0.5444=1.837 (M2°—N2°) (107)
As is readily shown, the parallax age gives also the interval between
lunar apogee and apogean tides.

80

For example, at Fort Hamilton, New York Harbor, M,°=221°
and N2°=204°. The parallax age is therefore

1.837 (221°—204°)=31 hours.

Perigean tides occur at this station, therefore, a little more than one
day after lunar perigee; and apogean tides a little more than one day
after lunar apogee.

The parallax age at stations throughout the world ranges up to
3 days. In some regions it has a negative value.

147. Combined effect of S, and Nz components.—Perigean and apogean
tides tend to obscure the spring and neap tides at stations at which

FIGURE 31.—Predicted high and low waters at Boston, Mass., January 1937.

the amplitude of the N. component exceeds that of the S, component.
A typical monthly variation of high and low water at such a station
is shown by the plot, in figure 31, of the predicted tides during January
1937, at Boston, Mass., where the amplitude of the N, component is
the larger. These tides may be contrasted with the predicted tides
at Sheerness, England, during the same month, shown in figure 32.

&

09° 9% 9000 26° 9%
Coon 05°° o.6€U°8 S oO Pio} 00 °°? fe) ap

1s fore) So
905 095 foray) oe

onne 004, . i l
00 ORONO elon 02,9 g 6 9 2

Day 1 = 10 1S 20 25 3t
FICURE 32.—Predicted high and low waters at Sheerness, England, January 1937.
At Sheerness the amplitude of the S, component considerably exceeds
that of the N. component. The figures illustrate quite strikingly

the reason why the terms “spring” and ‘‘neap’’ tides are commonly
used in England, but not in the United States.

81

The tidal datum plane at Boston is mean low water, and the times
of high and low water are on the standard time of the 75th meridian.
At Sheerness the tidal-datum plane is mean low water of spring
tides, and the time is Greenwich time.

The harmonic constants of the three principal semidiurnal com-
ponents at these stations are taken as follows from table V, paragraph
134. The last two columns show the phase and parallax ages, com-
puted from the epochs as indicated in equations (106) and (107).

Amplitudes Epochs Ages in hours
Station =

M; Sy N: | M.° | 8° | No | Phase | Paral:
TBO SHOTS os ne 4.371 0. 699 0.995 330° Be 300° 34.4 5b. 2
ISheGnmesseee ee Se Se re Pe ese 6.297 | 1.750] 1.046 il? 56° 337° 54.1 44.1

The Greenwich times of the moon’s phases, apogee and perigee, in
January 1937 were, from the Nautical Almanac:

Moon’s phases
= Apogee Perigee
Last quarter Change First quarter Full

Day Hour | Day Hour Day Hour Day Hour Day Hour | Day Hour
4 2:22 p.m.| 12 4:47p.m.] 19 2:02p.m. | 26 5:15 p.m. 6 &p.m. 22 3a.m.

——

The times of spring, neap, apogean and perigean tides at Sheerness.
are immediately determined from these astonomical data by adding
the tidal ages. For Boston, they are similarly determined after
correcting the times for the difference in longitude by subtracting 5
hours. The times of these tides then are:

Station Spring tide Neap tide Apozgean Perigean

Day Hour Day Hour Day Hour | Day Hour
14

SHECENICSS werner: fot Thy oe ee 11 p.m. (GQ Dios, |) sy alilmssens ye GAO wiljop sen
7).3ae oe 0 os PP) er Aire leg 2 Ole

ESOS OTe esa Ses ree ee oe at ee oe 13 10 p.m. Se Sipamle|) Soe ope miel 245 oie m7.
Pay “Nil yo), aoa PAL «VG, adele

These times are indicated in figures 32 and 33.

148. Exceptionally high and low waters are to be anticipated when
the perigean and spring tides nearly coincide. Since the next succeed-
ing apogean tide occurs one-half of an anomalistic month, or a little
less than 14 days later, and the next succeeding spring tide one-half
a synodic month, or a little more than 14 days later, it follows that
when the tides after say the new moon are especially large, those after
the next (or preceding) full moon are not.

82

As the length of the anomalistic month is approximately 27% days
while that of the synodic month is 29 days, lunar perigee gains 2
days a month on the moon’s phases. It follows therefore that perigean
tides will most nearly coincide with spring tides at intervals of 7
months. Similarly, at stations having tides of the semidiurnal type,
an exceptionally small tidal range is to be anticipated once during
the month at 7 month intervals, occuring half way between the
months at which exceptionally high and low waters occur.

EFFECT OF THE PRINCIPAL DIURNAL COMPONENTS

149. The K, and O, components—tropic tides.—Since the difference
between the speeds of these components is relatively small, they
combine to form a diurnal tidal fluctuation with an amplitude ranging
from a minimum of K,—O, to a maximum of K,+0O,;. The period
from maximum to maximum is:

360°/(k;—0,) =360°/(15.041,068,6 — 13.943,035,6)
= 360°/1.098033=327.859 hours.

This period is one-half of the tropical month (par. 62).

When the amplitide of the resultant of these two diurnal compo-
nents is a minimum, the tides are called equatorial tides, since the
moon is then near the Equator. When it is a maximum, the tides are
called tropic tides, since this maximum results from the maximum in-
equality of the two daily tidal impulses, and therefore occurs when
the moon has its greatest declination, near the celestial Tropics (par.
40).

150. Effect of the diurnal components on high and low waters.—Since
the diurnal part of the tide rises once and falls once daily, it has a zero
elevation (at mean sea level) at semidaily intervals approximating the
period of the semidiurnal components. If the epochs of the Ky, Oi,
and M, components are such that the resultant ordinate of the diurnal
components is nearly zero at the two daily low waters of the M, com-
ponent, the diurnal part of the tide evidently increases one of the two
daily high waters and decreases the other, producing a diurnal in-
equality of the high waters, as may perhaps be seen more clearly by
turning back to figure 13, page 22. Similarly, if these epochs
are such that the diurnal part of the tide is nearly zero at the two daily
high waters of the M, component, a diurnal inequality of the low waters
is produced. Obviously, both the high and the low waters usually
will show an inequality because of the diurnal components, but the
inequality of the high waters is not, in general, the same as that of the
low waters. As has been shown, these inequalities in the two daily
tides vary from a minimum at the time of equatorial tides to a maxi-
mum at the time of tropic tides.

83

A characteristic fluctuation of tides of the mixed type is exemplified
by the tide curve at San Francisco, Calif., at the time of a tropic tide,
June 29, 1935, shown in figure 33.

Hours.

FIGURE 33.—Tropic tide, San Francisco, June 29, 1935.

A is the lower low water (LLW), B the lower high water (LHW), Cthe

lagher low water (HLW), and D the higher high water (HHW).

151. As shown in paragraph 68, the mean speed of the lunar diurnal
part of the tide is m,, the speed of the lunar day. It is therefore
exactly one-half of the mean speed, mz, of the lunar semidurnal part.
Consequently, the resultant of the lunar diurnal components keeps in
general step, from month to month, with the resultant of the lunar
semidiurnal components. In most regions, the lunar components
are so much larger than the solar that they determine the general shape
of the daily tide curves. Usually, therefore, the higher and lower high
and low waters at a tidal station always follow one another in the same
sequence. If the higher high follows the lower low water, the lower
high must follow the higher low, and vice versa, so that the sequence
is established either as “HHW to LLW” or as “LLUW to HHW.”
At San Francisco, for example, the sequence is HHW to LLW. At
some stations, but exceptionally, the sequence changes during the
year. Such a condition is to be anticipated when the principal solar
diurnal component, P,, is relatively large.

152. Tropic and diurnal ranges, high- and low- water inequalities.—
The average difference, from month to month, in elevation between
the higher high and the lower low waters of tropic tides is called the
great tropic range, and the corresponding difference between the lower
high and the higher low waters of tropic tides is called the small tropic
range. ‘The difference in the average heights of all higher high waters
and the average heights of all of the lower low waters from day to day

84

for one or more tropical months is called the great diurnal range, or
the diurnal range. The corresponding difference between the average
heights of all of the lower high waters and the higher low waters is
called the small diurnal range. These are all called dechinational
ranges, since they depend on the declination of the moon.

The diurnal high water inequality (DHQ) is defined as the difference
between mean higher high water and mean high water. The diurnal
low water inequality (DLQ) is similarly the difference between mean
low water and mean lower low water. It is apparent that the great
diurnal range is equal to the mean range plus (DHQ+DLQ) and the
small diurnal range is equal to the mean range minus (DHQ+ DLQ).

153. Diurnal age.-—The diurnal age is the interval between the in-
stant at which the moon is at its maximum monthly declination, either
north or south of the Equator, and the time of tropic tides.

Since the K, and O,; components of the equilibrium tides are in con-
junction when the moon is at its maximum declination, the phases of
these components of the actual tides then differ by the difference of
their epochs and these components of the actual tides are in con-
junction (K,°—O,°)/(ki—o;) hours later. Therefore:

Diurnal age (in hours) = (K°;—O,°)/1.098=0.911(K,°—O,°) (108)

For example, at Fort Hamilton, New York Harbor, K,°=104°
and O,°=98°. The diurnal age at this station is therefore

0.911 (104-98) =5.5 hours.

The diurnal age at a station may amount to several days, and not in-
frequently is negative.

154. As was shown in paragraph 40, the amplitude of the semi-
diurnal part of the tide decreases as the declination of the moon
increases, while that of the diurnal part increases with the declination.
As a consequence the mean daily tidal range tends to decrease with
the declination, but this decrease is overshadowed by the increasing
range from lower low to higher high water. In tides of the semidiurnal
type, the diurnal components do not obscure, to any marked degree,
the spring and neap, perigean, and apogean variations due to the
S, and N. components. In tides of the mixed type the variations
in higher high and lower low waters, culminating twice a month in
the tropic tides, become the outstanding characteristic, and obscure,
more or less completely, the spring, neap, perigean, and apogean
tides. In tides of the diurnal type, the diurnal components completely
dominate the semidiurnal during a considerable part of the month.
The fluctuations of the diurnal tides are, however, frequently so small
that meteorological disturbances become their outstanding
characteristic.

85
EFFECT OF OVERTIDES

155. Since the periods of the lunar overtides are one-half, one-third,
and one-fourth of the period of the M, component, they unite with
the latter to produce a tidal curve which is distorted from a sinusoidal
curve, but is repeated without change in each successive period of that
component. The form of the curve resulting from the combination
of the M, and M, components at Philadelphia is shown in figure 34.
At this station the amplitude of the M, component is 2.367 feet, and of
the M, component 0.368 feet, their epochs being 49° and 7°, respectively.

+3

+2

-—3 6 2 1S 24

Luner Hours.

FIGURE 34.—Resultant of Mz and Ms components at Philadelphia.

The effect of the overtide in this case in increasing the interval from
high water to low water and in decreasing the interval from low water
to high water is apparent.

156. If a high water of the M, component nearly coincides with a
high water of the M, component, the next high water of the overtide
will nearly coincide with the low water of the primary component.
The overtide will therefore raise the elevation of both the high and
low waters with respect to mean sea level. Similarly if the epochs
are such that a low water of the overtide nearly coincides with the
high and low waters of the primary component, it will lower both the
high and low waters of the resultant tide with respect to mean sea
level. In the case illustrated in figure 34, the epochs differ by about
45° and the overtide has little effect in altering the relation of the
high and low waters of the resultant with respect to mean sea level.

CHAPTER IV

TIDAL DATUM PLANES

Paragraphs
ancipaletidal datums and ranges_...-.-..-. 222252222520. see leek 157-158
fee Cae Ne MerreeE es Mere Bee ee ee 159-163
Base Ileus. 2S 5 ee ee 164
Mowaandebiohuwater datums im general 99822 2-2 22 165-168
Mieauplonveam@entobewater <2. 922... 22222 eo ei See eee eke ek 169-180
Worrechon Tor lonsitude of moon's node=_--2=-_ 22-22 ==) 8 171-180
Mean low and high water of spring tides________________________=__ 181-185
Mean low and high water of neap, perigean, apogean, and tropic tides_ 186
Mean lower low and higher high waters.__...__._________________-- 187-192
Se EEG OLAN eStore Sane a ao te Ce Tee ee ee OBE 194
iypicaarelation between datums —.._____--5..2.2-20. 2212 eee le 195
Wetermimation of datums by comparison. 22-2222 22) aise ee 196-207
pisces COMMON aR LUNI Sree =, tReet i a ae ey ee OS ee 208-211
ITiGinll ClOSCINVARALOTNS SE © Ne AR ree res ete es ee ey ny 212-215

157. Principal tidal datums.—The mean height of sea level, and the
mean heights of low or high waters of various descriptions, afford the
datums to which the elevations of upland areas, and of the bottom of
the sea and of tidal waterways, ordinarily are referred. The datums
which need be especially considered, and the abbreviations by which
they are designated, are as follows:

Mean sea level, MSL.

Half tide level, HTL.

Mean low water and mean high water, MLW and MHW.

Mean lower low water and mean higher high water, LLW and
HHW.

Mean low and high water of spring tides. In England, these
datums are taken as mean low and high water of ordinary
spring tides, after rejecting any spring tides which differ
substantially from the usual, and are designated as LWOST
and HWOST respectively.

In some cases channel depths at foreign ports are referred to mean
high or low water of neap tides. Mean low and high waters of peri-
gean, apogean, and tropic tides are rarely if ever used as a reference.

158. Tidal ranges—The symbols conventionally assigned to the
tidal ranges determined by the datums listed in the preceding para-
eraph, are as follows:

Mean range, Mn=MHW—MLW.

Diurnal or great diurnal range, GG=HHW—LLW.

Spring range, Sg, mean low water to mean high water of spring

tides.
(87)

88

MEAN SEA LEVEL

159. Use.—This datum is the basic plane of reference and the zero of
the ordinates of the harmonic components of the tide. It is deter-
mined by averaging the observed hourly tidal heights, measured from
a fixed bench mark, over a sufficient period of time. Because of the
variation in the density of the waters of the oceans with changes in
their temperature and salinity; because of the variation in the mean
barometric pressure upon them; and because of the effect of winds,
evaporation, and precipitation; mean sea level at different tidal sta-
tions may not be on precisely the same geodetic level surface. Thus
mean sea level at Balboa, at the Pacific entrance to the Panama Canal,
as determined from observations extending over 25 years, is nearly 0.7
foot higher than at Cristobal at the Atlantic entrance. In general,
however, mean sea level at tidal stations which have a free connection
with the sea, when determined from observations extending over a
number of years, are so nearly on the same level surface that the
difference between the elevation of any point on land above mean
sea level at one station, as determined by a line of levels from that
station, and the elevation of the same point above mean sea level at
another station, is within the error inherent in long lines of levels.
Mean sea level is therefore the standard reference datum for land
elevations. At tidal stations on tidal rivers, or on land-locked bays
and sounds with restricted entrances, the mean tidal height may be
above mean sea level and is more correctly designated as mean river (or
bay) level.

160. Fluctuations in mean sea level—Small fortnightly, monthly,
and semiannual fluctuations of mean sea level result from the long
period harmonic components established by the attraction of the sun
“and moon (par. 71). These are, however, completely overshadowed
by the disturbances resulting from storm tides, and smaller systematic
meteorological disturbances.

161. Storm tides—Occasional violent fluctuations of the water
levels at a tidal station result from strong onshore or offshore winds.
When these are of hurricane velocity, the water may be raised many
feet. In the long run, storm disturbances raise (or lower) both the
high and the low waters by substantially the same amount, and may
be considered, therefore, as affecting primarily the heights of mean
sea level.

162. Systematic meteorological variations in mean sea level —lLesser
atmospheric disturbances produce a less apparent, but more con-
tinuous, variation in mean sea level. The seasonal variations in the
density of the water on the continental shelf and in the mean baro-
metric pressure over wide areas of the oceans, with concurrent varia-
tions in the prevailing winds, and perhaps other meteorological causes,

89

result in fairly regular and consistent seasonal variations in the
monthly mean sea level, even at stations not affected by the varying
inflow from large rivers. <A study of these variations at stations on
the coasts of the United States, contained in Special Publication, No.
135, United States Coast and Geodetic Survey, shows that at North
Atlantic ports mean sea level is quite consistently 0.2 feet or more
higher during the summer months than during the winter months.
The annual variation im mean sea level at the South Atlantic ports of
Charleston and Fernandina approximates a foot, the highest eleva-
tions occurring in the fall. On the Gulf coast, the annual variation is
about three fourths of a foot; and on the Pacific coast about half a
foot. A comparison between the monthly changes in mean sea level
at Portsmouth, N. H., and at Ketchikan, Alaska, and the monthly
mean barometric pressure in the two regions, shows a striking corre-
spondence in each case (Special Publications, No. 150 and 127, U.
S. Coast and Geodetic Survey). At Balboa, at the Pacific entrance
to the Panama Canal, an annual variation of about a foot in the eleva-
tion of the monthly mean sea level follows closely an annual variation
of about 15° F. in the monthly mean water temperature. These
seasonal variations are reflected in the values of the long-term com-
ponents Sa and Ssa derived from harmonic analysis.

163. Variations from year to year.—Because of varying occurrence
of storm tides from year to year, and the varying intensities of the
causes of the seasonal variations in mean sea level, the mean annual
sea level at a station varies from year to year. At stations on the
coasts of the United States, where long-term observations have been
made, these variations are, however, not often greater than 0.1 feet.
The variations from year to year are quite uniform at all stations on
the same sea coast.

Because of these small meteorological variations, mean sea level at
a tidal station cannot be expected to be identically the same during
any two periods, no matter how long these periods may be. A 9-year
average is accepted by the United States Coast and Geodetic Survey
as a primary determination which gives the elevation of mean sea
level with sufficient accuracy for all practical purposes of that
survey. Strictly speaking, however, it should be designated as the
mean sea level during the particular period from which it was derived

HALF TIDE LEVEL

164. This datum is the elevation midway between mean low water
and mean high water. Because of the distortion of the tide curves
by the diurnal components and the lunar and solar overtides (par.
156), half tide level generally does not coincide exactly with mean sea
level. On the Atlantic coast of the United States it usually is below

90

mean sea level, while on the Pacific coast, except in Alaska, it usually
is above. Thus at Fort Hamilton, New York Harbor, it is 0.05 feet
below mean sea level, at Philadelphia 0.16 feet below, and at San
Francisco, 0.06 feet above that datum. Quite obviously, half tide
level varies from month to month and from year to year by amounts
that closely approximate the variations in mean sea level. Half tide
level is rarely if ever used as a datum plane for land elevations and
soundings, but affords a convenient reference for the correction of
mean high and mean low waters.

LOW AND HIGH WATER DATUMS IN GENERAL

165. Since it clearly is desirable that the soundings on navigation
charts, and the designated depths of improved channels, show the
depths that generally can be counted on by navigators, they ordinarily
are referred to one of the low water tidal datums, and not to mean sea
level. Different low water datums are used for this purpose in dif-
ferent countries. The datums adopted in the United States are the
most definitely determinable, but are not as low as those generally
used in other countries. When comparing the channel depths in
foreign ports with those in this country, the respective datums must
be taken into consideration. Thus a channel 28 feet in depth at the
adopted datum in a Canadian port might be 30 feet or more in depth
if referred to the low water datum officially adopted in the United
States for the region in which the harbor lies.

High water datums, while not suitable for charting, establish the
tidal ranges, which are usually noted on charts to indicate the depths
available at high water. In regions where the range between spring
and neap tides is considerable, the elevation of neap high water is of
especial importance, since it indicates the least depths at high water
which can be counted on throughout the month.

166. Low and high water datums do not establish a level surface.—
Obviously, as any of the several low and high water datums may be at
a different height below or above mean sea level at different stations,
these datums do not establish the same level surface from station to
station, and are applicable only to the area in the vicinity of each
station. Thus mean low water at the head of the Bay of Fundy is
some 15 feet below the level of mean low water at the entrance to the
bay. The change in the elevation of each of these datums, from
station to station is, however, generally so gradual as to present no
practical complications.

167. Meteorological variations in high and low water datums.—The
variations in mean sea level from month to month, and from year to
year, produce nearly identical variations in the several low and high
water datums. They do not, however, produce any substantial

91

variation in the height of these datums with respect to mean sea
level or to half tide level. The variations peculiar to the low and
high water datums, and the corrections to be made therefor, may
therefore be determined by taking the height of these datums with
respect to mean sea level, or half tide level, during the period of the
observations. The corrected heights, subtracted from or added to
the elevation of mean sea level, give them the corrected elevations of _
the datums.

168. Changes caused by channel improvements—A major enlarge-
ment of a tidal waterway, by dredging for the improvement of its
navigability, may change materially the elevations of the low and
high water datums along it. A classic example is the effect of the
improvement of the Clyde in Scotland, which during the last century
was converted from a shallow stream, fordable at low tide, to a water-
way for deep-draft vessels. The enlargement lowered the low water
levels at Glasgow by more than 8 feet, raised the high water levels by
some 2 feet, and consequently lowered the midtide level by 3 feet.
In exceptional cases, the low water datum may even be raised by
channel enlargement. Extensive channel improvements may there-
fore require a revision of established tidal datum planes.

MEAN LOW AND MEAN HIGH WATER

169. Definition.—-Mean low water is, as its name implies, the
average height of ali low waters over a long period of time, and mean
high water is the average height of all high waters. Because of
variations in the heights of high and low waters between springs and
neaps in the half synodic month, between perigean and apogean tides
in the anomalistic month, and between tropic and equatorial tides
in the half tropic month, a determination of mean high or low water,
with respect to mean sea level, from observations extending over a
day or a week, might differ quite widely from the long time mean.
To eliminate these variations the tides must be averaged over a
period in which these variations go through almost, if not quite, their
entire range one or more times. For a determination of mean low
or mean high water, the shortest period suitable for this purpose is
29 days. This period, which is sometimes called a /unation, is so
close to the synodic month of 29% days as to practically eliminate the
spring and neap variations. It is sufficiently close to the anomalistic
month of 27 days, 13 hours, to nearly eliminate the perigean variation,
and to the tropical month of 27 days, 8 hours, to nearly eliminate the
declinational variation. Longer periods theoretically should be multi-
ples of 29 days. For convenience of computation it is more usual to
take successive 29-day periods, beginning say on the first of each

192750—40—_—7

92

month. If the observations extend over a year, no sensible error is
introduced by taking all of the low or high waters for the 365 or 366
days.

170. Use.—Mean low water datum is the most readily determined
of the several low water planes, and adequately serves as a plane of
reference for navigation charts and for the designation of channel
depths when the tidal range is moderate. It is the official reference
plane for navigation charts and for federally improved channels on
the Atlantic and Gulf coasts of the United States. Obviously low
water of the varying tides is as often as not below this datum. At
Eastport, Maine, where the mean tidal range is 18.2 feet, the normal
tide occasionally falls 3 feet or more below mean low water datum;
but at most of the other stations on the Atlantic coast of the United
States, where the tidal range is much less, such minus tides (except
those due to storms) do not often exceed 1 or 2 feet, and ordinarily
are less.

171. Correction for longitude of the moon’s node.—Because of the
variation in the amplitudes of the tidal components with the chang-
ing inclination of the moon’s orbit to the Equator (par. 102), the
several tidal ranges and high and low water datums go through a
small variation in a period of 19 years. The mean range derived
from observations extending over a month or a year may be reduced
to its true mean value by applying a reduction factor, conventionally
designated as F(Mn). The corrected mean low water datum is
then found by subtracting one-half of the corrected mean range from
half tide level; and the corrected mean high water datum by the
corresponding addition. ‘These corrections are called the corrections
for the longitude of the moon’s node, since this longitude determines
the inclination of the moon’s orbit.

172. The numerical values of the reduction factor /(Mn) are
derived by deducing an expression for the mean range, Mn, in terms
of the amplitudes of the tidal components, and applying to these
amplitudes the reduction factors derived in paragraphs 124-126,
determined by the inclination, J, of the moon’s orbit during the
period of the observations. To simplify the correction, the ampli-
tudes of the semidiurnal components are assumed to be proportional
to the mean values of the coefficients of their equilibrium components,
given in table IV, paragraph 129; and the amplitudes of the diurnal
components also proportional to the mean values of the correspond-
ing coefficients. The relation between the amplitudes of the diurnal
and semidiurnal components is established by the ratio, (K,+0,)/Mao,
of these actual components at the station, as determined by harmonic
analysis, or inferred from available data. The derivation of these
factors is explained at length in appendix II.

93

173. The accepted values of /(Mn) corresponding to successive
values of J, and of (K,+0,)/Mz ranging from 0 to 1, are shown in
table VI. This table is abstracted from table 14 of the Manual of
Tides by Harris, published in the Report of the United States Coast
and Geodetic Survey for 1894, part IT. .

TaBLeE VI. —Factor F(Mn) for correction of Mn for longitude of the moon’s node
I= TS7-51 192 ||) 2028!) BIS" B20" |e eage 4) 99457 || “one! |) ogo =| 979) || ogo | Role

Ki F019. _| 0.970 | 0.972 2 2 2
Some 00 --| .972 | 0.977 | 0.982 | 0.988 | 0.994 | 1.000 | 1.006 | 1.012 | 1.019 | 1.026 | 1.029
2 .2,-| .971 | .973 | .978| .982| 988 .994 | 1.000 | 1.006 | 1.012 | 1.019 | 1.026 | 1.029
4| .972| .974]| .979] .983| .988| .994 | 1.000 | 1.006 | 1.012 | 1.018 | 1.025 | 1.028
6.__| .974| .976 | .980] .984| .989 | .994 | 1.000 | 1.006 | 1.011 | 1.017 | 1.023 | 1.026
8.__| .977 | .979 | .982| .986] .990 | .995 | 1.000 | 1.005 | 1.010 | 1.015 | 1.020 | 1.022
1.0.--| .980 | .982| .985 | .989| .992| .996 | 1.000 | 1.004 | 1.008 | 1.013 | 1.017) 1.019

174. The values of J at the middle of each calendar year from 1890
to 1969 are shown in table VII.

TaBLE VII.—/nelination, I, of moon’s orbit at middle of year

Year 0 1 2 3 4 5 6 7 8 9
{SQ meee ee eS. PEI) | AOoal ||— 27eeh |) A | PRG) PN PYESE) | AGRO) || Baieh by 23°. 9
POG eee Se ee PRN PUP Ne ae 18°. 4 18°.4 | 19°.0 Zirers |) PALS GS |) “PRE, DE Be
ef Qin ee ee AEG || BE oi) eee | EEG | PS | aa | ieee eb |p ee) |) Reo) ions
Qe eee eee Se = 20°. 0 18°.8 18°.3 Wes 5S || Teed AOS |) 2ASa |) 2EoSe |) SARS Zick
19 5— ee Ses 28250) || 282.70) |) 28°75) |) 28°50 ZUBIN PAB PEGS || SORE |) OF50) 19QR5
Cys eee ee ASSIA G Loan TEES || UGE @ |) Palais || PREY AEG) Bie ey | Bye 28°.3
UGG Be Se es ASG |) Boe | PSoS |) AGsoy | Zao | PBER@ |) Pies) |) Pe 8} 19°.0 18°. 4
OG—— Pe 28 sees. 18°. 4 1G? 2 |) AUP |) 2 BRC as Gs || AR) 790) || SOG 28°. 6

175. Application of corrections —The correction to the observed
tidal range in any month or year is readily determined from tables
VI and VII if the harmonic constants M:, K,, and O, have been
determined for the station. If these constants are not available, the
ratio of (K,+0,)/M, at the nearest station for which they are deter-
mined may be used, if the tide is of a similar type. Otherwise the
ratio is taken as equal to 2 (DHQ+DLQ)/Mn, as explained in
appendix IT.

176. Example —At Fort Hamilton, New York Harbor, K,;—0.322
feet, O,=0.172 feet, and M,—2.210 feet. The value of (K,+0,)/Msg is
then 0.22. For comparison the great diurnal range is 5.29 feet and
the mean range is 4.73 feet. The value of 2(DHQ+DLQ)/Mn is then
0.24. The mean tidal range for 1902 was 4.79 feet. The value of J
for 1902 is, from table VII, 19°.2. The corresponding value of F(Mn)
for (K,+0,)/M.=0.2 is, from table VI, 0.974. The mean range cor-
_rected for the longitude of the moon’s node is then 4.79 < 0.973 =4.67;
which is 0.12 feet less than the observed range. Observed mean low
water for the year was 2.44 feet below mean sea level, and mean high
water 2.35 above mean sea level. Applying half the correction to
each, these elevations, corrected for the longitude of the moon’s node,
become respectively 2.38 feet and 2.29 feet.

94

177. The annual mean tidal ranges at Fort Hamilton, as observed,
and after correction for the longitude of the moon’s node, for the years
1893 to 1932 are plotted in figure 35 from the data given in Special
Publication, No. 111 of the United States Coast and Geodetic Survey
(1935 edition).

The variation in the observed annual ranges with the longitude of
the moon’s node, N, is apparent in the figure. The increase in the

Mean Range
Feet

Observed Range o Corrected +

FIGURE 35.—Observed and corrected annual mean tidal range Fort Hamilton, New York, Harbor.

corrected ranges between 1902 and 1912 coincides with the major
enlargement of the harbor entrance in the dredging of the Ambrose
Channel during this period.

178. Table VI is not extended to give the values of F(Mn) for values
of (K,+0,)/M, in excess of unity. When this ratio exceeds unity the
tides are decidedly of the mixed type, and the diurnal inequalities
become their important feature. A study included in Special Publica-
tion, No.115, United States Coast and Geodetic Survey, of the annual
mean tidal ranges at San Francisco, Calif., where the ratio (K,+0,)/M,
is 1.1,shows that any effects of the longitude of the moon’s node on the
annual mean tidal ranges during the 26-year period from 1898 to 1923
are completely overshadowed by accidental variations in the ranges,
and that the correction of the observed ranges for the longitude of
the moon’s node serves little purpose in reducing the observations to
better concordance.

179. The correction for the longitude of the moon’s node is applied
only in an independent determination of mean low and high water
datums at a station. Ordinarily these datums are determined by a
comparison of the high and low waters with those at an established
base station (par. 196 et seq.) at which the correction already has
been applied. The approximations inherent to this correction are
practically eliminated when the datums are determined from observa-
tions extending over nine years, the period in which the nioon’s node

95

retrogresses through substantially 180°. The corrections are of im-
portance in a close study of the effect of a channel improvement upon
the mean tidal ranges.

180. Precision of observations —Observations extending over 9 years
are considered by the United States Coast and Geodetic Survey to
afford a primary determination of the mean low and high water datums
at a tidal station. In general observations for a year, corrected for
the longitude of the moon’s node, determine the relation of these
datums to mean sea level, or half tide level, at the station, within
0.05 foot of the 9 year mean; and observations for a month within 0.1
foot (Special Publication 135, U. S. Coast and Geodetic Survey,
p 107).

MEAN LOW AND HIGH WATERS OF SPRING TIDES

181. Differing definitions —Spring low waters and high waters are
most accurately defined as the low waters and high waters nearest
the time of conjunction of the principal lunar and solar semidiurnal
components of the tide, M, and 8, (par. 143); but may be more loosely
taken as the lowest low waters and highest high waters occurring
semimonthly soon after new and full moon. At English ports and in
other regions where the lowest low waters and highest high waters
follow consistently the conjunction of these components, and the
tides run through a regular variation from springs to neaps, with small
diurnal differences, spring tides are readily identified in the recorded
low and high waters, and their means over a number of months afford
fairly definite datums. Because of the small number of spring tides
in a half year or a year, a single abnormal tide would have a relatively
large effect on the mean value. Thus a storm disturbance of 3 feet
would change the mean low water of spring tides during a 6-month
period by a quarter of a foot, while it would change the mean of all
low waters during the same period by less than one hundredth of a
foot. It is therefore the English practise to reject abnormal spring
tides from the computations, and to designate the datums as mean
low and high water of ordinary spring tides. These datums depend
to some extent, consequently, on the judgment of the computer.

182. In regions where the tides have a considerable variation from
apogee to perigee, or from equatorial to tropic tides, the spring tides
may not be as readily identified; and when the diurnal components of
the tide are large, the low and high waters occurring next before the
time of conjunction of the M, and S, components may differ widely
from those next following. In some countries the datums designated
as mean low and high waters of spring tides are in fact the means of
the lowest low and highest high waters which occur soon after the
successive full and new moons. Such determinations obviously are
somewhat haphazard, but afford a low water datum below which the

96

tide does not often fall. It might be better named the mean lower
low water of spring tides.

183. In the United States, mean low water of spring tides is taken
by the Coast and Geodetic Survey as the average of the two low waters
nearest the successive times of spring tides, these times being deter-
mined by adding the phase age (par. 144) to the hour of new or full
moon. The variations in the diurnal inequality at the times of spring
tide are thereby eliminated. With this procedure four tidal heights
per month enter the computation both of mean low water and of mean
high water of spring tides; but observations must extend over a long
period to afford a mean in which the other systematic and the acci-
dental variations in the tide are satisfactorily eliminated.

184. Approximate values—It is shown in appendix II that the
height of mean high water above half-tide level, and of mean low
water below half-tide level is equal to the amplitude of the M, com-
ponent increased by a relatively small correction due to the displace-
ment by the other components of the time of high and low water.
The S, component does not have, therefore, any large effect on the
elevation of mean high or low water or the mean tidal range. At the
time of spring tides, however, the S,; component is in conjunction with
the M, component, and the height of high water is increased, and of
low water decreased, by substantially its amplitude. It follows there-
fore that mean high water of spring tides, as the term is used in the
United States, is closely approximated by adding the value of S., as
computed by harmonic analysis, to the corrected elevation of mean
high water, as determined by observation; and mean low water of
spring tides by subtracting this value from the established mean low
water. In other words:

LWOST=MLW-—S, (109)
HWOST=MHW-+S, (110)
Se—Mn+28, (111)

For example, the elevation of mean low water below half-tide level
at Fort Hamilton, New York Harbor, is determined from observations
extending over a long period, to be 2.37 feet, and the value of S, at this
station is 0.44. The elevation of low water of spring tides, from equa-
tion (109) is then 2.81 below half-tide level. Its value computed
directly from observations extending over several years is 2.79. Sim-
ularly at the Presidio of San Francisco, the elevation of low water of
spring tides from equation (109) is 2.37 feet below half-tide level,
while its elevation from direct observation, is 2.36 feet. The corre-
spondence is, therefore, very close at these stations.

OE

If then the harmonic components at a station have been computed,
and a good determination made of the mean low water datum,
formula (109) generally affords a satisfactory determination of the
mean low water of spring tides. After a satisfactory determination of
mean low water of spring tides has been made at one station, that at
other stations in the vicinity may be derived by comparison (par. 202).

185. Use.——Mean low water of ordinary spring tides is the reference
plane for the British Admiralty charts and generally for works of
harbor improvement in the British Empire. It is used in some other
countries as well. In Canada, low water datum is taken as from 0.5 to
1.5 feet below the mean of the lowest low waters of spring tides. In
the United States, mean low water of spring tides is used as a datum
by the Coast and Geodetic Survey only on the Pacific coast of the
Panama Canal Zone, where the range from springs to neaps 1s marked
and regular.

MEAN HIGH AND LOW WATERS OF NEAP, PERIGHAN, APOGEAN AND
TROPIC TIDES

186. These datums are determined in the same manner as the high
and low waters of spring tides. Thus the mean high water of neap
tides is taken as the mean of the successive pairs of high waters nearest
the time of neap tides, and is approximately equal to MHW—S,, the
neap range being approximately equal to Mn—2S,. Mean high
water of perigean tide is similarly the mean of the successive pairs of
high waters nearest the time of perigean tide, as determined by adding
the parallax age to the time of lunar perigee. It is approximately
equal to MHW-+N,, while mean high water of apogean tide is approxi-
mately equal to MHW—N>. The lower low, higher high, higher low,
and lower high waters of tropic tides are the averages of the lower low,
higher high, higher low, and lower high waters at the time of tropic
tides as derived from the diurnal age. As has been stated, these
datums are rarely if ever used as reference planes for charts. The
elevations of mean high and low waters of neap tides, are however of
importance at stations having a marked and regular range from springs
to neaps, and especially at ports where navigation is on the tide.

MEAN LOWER LOW AND HIGHER HIGH WATERS

187. These planes are sometimes called declinational planes, since
the lower low and higher high waters vary with the declination of the
moon and sun. Mean lower low water is the average height of the
lower of the two daily low waters of tides of the semidiurnal and mixed
types. Since the lunar day is longer than the calendar day, occasion-

98

ally but one low water occurs (about noon) during the calendar day
even when the tide is wholly semidiurnal. It is included in, or ex-
cluded from, the summation according to its relation to the preceding
low water. If two low waters of the same height occur on a calendar
day, but one is included. When, however, the tide becomes tempo-
rarily dirunal, each low water is included in the summation. Mean
higher high water is similarly computed.

188. Use——wWhere the tides are of the mixed type, mean lower low
water affords a more suitable reference plane than mean low water,
and is the official reference plane for navigation charts and channel
improvements on the Pacific coast of the United States. While this
datum is below mean low water (by as much as 1.8 feet at Seattle) yet
one of the two daily tides is as like as not to fall below it, sometimes
considerably. Thus at Seattle normal tides occasionally fall as much
as 3 feet below mean lower low water.

At localities having a tide which is wholly diurnal, mean low water
and mean lower low water become synonomous. On the Gulf of
Mexico, where the tides are generally of the diurnal type, but small and
irregular, mean low water affords a more satisfactory reference plane
than mean lower low water, and is the officially adopted plane in the
United States.

189. Corrections to short term determinations—An independent
determination of mean lower low or higher high water at a station,
like that of mean low or high water, must extend over a minimum
period of 29 days to eliminate the monthly variations in the tidal
range. Furthermore, the elevations of mean lower low and higher
high waters vary with the changing declination of the sun from month
to month during the year, as well as varying with the changing incli-
nation of the moon’s orbit during a period of 19 years. The correc-
tions to reduce to their true mean values, determinations based on
observations during a month or a year, are derived by applying a
reduction factor, conventionally designated 1.02 F;, to the diurnal
low and high water inequalities, DLQ and DHQ (par. 152). The
corrected diurnal low water inequality is then subtracted from the
corrected mean low water datum, derived as explained in paragraphs
171 to 177; and the corrected diurnal high water inequality added to
the corrected mean high water datum.

190. The derivation of the reduction factors, 1.02 F,, is explained
in appendix IJ. The computed values for each month of the year
from 1891 to 1950, are given in table 7, Special Publication No. 135,
United States Coast and Geodetic Survey (Tidal Datum Planes),
pages 114-115. The values from 1921 to 1950 are extracted there-
from in the following table:

99

TasLe VIII.—Factors 1.02 F,, for correcting diurnal inequality to mean value

Year Jan. | Feb. | Mar.| Apr. | May | June | July | Aug. | Sept.} Oct. | Nov.| Dec. | Mean
1.18 | 1.42 | 1.31 | 1.06 | 0.95 | 0.99 | 1.21 | 1.46 | 1.32 | 1.07 | 0.96 1.160
1.22 | 1.46 | 1.34 | 1.08 SOG AO0n eo2a e427 | e330) 107 . 96 1.177
1.22 | 1.46 | 1.34 | 1.08 SOG SOOM te 20n | teAGa ele ole 06 .95 1.171
12200} 1242) 15300) 1-'05 . 94 OSs ee | 39) | e267 02 .92 1.138
TRE aR) |p alloy U8) . 90 OE Maal) tei | i ike) 98 . 88 1. 084
ess |) GER |) th ity . 96 . 87 89) || 1506) 12:23) )) Tas 93 . 85 1.029
VS(ORS |) WG wh |) ale aN) 91 . 83 86 | 1.00 | 1.16 | 1.07 89 . 81 . 978

AE) |) aA Tt al e(0 88 - 80 83 296 he LOE L022 . 86 .79 . 938
.95 | 1.08 | 1.01 85 78 80 OSM ON .99 . 84 5 el . 908
925) 1805 .99 . 83 . 76 79 SO aEO# .97 . 82 . 76 . 887
.91 | 1.03 97 82 . 76 78 .91 | 1.04 . 96 . 82 . 76 . 879
nhl |) si (083 .97 82 . 76 78 .91 | 1.04 . 96 . 82 . 76 . 879
.91 | 1.04 . 98 . 83 ahs) 79 .92 | 1.05 .98 . 83 = (i . 888
OD) LOOM 00 84 .78 80 .94 | 1.08 | 1.01 85 . 78 . 905
om melee! OMe Os . 87 . 80 83 OTe leew ieueOo 88 .81 . 937
(0) |) we wes) ss 90 . 83 86 | 1.01 | 1.19 | 1.10 92 . 84 .978
1.04 | 1.22 | 1.14 94 . 86 90 | 1.07 | 1.26 | 1.16 96 . 88 1.026
ILS) 1] ale eka) |] ae Pal 99 . 90 OAT soa lel 4s leone Ol 392 1. 082
aby Wy WEB Wey leg! . 94 ORE WTS ei) 1425) 12:29) 05 . 95 1.136
WGA || WZ!) TIER BY |) ers AOD | We 1) 1E GA) |) eet eS Rs Ue 07 . 96 eA.
1,22) | 1.46) 1.34) 1.07 58 OD |) eae zy tae BBY! IL yy . 96 1.176
UL PAL |) Woetsy || ilbseR) |) Ue @y/ . 96 99 | 1.20 | 1.43 | 1.30 | 1.04 94 1.161
TSTSR le 39) | ae 28e EOS - 93 96 | 1.15 | 1.36 | 1.24 | 1.00 .9L 1.118
Usa} ales || ileal 99 . 89 OPN LOOM Pe 2Ont Leal? . 96 . 87 1.065
LS OG ele le 94 . 85 88 | 1.04 | 1.20 | 1.10 .91 . 83 1. 007
TE Con |) We als}. |) SIs 90 . 82 84 OO) elena pelOb . 88 . 80 . 962
alii |) Usa |) aOR} 87 79 82 95 | 1.09 | 1.01 . 85 . 78 925
.94 | 1.07 | 1.00 84 his 80 93 | 1.06 98 . 83 = Ns) 899
-91 | 1.04 . 98 . 83 76 78 91 | 1.04 96 . 82 76 882
.90 | 1.03 97 . 82 76 78 91 | 1.04 96 . 82 76 878

The mean annual values of the correction, given in the last column,
are the reduction factors to be applied the diurnal inequalities derived
from observations extending through the year.

The approximations introduced by these corrections are practically
eliminated in a determination of these datums from observations
extending over 9 years.

191. Example.—The application of the reduction factors to obtain
the corrected mean lower low and mean higher high waters is ilus-
trated by the determination of these corrections to the observed
annual mean tidal heights above an arbitrary datum plane at Ketch-
ikan, Alaska, in 1922, given in Special Publication, No. 127, United
States Coast and Geodetic Survey (Tides and Currents in Southeast
Alaska).

The observed heights are:

Mean higher high water (HHW) =21.55.

Mean high water (MHW)=20.75.

Mean lower low water (LLW)=6.06.

Mean low water (MLW) =7.43.

Annual half tide level (HTL)=% (20.75-+-7.43)=14.09.
Mean high water above HTL=6.66.

Mean low water below HTL=6.66.

Mean range (Mn)=20.75—7.43=13.32.

Annual high water inequality (DHQ)=21.55—20.75=0.80.
Annual low water inequality (DLQ)=7.43—6.06=1.37.

100

From table V, paragraph 134:
K,=1.648 O,=1.014 M,.=6.138
Whence (K,-+O,)/M2=0.43.
From table VII, paragraph 174, 7=18°.3.
From table VI, paragraph 173, #(Mn)=0.971.
Corrected Mn=13.32 * 0.971=12.93. ,
Correction to MHW and MLW=4%(13.32—12.93)=0.20.
Corrected MHW above HTL=6.66—0.20=6.46.
Corrected MLW below HTL=6.46.
1.02 F, rom table VIII)=1.177.
Corrected DHQ=0.80 < 1.177=0.94.
Corrected DLQ=1.38 X1.177=1.62.
Corrected HHW above HTL=6.46+0.94=7.40.
Corrected LLW below HTL=—6.46+1.61=8.08.
Corrected HHW on staff=14.09+7.40=21.49.
Corrected LLW=14.09—8.08=6.01.

It may be noted that in this case the corrections to DHQ and
DLQ nearly counterbalance the corrections to Mn. The correction
factor 1.02 F, to the mean annual diurnal inequalities decreases with
I, while the correction factor F(Mn) to the mean range increases
with that angle. <A glance at table VIII shows, however, that the
plane of lower low water goes through marked variations from month
to month.

192. Precision of determinations.—As with the other datum planes,
a determination of mean lower low or higher high water from corrected
observations extending over a period of 9 years is considered by the
Coast and Geodetic Survey as a primary determination. In general,
observations for a year, similarly corrected, determine the relation of
these datums to mean sea level, or half tide level, within 0.1 foot of
the 9 year determination; and observations over a month with a
quarter of a foot. At least 3 days observations should be used to
determine this datum within a foot of the long term value. (Special
Publication 135, U. S. Coast and Geodetic Survey, p. 124.)

OTHER DATUM PLANES

193. Harmonic tide plane.—A tidal plane often referred to, and used
at some ports in India, is that at an elevation of M.+S,+K,+0,
below mean sea level. It nearly coincides with what might be called
tropic lower low water of spring tides. It has the advantage of being
so low that normal tides rarely fall below it.

194. Arbitrary datum planes.—As will later be shown, the tidal
datums herein before listed, after being determined from a more or
less extended set of observations, are referred to standard bench marks
which thereafter become the controlling reference for charts, tide
tables, and channel depths. In some countries local datum planes,

101

established by reference to such a bench mark, are arbitrarily adopted
for these purposes, without particular relation to one of the character-
istic tidal planes.

TYPICAL RELATIONS BETWEEN DATUM PLANES

195. The relations between these planes at Fort Hamilton, New
York Harbor, where the tide is of the semidiurnal type, and at the
Presidio, San Francisco Harbor, where the tide is of the mixed type,

are as follows:
Elevations below mean sea level (feet)

New York San Francisco
(1912-30) (1898-1923)

INeanplOowawateree ne. . secre won LEP eet ee ees ORD ils SY7/
Mieanslower low waters ===) 222 ese Ses 2. 64 3. 02
Mowawater of spring tidesss i= eo be he se 2. 88 2. 26
Fanmoniestideplane ost is 21 Ys es yt” a0) 4.14

DETERMINATION OF TIDAL DATUMS BY COMPARISON

196. Because of the variation from day to day, from month to
month, and from year to year in the elevation of mean sea level, and
the periodic variations in the height of the successive high and low
waters with respect to mean sea level, long-continued observations are
_ necessary to establish, with good precision, the several tidal datums
at a station; but after these datums have been established at one
primary or base station they may be determined at other stations in
the same region, where the tidal variations are due to like causes, by
comparing, during a relatively short period, the high and low water
elevations at the secondary station with those at the base station.
This method is applicable only when the tides at the base and second-
ary stations are similar; 1. e. when the ratio 2 (DHQ+DLQ)/Mn at
the two stations is substantially the same, and the higher high and
lower low waters are in the same sequence (par. 151). Such condi-
tions are to be anticipated at stations on the same general embayment
of the coast line, and with free connections with the sea. They may
be fulfilled at stations several hundred or even a thousand miles apart.
The method of comparison is not applicable to stations on tidal rivers
and estuaries in which the water levels are sensibly affected by the
inflow from large rivers.

197. Establishment of half-tide level by comparison.—While the violent
disturbances produced by storms may vary considerably even at
stations in the same bay, the ordinary fluctuations of mean sea level,
and of half-tide level, when averaged over a sufficient period of days,
generally affect the elevations of these datums by substantially the
same amount over quite extensive areas. ‘To determine the half-tide
level at a secondary station from the established datum at a base
station, concurrent observations are therefore made for a suitable

102

period of the heights of high and low waters above arbitrarily selected
zero elevations at the two stations. These heights are called the
respective high and low waters on the staff. The mean high and low
waters, and the half-tide levels on the staff at the two stations during
the period of observation are computed. The difference between the
established half-tide level and the observed half-tide level at the pri-
mary or base station gives the correction to be applied to the observed
half-tide level at the secondary station.

198. Mean sea level by comparison.—lf the base and secondary sta-
tions both have a free connection with the sea or are freely connected
with each other by deep water, so that they both may be presumed to
have the same overtides, the difference between mean sea level and
half-tide level should be the same at both. This difference, as deter-
mined at the base station, applied to the corrected half-tide level at
the secondary station, gives mean-tide level at the secondary station.

199. Mean high and low waters by comparison.—While the tidal
range often varies materially from station to station in the same region,
the heights of the successive low and high waters with respect to half-
tide level at one station are proportional to those at another if the
amplitudes of the components of the tide at one of the two stations
have a constant ratio to those at the other, and the epochs of the
several components at one station differ from those at the other by
a constant angle. These conditions are to be expected when the tides
at the two stations are both produced by the same offshore fluctua-
tions of the ocean. They are exemplified by the relationship of the
principal tidal components at stations on the New England coast
north of Cape Cod. Harmonic constants have been determined at
Portland, Maine, at Pulpit Harbor, 80 miles to the northeast, at East-
port, 190 miles northeast, and at Boston, 90 miles to the south of Port-
land. The ratio of the amplitudes of the principal tidal components at
these stations to those at Portland, and the difference in the epochs
of the respective components, are shown in the following tabulation,
prepared from the data set forth in table V, paragraph 134. To
extend the comparison, the ratios of the amplitudes of the principal
components at Fernandina, Fla., 1,000 miles to the southward, to
those at Portland are added, together with the difference in their
epochs.

Pulpit Harbor Eastport Boston Fernandina

H/Ho | x—xo | H/Ho k—xo | H/Ho k—xo | H/Ho K—K0

M2 1h, 2 —4° 1. 96 2° 1.00 6° 0. 65 She
S2 1.11 —5° 2. 00 6° 1.00 5° 73 —13°
N2 1.11 —4° 1. 82 6° 1.05 8° 62 =i
Ky 299 —3° 1.04 | —3° o Be 22 ad —5°
O1 1.03 —3° 1.07 0° 58) 6° a ((il seit

103

It may be noted that the ratios of the amplitudes of the semidiurnal
components at Eastport to those at Portland are quite consistent, as
are the ratios of the diurnal components, but the ratios of the semi-
diurnal differ widely from the diurnal. At Fernandina a similar
divergence occurs in the differences of the epochs.

200. Since the successive heights of high waters and low waters
with respect to half-tide level at one station are found to have a
substantially constant ratio to these heights at another station in
the same region, the long-term means of the high and low waters at
the two stations are proportional to the respective mean values
during any period of concurrent observations. The ratio of the
mean range at the primary station during a period of concurrent
observations to its established long-term mean, applied to the mean
range during the same period at the secondary station, gives therefore
the corrected mean range at the secondary station. The corrected
heights of mean high water and mean low water on the staff are then
obtained by adding and subtracting one-half of the corrected mean
range to the corrected height of half tide on the staff.

201. It may be observed that a comparison, if based on a fairly long
set of concurrent observations, will give reliable results even when
the timing of the components is not the same at the two stations,
for, as shown in appendix IJ, mean range at each station depends on
the M, component and the ratios of the other components thereto,
and not on the epochs of these components.

202. Mean low water and mean high water of spring tides by com-
parison.—it has been shown (par. 184) that the spring range may be
taken as Mn+28,. Since the amplitude, S, ordinarily has a constant
ratio to the amplitudes of the other principal components at stations
in the same region, and hence to the respective mean ranges at these
stations, the ratio of the spring range to the mean range should be
the same at all such stations. After this ratio has been determined
at a base station, it may be applied to the corrected mean range at
any secondary station, as derived by comparison, to determine the
spring range at the secondary station. Mean low water of spring
tides at the secondary station is then one-half the spring range below
the corrected half-tide level, and mean high water of spring tides
one-half of the spring range above the corrected half-tide level. At
stations on the Pacific coast of the Panama Canal Zone, for example,
the ratio of spring range to mean range is 1.26, and the elevation of
low water of spring tides is taken as HTL—0.63 Mn.

203. Mean lower low and mean higher high waters by comparison.—
It has been seen that the elevation of mean higher high water exceeds
that of mean high water by the diurnal high water inequality, DHQ,
and the elevation of mean lower low water is that of mean low water
less the diurnal low water inequality, DLQ. The elevation of mean

104

high water and of mean low water depends principally on the semi-
diurnal components of the tides, while the diurnal inequalities depend
wholly on the diurnal components. Since the ratio of the amplitudes
of the diurnal components at two stations tends to differ from the
ratio of the amplitudes of the semidiurnal components, even when
the two stations are in the same region, the diurnal inequalities are
separately compared. The ratios of the observed mean inequalities
at the base station, during a period of concurrent observations, to the
established long term mean values of these inequalities at the base
station, applied to the observed mean inequalities at the secondary
station, give the corrected values of the inequalities at the secondary
station. These, added to and subtracted from the corrected mean
high and low waters on the staff, give the corrected mean higher
high and lower low waters on the staff at the secondary station.

204. Example-——The computation of the mean lower low water
datum at Anacortes, Wash., from concurrent observations extending
over 7 days at this station and at a base station at Seattle is shown
below. The ‘‘accepted datums” in the second column are the estab-
lished long-term means at Seattle.

Seattle Anacortes
Observed | Accepted ae aulO Oe Observed | Corrected

HHW 18. 40 S74 pads |e = epee eae 22:94 7% |: Soe ae
MHW 17. 84 BSR walt See Ate ae ee 22.47 Oe ele
MLW 10. 35 LON ZAS 7 eee ee eee ae L780" ~ -|| 2-5 es
LW 7.86 (ee IU Ae eee Oe 15 57 | eee

Mn 7.49 7. 64 1. 020 P60! See
4Mn 75 3. 82 1. 020 Ph, Bali) 2. 38
13k 14.10 14. 06 —.04 20. 14 20. 10
DLQ 2. 49 2. 83 IE ati 2e23) 2. 54

Mean lower low water at Anacortes, corrected =HTL—14Mn—DLQ=20.10—2.38—2.54=15.18

205. In the usual form of computation, the difference in the height
and time of each tide at the two stations is computed, and any tide
showing an abnormal difference in height or time is rejected. When
a record of the actual high and low waters at a suitable base station is
not available, a comparison based on the predicted tides at a suitable
station, as given in the tide tables, affords a better determination than
a short-term record at the secondary station alone, although clearly
not as reliable as a comparison with the actual tides at the base station.

206. Precision of a determination by comparison.—The precision of a
determination of tidal datums by comparison depends on the water
distance between the stations, the freedom of the movement of the
tide between them, and the length of the observations. When the
secondary station is only a few miles from the base station, on an open
and unrestricted waterway, observations extending over a short time

105

will give the datum with the precision to which the gages can be read.
To establish the datum for a project survey of a harbor or waterway
where no reliable datum is available, the observations should extend
over at least one period of 29 days. Such a comparison should es-
tablish the datum within a tenth of a foot if the base station is not too
remote. If less accuracy is needed, a comparison for a week may be
sufficient.

207. In general a comparison with a suitable base station extending
over a year will give a determination of mean sea level within 0.05
foot of the long-term mean at the secondary station, and a comparison
extending over 4 years within 0.02 foot. (Special Publication 135, U.S.
Coast and Geodetic Survey.) The determination of mean sea level
at a station where a long record is not available is always improved by
comparing it with a primary station.

FIXATION OF DATUM PLANES

208. It has been seen that even the long-term means of the eleva-
tions of the various tidal datums change slightly as the records are
extended. Since changes in the datum on which successive surveys
are based tends to confusion and error, and makes a comparison be-
tween surveys a difficult and laborious process, an accepted elevation
of the adopted datum is established with respect to a stable bench
mark, or preferably a group of bench marks, as soon as this datum is
determined with sufficient precision. This datum is not thereafter
changed, unless new conditions make it grossly erroneous.

209. Accuracy required. —So far as the usual purposes of navigation
and of harbor improvement are concerned, no high degree of precision
is required in the determination of a reference datum. The surface
of tidal waters is constantly changing in elevation, and may occasion-
ally be a foot or more below any of the datums used in the United
States. The squat of a vessel underway, and its pitch in rough water,
also render useless any refinements in the indicated depths. Hydro-
graphic charts therefore show the depths of inshore soundings to the
nearest foot, and offshore soundings in shoal areas to the nearest
quarter fathom (1.5 feet), and to the nearest fathom in deep water.
Channel depths are usually laid out to the nearest foot, although ordi-
narily the sounding from which estimates of dredging are prepared
are taken to the nearest tenth of a foot. The fixation of the reference
datum within a tenth of a foot or more of its true long-term mean is
therefore ordinarily sufficient. The stabilization of the datum is more
important than its inherent accuracy.

210. Datums for dredging contracts.—In the administration of dredg-
ing by contract the definite fixation of the datum plane to cited shore
bench marks is essential. If the material removed is measured and

106

paid for in place, as computed from soundings taken before and after
dredging, the systematic error introduced by the variation of even a
tenth of a foot between the datums of the two surveys might, in a large
contract, result in overpayments or under payments amounting to
thousands of dollars. Even when the material is measured in the
scows into which it is loaded, the difference in the deductions made for
material removed below the plane of tolerance for payment might.
amount to a considerable sum. The reference bench marks cited in
the specifications should be verified before the specifications are issued.

211. Unless the tidal range differs materially in different parts of a
harbor or waterway, the datum for harbor improvement is taken as
that at one selected tidal station. The datum at all other points is
then taken as at the same elevation, this elevation being determined
either by lines or levels on shore, or by water levels established by the
half tide level corrected by comparison with the base station. When,
however, the tidal range, and consequently the elevation of the adopted
low water datum below mean sea level, differs materially along the
waterway, a succession of reference planes should be used, each applhi-
cable to definitely defined sections or areas, and all correlated to a
common datum, preferably mean sea level. Thus on successive sec-
tions of the East River, N. Y., some 8 datum planes of mean low water
are used, varying in elevation from 1.96 to 3.51 feet below mean sea

level.
TIDAL OBSERVATIONS

212. Staff and automatic gages.—Tidal observations to establish
tidal datums, to provide the data for the harmonic analysis of the
tide, or to show the varying height of the water with respect to the
datum during surveys and dredging operations, are taken on staff
or on automatic gages. The staff gage is a graduated board, usually
set vertically, on which the height of tide is read by an observer. It
is ordinarily graduated in feet and tenths, with bold markings so
that it can be read at a distance. An automatic gage is a device by
which the elevation of a float is recorded, on a reduced scale, on a
moving paper driven by clockwork. The float is enclosed in a box
or pipe, with a restricted entrance near the bottom, to dampen the
fluctuations due to wind waves. In cold climates this box is filled
with kerosene to prevent freezing. A staff gage is always installed
with an automatic gage, the zero of the staff establishing by direct
comparison the zero of the record. Two types of automatic gages
have been developed by the United States Coast and Geodetic Survey,
one a more elaborate instrument for permanent stations at which
long-term records are maintained, and the other a portable type for
the temporary occupation of a station.

107

213. Uses—The automatic gage is especially useful for the estab-
lishment of tidal datum planes, for securing the data for the harmonic
analysis of the tides, and for hydrographic surveying when it is not
possible to establish a gage within sight of the area surveyed. The
staff gage is usually more convenient for determining the varying
elevation of the water during surveys made in the vicinity of the
gage, and for regulating the operation of dredges, since the elevations
are immediately available and can be read from a distance. For
establishing a low water datum by comparison, only the heights of
the high and low waters need be taken off the record; although the
times of their occurrence may be taken off also as a check, or to
determine the lunitidal intervals, if a determination of the latter is
desired.

214. Reference to bench marks.—A staff gage is easily destroyed and
usually lasts for but a short time, unless at least it is built into a per-
manent structure. Even in the latter case the structure may settle
or suffer enough disintegration to displace the zero of the gage. The
record of an automatic gage is dependent on its accompanying staff
gage. No tide gage serves much useful purpose, therefore, unless its
zero is referred to stable shore bench marks, and if a valuable record
is desired it should be referred to at least three bench marks well
separated from each other. Staff gages for surveys and for the opera-
tion of dredges are ordinarily set from bench marks, with their zero
at the established datum.

215. Operation of an automatic gage.—No clock keeps perfect time,
and a clock mechanism driving a relatively heavy recording device
cannot be expected to. The registering apparatus of the gage may
bend or lag or get out of order, the intake and the well may clog, the
float may leak and the wharf or other structure on which the gage is
installed may settle. An automatic gage must therefore be tended
daily to see that it is functioning properly, and the height of the tide
on the staff, with the time at which it is taken, inscribed on the record
of the gage. The gage must be inspected by an engineer at intervals,
and the zero of the staff gage checked against the reference bench
marks. The detailed technique for the installation and operation of
automatic tide gages and the tabulation of the record, is given in a
Manual of Tide Observation, Special Publication 196, United States
Coast and Geodetic Survey. Because of the cost of securing them,
reliable records of the tide over any considerable period are available
only at a relatively limited number of stations in the harbors of the
United States; but these are sufficient to afford a good determination,
by comparison, of the datums at any point on the coast line.

192750—40——8

TIDAL CURRENTS

CHAPTER V

RELATION OF CURRENT TO SURFACE SLOPE

Paragraphs
Senvecalequatviom Of tidal mOtlOns.--==252= a2 25s 2 Fee eS 216-223
Friction term for harmonically varying current-_-__-______________-___ 224-226
Surface, velocity, acceleration and friction heads__-______-___----___- 227-233
Hnurancerand recovery headss222 020 2 UY lee ee See ee 234-235
Wontrachloneneads ssa - a Se eo ee eee ee i Ee ee oe BS 236
Currents due to harmonically varying head___________._______-____- 237-253
Pe EC a RC ULTCMUS. tae... Sauer. MEET! Zonk a Ee seein ss Sols ee ae Se eet 254-255
Sigel iG: vOrirICtLOneal OW. Sevens 55 S12 2Nee i Gedo Si os eee 256
MeO esshtl Owe eee a Sie e ee ie a Se Se ee ee ee 257-258
Distortions of primaryeCuLrenuos === = 22a Ss eee eee ae 260-277
CHU elOCLLYZCUIVCS . seas = Beat Ree eee oes Fee See eae 278-282
Limitations on computation of current from surface heads____________ 284-286
Relation of current to surface head and slope when flow is frictionless__ 287—290
MPEP EMR CULTCIUS =i = ey 8 ea ee SPs es et Spee 291-294

216. General equation for varying flow in a channel.—The velocity
of the current in a tidal channel is continuously increasing or decreas-
ing, and the direction of the flow is periodically reversed. ‘To become
applicable to tidal flow, the familiar equations for steady flow must
therefore be elaborated to account for the work done in the accelera-
tion and deceleration of the current. The most casual consideration
of tidal flow shows, however, that in a channel whose width and
depth are small in comparison with the length, the lateral and vertical
movements of the water may be neglected, as they are in the equations
for steady flow. Similarly, in the derivation of the equations for tidal
flow, the velocity at a given instant may be taken as of the same
value throughout a cross section of the channel perpendicular to the
channel axis.

217. Units—lIn the ensuing development and application of the
equations for tidal flow, the time, f, will be expressed in seconds, unless
otherwise stated; lengths, heads, and other dimensions in feet; veloci-
ties in feet per second and acceleration in feet per second per second.
Conforming to these units, the speeds of the harmonic components
(par. 49) are derived in radians (or degrees) per second; but in the
application of the formulas, it ordinarily will be more convenient to
convert these speeds into degrees per hour.

(109)

110

218. Derivation of equation of motion.—Taking the X axis of coordi-
nates as a horizontal line in the direction of the axis of the channel, and
the Y axis as vertical, let:

» be the velocity of the current at the time f, and at a cross
section of the channel distant x from the origin of coordinates;
» is taken as positive when the direction of flow is in the
positive direction of z, and negative when the flow is in the
opposite direction.
Ov/dt, the acceleration of the velocity at a given cross section
with respect to time.
07/Oxr, the rate at which the velocity is increasing (algebraically)
at a given instant with the distance of the cross section from
the origin.
dx, the distance, along the direction of the X axis, traveled by
a particle of water during the elementary time interval di.
(Oy/O0xr)dxr, the (algebraic) increase in the elevation of the water
surface in the distance dz; this increase being positive if the
slope of the water surface is upward, and negative if down-
ward in the direction x positive.
X, the area of the cross section of water prism of the channel,
at the point z, and at the time f.
Q, the discharge through the cross section.
w, the weight of 1 cubic foot of water.
g, the acceleration due to gravity.
m, the mass of the water discharged through the cross section —
during the time interval dt.
r, the hydraulic radius of the channel at the section under
consideration.
C, the Chezy coefficient applicable to this section.
hens v— dads
Q=Xv= Xdz/di.
The volume of the discharge, during the time dt is Qdt = Xdz, and
its mass, m, Is:
m=wQdt/g=wXdzr/q.
The mass of the water in an elementary section of the channel of
length dz is also:
wXdr/g=m.

219. During the time interval dt, work is done in an elementary
section of the channel of length dz:

(a) In raising the mass of the discharge the distance (dy/dx)dz in its
passage through the channel.

The work so done is:

mg (oy/Ox)dx.

lt

(b) In increasing the kinetic energy of this mass because of the
increase (0v/Oz)dz, in its velocity in the distance dz. The kinetic
energy of the moving mass is mv?/2. The force required to increase
this energy is:

O(mv?/2)/Ox=mvov/dxr
and the work done by this force over the distance dz is:
mv(Ov/Ox) dz.

(c) In the acceleration of the mass of water in the section, with
respect to time. The work so done is:

m(Ov/Ot)dz.

(dq) In overcoming frictional resistance in the section. The fric-
tional resistance in a channel is due to the turbulence which the flow
produces and is dependent upon the velocity of the current. The
turbulence created at any instant by the slowly varying velocity in a
tidal channel cannot differ sensibly from that which would be produced
by the same constant velocity. The work done in overcoming fric-
tional resistance in the section of length dz may then be taken as that
developed from the usually accepted formula for steady flow. This
work is mg(v?/C?r)dx. To become applicable to the reversing flow in
tidal channels, the algebraic sign of this expression must be considered,
since the work is positive when the flow is in the positive direction of
a, and negative when the flow is in the opposite direction. Since v?
does not change its sign in passing through zero, and the other quanti-
ties are not directional, this item of work will be written:

+ mg (v?/C?r) dz.

The positive sign is to be applied when vis positive, and the negative
sign when vis negative.

220. Each of the items of work developed in the preceding para-
graph may be either positive or negative. The work done in raising
the mass of the discharge (item @) is positive if 0y/Oz is positive, and
negative if negative. Item (6) is positive if the kinetic energy is
increasing in the positive direction of x, and negative if decreasing,
while item (c) changes its sign with 0v/dt, and item (d) with v.

Since no external work is done by the flow in the channel, the sum
of all of the items must be zero, giving:

mg (Oy/Ox)dx-+ mv(dv/dx)dx-+ m(dv/ Ot) dx + mg (v?/C?r) dr =0
Dividing by mgdz, this equation reduces to:
oy/Ox+ (v/g) Ov/Ox-+ (1/g) Ov/ Ot + v?/C?r=0. (112)

Thus is the basic equation of motion in a tidal channel.
221. Discussion.—The first term, dy/Oz, in equation (112) is the
slope of the water surface in the channel at the given cross section and

112

at the given time. The second term, (v/g)0v/0z, is the rate of change
of v?/29 and may therefore be regarded as the component slope due to
the velocity head. The third term represents the effect of the accelera-
tion or deceleration of the current, and the last term is due to the
frictional resistance.

The velocity, v, at a given instant, varies in fact from point to
point in a cross section of a channel carrying tidal flow, as it does
in a cross section of a channel when the flow is steady. In both cases,
vis taken as the mean velocity at the section.

In the derivation of equation (112) the flow is regarded as contin-
uously turbulent, even during the short interval in which the velocity
becomes very small in passing through zero, as the current reverses.
It is evident, however, that a change in the character of the flow during
so brief a period may be disregarded, even if such change in fact
occurs.

222. Application of general equation to steady uniform flow.—When
the flow is steady and uniform, the velocity throughout the channel
remains constant, and 0v/Oz and Ov/ot are zero. Taking the velocity
as in the positive direction, equation (112) reduces to

oy/Oxr+v?/C?r=0. (113)

Designating the slope of the water surface as s, and observing that
when the flow is steady the slope is downward, so that 0y/0x=—s,
equation (113) becomes:

—st7?/C’r=0.
Whence
v=COyrs. (114)

Equation (114) is the generally accepted basic formula for steady
flow, in which the Chezy coefficient, C, may be determined from the
Kutter, Bazin, Manning, or other formulas.

223. Selection of Chezy coefficient for tidal flow.—It is apparent from
the preceding discussion that the value of C to be used in equation
(112) when the flow is tidal should be that applicable to the channel
were the flow steady. While the value of C determined from the
Kutter formula varies somewhat with the slope in the channel, and this
slope fluctuates between limits when the flow is tidal, this variation in
Cis so small with the slopes usually found in tidal channels that either
the maximum or the numerical mean or median slope during the tidal
cycle may be used in the application of the formula without affecting
the value of C to a greater degree than that inherent in the uncertainty
in the selection of the proper coefficient of roughness.

224. Expression for the friction term when the velocity has a harmonic
fluctuation.—It has been seen that the friction term in equation (112)
changes its sign in passing through zero, while the expression for the
friction term, v?/C?r, does not change its sign. A mathematically con-

113

tinuous expression for the friction term may be derived when the
velocity has the simple harmonic variation: .

v=B sin (at+ 8), (115)

in which B is the maximum numerical value of v during the tidal cycle.
Designating the friction term as F’, then:

F=-+ (B?/C’r) sin? (at+8). (116)

The positive sign is to be applied when (at+ 8) has values between
0 and a, 27 and 37, 47 and 57, etc.; and the negative sign when (at-+ 6)
has values between z and 27, 3m and 47, etc.

The graph of such a function is shown by the solid line in figure 36.

=sin *(at+B)
\\ 3rsin (at+B)

FIGURE 36.—Graph of friction term of an harmonic current.
225. Placing, for convenience, at-+ 6=z, the function +sin? x is by

definition such that sin? (—z)=—sin? z. By Fourier’s theorem it
should therefore be expressed by the series:

A, sin z+ A, sin 27+ A; sin 82+ ...A,sinnx... (117)

In which the coefficients, for values of « between 0 and 7, are:
A,= elm) f Siierhcinny Ga, AG — in) | sin? x sin 22 dz,
0 0

A, —(2/m) j, sin? z sin nx dr.
J0
And, for values of « between 7 and 27, the coefficients are:

20 Qr
A= ein) f Sina sii de, “As ln) { sin? x sin 2x dz,

20
A= ela) | sin? z sin nz dr

114
In these expressions, n is any integer. Since:

sin? r= % (1—cos 2z), and cos 2z sin nr==\ sin (2-+-n)r— sin (2—n)a,

A,,= (2/7) i) eae x sin nx dz
0
= c/n) {sin nx dx— A/a) {cos 2x sin nx dr
0 0

= (fn) | sinnrde— 12x) { “sin (2+n)x dx
+ (1/2) { “sin (2—n)x dx

__ COS NX cos (n+2)x]|*, cos (n—2) a |F (118)
Sn [+ 2(n+2)a if 2(n—2)r ]

The values of cos nz, cos (n+2)x and cos (n—2)x are +1 when
x=0. If nis odd their value is —1 when r=z7; but if n is even their
value is +1 when z=. Therefore, for values of x between 0 and z,
the value of A, is, when n is odd:

2 1 1 8 (119)

Aree, (n+2)r (n—2)r n(n? —4)

but when n is even, A,=0.
Substituting successive odd values of n:

A,=8/(37), A;s=—8/(157), A;=—8/(1057r), A,=—8/(3157), ete.
For values of z between 0 and 7, therefore:

sin? x=(8/37) (sin r—1/5 sin 3x—1/35 sin 5r—1/105 sin 7z. . .) (120)
Similarly, for values of x between a and 27:

i oes “| cos al cos eel
i Ne nT 2(n+2)r |, 2(n—2)r |,

(121)

Since, when n is odd, the functions cos nz, cos (n+2)x and cos (n—2)x
have a value of —1 when z=z and of +1 when r=2r7, the value of
A,, becomes, between these limits.

8
A,= = —4) 7 Uae
But when 7 is even the values of these cosine functions is -+1 both
when z=7 and when r=2z, and the coefficient is zero.

For values of « between a and 27, therefore;

115
sin? = — (8/37) (sin z—1/5 sin 3x—1/35 sin 52—1/105 sin 7/x. . . )(123)

It similarly may be shown that for values of « between 27 and 3r
the expression for sin2z is given by equation (120); for values of z
between 37 and 47, by equation (123) and so on.

226. Therefore, when v=B sin (at+ 8) the value of Fis represented
by the continuous function:

F= (8/37) (B?/C’r)[sin (at+ 8)—1/5 sin 3(at-+ 8)
=i doysia oO) (Gt-19)).- (124)

The friction term is then the resultant of a principal component,
F,= (8/37) (B?/C’r) sin (at+ 8) = (8/32) Bo/C?r (125)

with the speed of the velocity, and minor components whose speeds
are 3, 5, 7, etc., times the speed of the principal component. The
correspondence between the principal component and the complete
value of F is shown in figure 36.

The derivation of a mathematically continuous expression for F
when the velocity is the resultant of two or more harmonic components
would be difficult, if not impossible.

SURFACH, VELOCITY, ACCELERATION, AND FRICTION HEADS

227. It is the generally accepted practice to apply the formulas for
steady flow to sections or reaches of a channel of considerable length,
even though the velocity is not entirely uniform throughout such
reaches because of a variation of successive cross sections of the water
prism in the channel. For the computation of the friction term, the
velocity throughout the reach is taken as the average velocity, as
determined usually by the discharge through the average cross sec-
tion. The error introduced by the assumption, as well as the error
introduced by considering the velocity at any point in the channel
as the mean velocity in the cross section, is generally small in com-
parison with the uncertainty in the selection of the proper coefficient
of roughness to derive the value of C. The equation for varying flow
may similarly be applied to sections of channel of considerable length,
so long as the velocity and the slope at any instant are tolerably con-
‘stant throughout the section. In deep channels these conditions are
fulfilled in sections several miles in length. Denoting the length of
such a section by J, equation (112) establishes the relationship:

loy/O0x+1(v/g) Ov/Ox-+ (L/g) Ov/ Ot Llv?/C’r=0. (126)

228. The end of the section from which distances in the section
extend in the positive direction may be designated the initial end.

116

The elevation of the water surface at the initial end of the section at
the time ¢ will be designated yo, at the other end at the same instant,
y,; the velocity at the initial end, v7, and the velocity at the other end,
V4.

229. In the first term of equation (126) the slope, dy/Oz, may be
considered the average slope through the section. This term is then -
the difference, y;—Yo, between the elevations of the water surface at
the ends of the section: it will be called the surface head and desig-
nated h;. Then

h,=loy/0x=Yi— Yo (127)

It may be noted that the surface head is positive when the water is
sloping wpward in the positive direction along the channel, and nega-
tive when sloping downward.

230. The second term of equation (126), (v/g)dv/ox is the change in
v?/2g between the ends of the section, and is therefore the velocity
head, h,, giving:

h,=l(v/9) 0v/or=v?2 /2g—v2/29 (128)

It may be noted that the velocity head is positive when the velocity
is increasing numerically in the positive direction, and negative when
the velocity is numerically decreasing in that direction.

231. The third term of the equation may be called the acceleration
head, and designated h,, giving .

ha= (l/g) Ov/ ot (129)

In which 0v/ot is the acceleration of the average velocity through the
section. ;

The acceleration head is positive when the average velocity through
the section is increasing algebraically with respect to time.

232. The fourth term is the friction head, h,, giving:

hy= £lv?/Or (130)

in which v may be taken as the average velocity through the section
at the given instant, C is the Chezy coefficient applicable to the sec-
tion, and 7 the hydraulic radius. The friction head has the same
sign as the velocity.
233. Substituting the symbols for the various heads in equation
(126):
ose Opens = 0 == 0 (131)

In this equation, h,, h,, and hy; have the same significance as in |
steady flow, except that A, and hy, as well as h, may be negative.
The acceleration head is the additional term which must be included

sly,

in tidal flow. It is readily evaluated when the rate of change in the
velocity is known. Thus if the velocity is algebraically decreasing
at a given time at the rate of 0.003 feet per second per second in a
‘section of channel 10,000 feet in length, the acceleration head is
— 10,000 X0.003/32.16= —0.93 feet.

234. Entrance and recovery heads—In steady flow, the head due to
the increase in velocity at the entrance to a contracted section of a
channel is termed the entrance head. Its commonly accepted

value is:
h=m(v2—2°)/2g (132)

In this equation 7% is the velocity in the approach to the contrac-
tion, 2, the velocity in the contracted section, and m a coefficient to
account for the increased turbulence at the contraction. The entrance
head is then the change in 2?/2g at the entrance, times a suitable
coefficient. If the velocity in the contracted section is not large, m is
often taken as unity.

At the outlet of the contracted section, the decrease in the kinetic
energy of the flowing water gives rise to a recovery of head whose
value is given by the same formula by taking v as the velocity in
the contracted section, and 2, the velocity in the expanded channel.
The recovery head is then also the change, in the positive direction,
of v?/2g, times a suitable coefficient. Since, however, the recovery of
energy is never complete, the value of m for the recovery head is
always less than unity, and frequently is taken as 0.5.

235. In reversing tidal flow each end of a contracted section of
channel is alternately the entrance and the outlet. At the initial end
the flow is into the contracted section when the velocity is positive,
and out when the velocity is negative; but since the numerical value
of the velocity remains the greater in the contracted section, the
change in v?/2g at the entrance is positive, whichever the direction of
the flow. At the other end the flow is out of the contracted section
when the velocity is positive, and into the section when negative;
but the change (in the positive direction) of v?/2g is always negative.
At both ends, therefore, the change (in the positive direction) of
v’/2g represents an entrance head when it has the same sign as the
velocity, and a recovery head, to which a reducing factor should be
applied, when it is of opposite sign to the velocity. On an expanded
section of the channel, the contrary condition obviously exists.

236. Contraction heads.—At a sudden local contraction in a section
of channel that otherwise may be taken as uniform, such as at a bridge,
the increased turbulence resulting from the increase and decrease of
the current produces a net head, of the same nature as an increase in
the friction head. Such a contraction head may be introduced in
computations of tidal flow by determining, from the applicable for-

118

_mulas developed for steady flow, its numerical value at the given
velocity in the section, and giving it the sign of the velocity.

CURRENTS PRODUCED BY A SIMPLE HARMONIC FLUCTUATION OF THE
SURFACE HEAD AND SLOPE IN A SHORT SECTION OF A CHANNEL

237. The surface head in a section of a tidal channel is the differ-
ence between the elevation of the tides at the ends of the section.
These tides are, as has been seen, the resultant of a number of har-
monic components, of various amplitudes, speeds, and phases, occa-
sionally modified by meteorological disturbances. Quite obviously,
the surface head likewise is the resultant of harmonic components of
the same speeds; but it does not follow that the amplitudes of the
components of the head are proportional to the amplitudes of the
tides. On the contrary, the amplitudes of the diurnal components of
the head may be, and frequently are, proportionally less than the
amplitudes of the diurnal components of tides of the mixed type; and
overtides of the head often are relatively more important than those
of the tides themselves. Generally, the head in a short section of a
tidal channel during a single tidal cycle does not depart widely from
a simple harmonic fluctuation with the speed, m:, of the principal
lunar component of the tides. The currents derived from such a
simple harmonic fluctuation of the head often afford a sufficient indi-
cation of the strength and timing of the actual currents; and in any
case provide a basis for a determination of the currents resulting from
a head which has a given variation from the simple harmonic fluctua-.
tion assumed.

238. Relation of surface head to a sumple harmonic fluctuation of the -
tides at the ends of a short section of channel.—lIf the tides at the ends
of the section are taken to have simple harmonic fluctuations of the
same speed, the head likewise has a simple harmonic fluctuation of
the same speed. Let O and A be two stations on a tidal channel, at
such a limited distance, /, apart that at any instant the variation in
the velocity and slope between the stations is immaterial. Let:

Up =A cos (at+ ao) (133):

be the elevation of the water surface at O, taken as the initial sta-
tion, and let:
Y,=A, cos (at-+ a) (134):

be the elevation of the water surface at A.

119
The surface head in the section is then:

h,=Yi—Yo= A; cos (at+a,) — Ay cos (at+ a)
=A, cos (at-+a,)+ Ap cos (at+a)+180°) (135)

Since two components of the same speed unite into a single com-
ponent of that speed, equation (135) may be written:

h,=H cos (at+H°) (136)

In which 7 is the amplitude, and H° the initial phase, of the fluctua-
tion of the surface head during the tidal cycle.

239. Computation of H and H°®.—The amplitude and phase of the
surface head in the section readily may be determined from the
amplitudes and phases of the tides at the ends of the section, through
the relation established in equations (135) and (136):

H cos (at+H°) =A, cos (at+a,)— Ay cos (at+ay) (137)

Equation (137) is identically true for all values of ¢. By placing
at=0, the equation of condition is derived:

H cos H°=A, cos a,— Ap COS ay (138)

and by placing at= 90°
Hsin H°= <A, sin a,— Ap sin ap (139)
The values of 7° and H may then be determined from the equations:
tanec EE cose” (140)
H=H sin H°/sin H°=H cos H®/cos H° (141)

The amplitude, H, is directionless. The quadrant in which H° lies
is determined by the algebraic signs of H sin H® and H cos H’°.

240. EKxample.—The curve showing the average height of the tide
at station 180-+30 on the Cape Cod Canal, after the time of a lunar
transit, prepared from observations during the period September 28
to October 6, 1932, is represented by the equation:

y=3.74 cos (m:t+58°34’)
and at station 225 by:

y=3.18 cos (m,¢+61°10’)

120

Taking station 180-+30 as the initial station:

A, cos o,=3.18 cos 61°10’=1.534 A, sin a,=3.18 sin 61°10’ =2.786
Ap COS a=3.74 cos 58°34’=1.950 Ay sin a=3.74 sin 58°34’=3.191

H cos H=—.416 H sin H°=—.405.
tan H°=—0.405/0.416=0.9736

The corresponding angle, from a table of natural tangents, is
44°14’. Since the sine and cosine are both negative, H° les in the
third quadrant, and is:

EP=180°+44°14’ =224°14’
and
H=0.416/cos 44°14’=0.58

The equation of the surface head in the section between the two
stations is therefore:

h,=0.58 cos (Mm t+224°14’)

241. Generating radius of head.—The rela-

Y tion between the generating radii of the

curves representing the tidal heights at

the two ends of the channel and that of the

head in the channel is shown in figure 37,

in which CP,)= A), is the generating radius

of the tide at the initial end, CP,;=A, that

c at the other end of the channel, CP2=P P;

is the generating radius of the curve showing
the surface head.

242. Equation of primary current.—As will later be made apparent,
the currents produced by a simple harmonic fluctuation of the surface
head in a short section of the channel depart somewhat from a simple
harmonic fluctuation. These distortions of the current are due to
the form of the velocity head term, (v/g) Ov/Ox, in the general equation
of motion, to the minor components of the friction term produced by a
harmonically varying current (par. 226) and to the variation in the
hydraulic radius and Chezy coefficient with the rise and fall of the
tide. Under usual conditions of tidal flow the velocity head in a
short section of a channel is relatively so small that the velocity head
term may be omitted. The other disturbing elements may be treated

Fic URE 37.—Relation of head to tides.

12]

as corrections to a harmonically varying primary current, BEES Se Ue.
by equation (115)
v=B sin (at+ 8)

243. Dropping the velocity head term, (v/g)0 v/dz, from equation
(112), and substituting for the friction term its principal component
for an harmonically varying current (equation 125), the differential
equation of the primary current is:

dy/dx-+ (1/9) dv/dt-+ (8/37) Bu/C2r =0 (142)

In this equation C and 7 are the values of the Chezy coefficient
and hydraulic radius at mean tide.
In equation (142):
oy/0x=h,/l=(H/l) cos (at-+-H°)=S cos (at+ HA), (143)
in which S=H// is the numerical value of the maximum slope in the

channel during the tidal cycle.
From equation (115):

0v/O0t=aB cos (at+ 8) (144)
Equation (142) then becomes:

S cos (at-+-H°) + (aB/g) cos (at+ 8) + (8/37) (B7/Cr) sin (at+ 8) =0
(145)
In which a is expressed in radians per second.

244. Solution of equation.—By placing at=0 and at=—7/2 in
equation (145), two equations of condition are established from
which expressions for B and 6 may be derived. When at=0, equation
(145) reduces to:

S cos H°-+ (aB/g) cos B+ (8/37) (B?/O?r) sin B=0 (146)
and when at=—7/2, to:
S sin H°+ (aB/g) sin B— (8/37) (B?/C?r) cos B=0 (147)

Multiplying equation (146) by cos 8 and equation (147) by sin B
and adding:

S(cos H® cos 6+sin H° sin 8)+ (aB/g) (cos? B-+sin2 8) =0
or:
S cos (H°— 8)+Ba/g=0 (148)

Multiplying equation (146) by sin 6 and equation (147) by cos 6 and
subtracting:

122

S(cos H® sin B—sin H® cos £)+ (8/37) (B?/C’r) (sin? B+-cos? B)=0

te _§ sin (EP—8) + (8/32) B/O%r=0 (149)
It is convenient to place:
H°—B=¢+7/2 (150)
So that equations (148) and (149) become:
S sin ¢=aB/g (151)
S cos ¢= (8/32) B?/C’r (5?)
Whence
B/C’r= (81/8) (a/g) cot ¢ (153)
Eliminating B from equations (151) and (153):
sin @ tan ¢= (32/8) (a/g)?C?r/S (154)
And from equation (152):
B=31/8 C-yrS yecos ¢ (155)
Or, from equation (151):
B=G/a) S sin ¢ (156)

It may be seen from equations (151) and (152) that both sin ¢ and
cos ¢ are intrinsically positive. @ is therefore an angle between 0
and 90°.

245. Computation of ¢ and B.—Equation (154) may be written:

(g/a)aJsin ¢ tan ¢=y37/8-CyrS/S (157)
Placing for convenience,
P=732/8 C yrS (158)
= 1.0854 OyrS
This equation reduces to:
(gia)ysin @ tan ¢=P/S 2 Nae)

The numerical values of P and P/S are readily computed from
the amplitudes, S, of the slope in the section during the tidal cycle,
and the Chezy coefficient C and hydraulic radius, 7, at mean tide.

123

TABLE IX
’ P/S log P/S 4/ 0s @ | log Vcos } fo) PIS log P/S
0 0 1 0
it 3,995 | 3.60147 1,000 9. 99997 46° | 197,500 | 5. 29563
2° 7,990 | 3.96253 1. 000 9. 99986 47° | 202,700 | 5.30681
32 11,990 | 4. 07848 . 999 9. 99970 48° 207,900 | 5.31790
4° 15, 980 | 4. 20369 . 999 9, 99947 49° | 218,300 | 5.32889
6° 19,980 | 4.30070 . 998 9. 99917 50° | 218,700 | 5.33980
( 23,990 | 4. 38002 . 997 9. 99880 one 224,200 | 5.35065
i 28,000 | 4.44710 . 996 9. 99837 io 229,900 | 5. 36144
8° 32,010 | 4. 50526 - 995 9. 99788 Dom 235, 600 | 5. 37220
g° 36, 030 | 4. 55660 . 994 9, 99731 54° 241, 500 | 5, 38292
10° 40,050 | 4.60257 . 992 9. 99667 oe 247, 500 | 6.39365
inl? 44,080 | 4.64420 991 9. 99597 56° 253, 700 | 5. 40437
i932 48,110 | 4.68226 . 989 9. 99520 57° | 260,100 | 5.41512
13° 52, 160 | 4.71731 . 987 9. 99426 58° 266, 600 | 5. 42589
14° 56, 210 | 4. 74980 . 985 9. 99345 59° 273,400 | 5.43672
15° 60, 270 | 4.78010 . 983 9. 99247 60° 280, 300 | 5.44762
16° 64, 340 | 4, 89850 . 980 9, 99142 61° 287, 500 | 5. 45861
ig. 68, 430 | 4.83522 . 978 9. 99030 §2° 294,900 | 5. 46971
18° 72,520 | 4.86046 . 975 9. 98910 63° 302, 600 | 5. 48094
19° 76,630 | 4.88438 972 9. 98783 64° 310, 700 | 5 49232
20° 80,750 | 4.90714 - 969 9. 98649 65° 319, 100 | 5. 50388
PANE 84,890 | 4.92883 . 966 9. 98507 66° 327, 800 | 5. 51565
222 89,040 | 4. 94957 . 963 9, 98358 67° 337, 000 | 5. 52767
Dae 93,210 | 4.96944 . 959 9. 98201 68° 346, 700 | 5. 53996
24° 97, 390 | 4.98853 . 955 9. 98036 69° 356, 900 | 5. 55257
5c 101,600 | 5, 00689 - 952 9. 97864 79° 367, 700 | 5. 56554
26° 105, 800 | 5.02459 . 948 9. 97683 “ae 379, 300 | 5. 57893
ie 110,100 | 5, 04168 . 944 9. 97494 ee 391,600 | 5. 59279
28° 114, 300 | 5, 05822 . 940 9. 97297 ABS 404, 800 | 5.60721
29° 118, 700 | 5.07424 . 935 9, 97091 74° 419, 000 | 5.62225
30° 123,000 | 5, 08978 . 931 9. 96876 Ue 434, 500 | 5.63803
31° 127, 300 | 5, 10489 926 9. 96653 76° 451, 500 | 5.65465
32 131, 700 | 5.11958 921 9. 96421 Uae 470, 200 | 5.67226
3B 136, 100 | 5. 13389 916 9. 96179 78° 491,000 | 5.69104
34° 140, 509 | 5, 14785 911 9. 95929 79° 514, 390 | 5.71123
Shi 145, 000 | 5, 16149 905 9. 95668 80° 540, 900 | 5. 73309
36° 149, 600 | 5. 17482 899 9. 95398 81° 571, 500 | 5. 75703
Byte 154, 100 | 5. 18787 894 9 95117 2° 607, 500 | 5.78356
38° 158, 700 | 5. 20065 888 9, 94826 83° 650, 700 | 5.81338
39° 163, 400 | 5. 21320 882 9. 94525 84° 704, 000 | 5.84758
40° 168, 100 | 5. 22552 875 9, 94212 Sie 772, 300 | 5.8877.
41° 172, 800 | 5. 23763 869 9. 93889 86° 864, 400 | 5.93673
42° 177, 600 | 5. 24955 862 9. 92554 Si 999, 000 | 5. 99958
43° 182, 500 | 5. 26130 855 9. 93206 88° |1, 224,300 | 6.08790
44° 187, 500 | 5, 27288 . 848 9. 92847 89° |1, 732, 200 | 6. 23859
45° 192, 500 | 5. 28432 . 841 9. 92474

log cos &

9. 92088
9. 91689
9.91275
9. 90847
9. 90403

9. 89943
9. 89467
9. 88973
9. 88461
9. 87929

9. 87378
9. 86805
9. 86210
9. 85592
9. 84948

9. 84278
9. 83580
9. 82852
9. 82092
9. 81297

9. 80466
79594
78679
77716
76702

75632
74499
73297
72017
70650

69184
9. 67604
9. 65894
9. 64030
9. 61983

{ OWwWOw sss 10

59716
57178
54295
50962
47015

42179
35940
27141
12093

DOOD ww s9 0

P=1.0854CyrS

log 1.0854=0.03559

246. In table LX the values of P/S=(g /a)sin ¢ tan @ with their
corresponding logarithms, are tabulated for each degree of ¢ from 0
to 89°, when a is the speed, ms, of the principal lunar semidiurnal
component, in radians per second.

the value of mz, in degrees per hour, is 28° .9841.

per second is then (28.9841/3600) (m/180)=0.000,140,52.

From table II, paragraph 75,
Its value in radians

This

124

value is used in the preparation of the table. The value of g is taken
as 32.16. By entering table 1X with the computed value of P/S or
of its logarithm, the value of ¢ for a simple tidal fluctuation with a
speed of mg, is readily determined by interpolation.

The value of B is then, from equation (155):

B= Py cos o , (160)

247. If the tidal fluctuation has a speed, a, differing from that of
the principal lunar semidiurnal component, m,, on which table [X is
constructed, equation (159) may be written:

(g/m>)+/sin ¢@ tan ¢=(a/m,) P/S (161

To determine the value of ¢, table IX is then entered with the
computed value of (a/m,)P/S. It may be noted that the speeds, a
and ms, may be expressed in any common units.

248. Value of 8—KFrom equation (150):

pB=M—¢—7n/2
Or, when angles are expressed in degrees:
Boo 90° (162
The equation of the primary current is then:
v=B sin (at+ H°—¢—90°) (163)

The values of B and ¢ are determined as shown in paragraph 246;
and H°, the initial phase of the head, is determined as shown in
paragraph 239.

249. Hramples.—The surface head between stations 180+30 and
225 in the Cape Cod Canal, at the time ¢ after a lunar transit, was
found in paragraph 240 to be:

h,;=0.58 cos (mst+224°147’)

The length of the section is 4,470 feet, giving: S=0.58/4470=
0.000,130. The hydraulic radius, at mean tide, at the time of the
observations, is given as 22.7 feet. As the bed is exceptionally rough,
an appropriate value of Kutter’s “n” is 0.030. Taking the mean
slope as 0.0001, the corresponding value of Cis 90. From these data:

P=1.0854Cy7S=5.31, P/S=40,820.

125

From table IX the corresponding value of ¢ is 10°12’, and from
equation (160):
B=5.31ycos 10°12’=5.27 feet per second.

From equation (163), the primary current, at the time ¢ after a
lunar transit is then:
p=5.27 sin (mt+124°02’)

250. The head and the primary current are plotted in figure 38..
As station 180+30 has been taken as the initial station, and as the

3)

1S)

es
rs) rs)
(os S)
a aha
—— !

3 S
= 9
(o>) a is
oh
9 -5-
>

Lunar Hovr after Transit.

FIGURE 38.—Primary current and head in section of Cape Cod Canal.

stationing is from east to west, westerly currents are positive, and
sasterly currents are negative.
251. As asecond example, the value of Sin a section of the Delaware

River near the mouth is 0.0000146, the hydraulic radius in the section,
The value of C corresponding to a.

at midtide, being 19.3 feet.
Then:

coefficient of roughness of 0.025 is 120.
P=218 P/S=149,300

o=36° B=2.18ycos 36°=1.95 feet per second.

126

252. The relation between the head in a section 5,000 feet in length
and the velocity is shown in figure 39, the origin of time being taken

oO

S) a2 s

a +0.05 ,,
YS

Qo S)
ele Bd

sae. (QO Q |
{ 2)

= QS: 8

Stes nO!

S =I

_O 3
iS)

Ps

Lunar Hour

FIGURE 39.—Primary current and head in section of Delaware River near entrance.

at a moment when the head turns from negative to positive.

253. For a third example, a channel may be taken with a hydraulic
radius of 100 feet, and a maximum slope, S, of 0.000,01 during the
tidal cycle. An appropriate value of Cis 150.

Then:

Paseo PG aaec0r
o—19- B=5.1354/cos 79° =2,24 feet per second.

The relation between the head in a 5,000-foot section of such a
channel and the velocity is shown in figure 40, the origin of time being
as in the preceding example.

ro)

rs)

7)

<

PS) ORO
Oo 3
AS an

tL 0 ;
t

mn g
ne Ss)

= =

isi — 0.05
S

=

Lunar Hours. -

FIGURE 49.—Primary current and head in deep channe!.

254. Lag of primary tidal current.—It may be observed that positive
directions have been so assigned to heights and velocities that the
water is running down hill when the head is positive and the velocity

127

is negative, or vice versa; and is running uphill when both the head
and the velocity have the same algebraic sign. While this convention
may appear unnatural, it removes the confusion that would result
were vertical distances taken as positive in a downward direction.
In each of the diagrams illustrating the preceding examples, the head
reaches a maximum at the time marked C, and the primary current
reaches its strength, in the opposite direction, at a subsequent time
marked ). The strength of the primary current in a tidal channel
therefore lags behind the maximum surface head and slope by the
time interval CD. The turn of the primary current lags behind the
turn of the surface head and slope by the equal interval AB.

Designating the time C as fy, and the time D as t,, then from the
equation of the surface head (equation 136):

h,=H cos (at+H°)

it is evident that

atj+H°=0.
From the equation of the primary current (equation 163):

v=B sin (at+ H°—¢—90°).
and
at, + H°— ¢@—90°=— 90°.
Whence
at,—at)=¢.

The intervals CD and AB are then equal to ¢/a; and the angle ¢ may
be designated the angular lag of the primary current.

255. Characteristics of tidal flow.—In each of the preceding examples
the water flows downhill during the intervals indicated as BA on the
diagrams. At the moments marked A the water surface is level, but
the momentum of the moving water continues to carry it in the direc-
tion of its motion. During the intervals from A to B the water flows
uphill until the momentum is checked. At the instants marked C,
when the current reaches its maximum velocity in either direction,
the acceleration is zero, and the velocity is determined by the slope
and frictional resistance only, and is the same as though the flow
were steady; but as the velocity lags behind the head, the maximum
velocity does not occur when the head is a maximum. When the lag
is very large, the maximum velocity occurs at a moment when the
head is very small.

128

256. “Hydraulic” or “frictional” flow—The first of the preceding
examples (fig. 38) shows that if the head in a tidal channel is sufficient
to produce strong currents, and the channel is not of great depth, the
lag of the current with respect to the head is small, and the current at
any instant is substantially the same as that which would be produced
by the instantaneous head were the flow steady. The flow under
these conditions is often termed ‘“‘hydraulic.”’ A better name is
“frictional tidal flow.’’ Currents of this character are found in the
East River, N. Y., and in other tidal straits of moderate depths which
are subject to a considerable tidal head.

If the lag is small, the value of cos in equation (155) is close to
unity, and the amplitude, B, of the velocity varies from day to day
-as the square root of the amplitude, S, of the slope, and hence as the
square root of the amplitude of the head, H, during the tidal cycle.
‘Since the tides at the ends of a tidal strait keep in general step as their
amplitudes change from day to day with the changing declinations
and distances of the moon, the daily variation in H/ is nearly propor-
tional to the daily variation in the tidal range. When therefore the
flow in a strait is largely frictional, or “hydraulic,” the ‘strength
of the current” in each section of the channel varies from day to day
approximately as the square root of the tidal range.

257. Frictionless tidal flow—The lag of the current increases as the
slopes in the channel and the current velocities decrease. It increases
also as the depth of the channel and the coefficient C increase. As
shown in the last example (fig. 40), the lag becomes very large in deep
channels with small slopes. Most of the potential energy due to the
head in the channel is then taken up in the acceleration and decelera-
tion of the current and little in overcoming frictional resistance. The
flow under these conditions is sometimes termed ‘‘tidal,” as distin-
guished from the “hydraulic” flow determined principally by fric-
tional resistance. A better name is “frictionless flow.”’

258. In a section of channel which is so deep, or in which the cur-
rents are so weak, that the flow is nearly frictionless, ¢ 1s nearly 90°.
A small error in taking off its value from table [X would then produce
a large error in the computation of the amplitude, 6, from equation
(160). When ¢ is large, the value of B is better derived from equa-
tion (156):

B=(g/a)S sin ¢

For tidal fluctuations having the speed of the M, component, and
for g=32.16, the value of g/a is 228,900; and its logarithm is 5.35958.

129

For wholly frictionless flow, 6=90°, and
B=gS/a=gH/al (164)

The amplitude of the fluctuations of the current then varies from
day to day directly as the head, and hence nearly as the amplitude
of the tide.

Obviously, in the fiords of Alaska, where depths of 1,000 feet are
common, and in other deep channels such as the Florida straits, the
tidal flow is essentially frictionless.

259. If the depths in a channel are small enough and the currents
sufficiently marked to be of consequence to shipping, the tidal flow is
not frictionless and the currents depend upon both the friction head
and the acceleration head, as indicated in the second example (fig. 40).
The so-called hydraulic state of flow is one of degree only, and merges
without distinction into conditions of flow in which the acceleration
head becomes of increasing importance. The maximum velocity, or
the “strength of the current”’ is always less than that which would be
produced by the maximum head were the flow steady. The accelera-
tion head acts as a brake on the currents as the friction head diminishes.

DISTORTIONS OF PRIMARY CURRENT

260. The primary current has been derived by taking the surface
slope as a simple harmonic fluctuation; dropping the velocity head
term from the general equation of motion (equation 112); substituting
for the friction term its principal harmonic component (8/37) Bo/C?r;
and taking the hydraulic radius, 7, and the Chezy coefficient, C, at
mean tide. The corrections for these approximations will now be
developed. These corrections produce a velocity-time curve which is
more or less distorted from the simple harmonic curve of the primary
current.

261. Corrections for the variation of frictional resistance with the re-
versing square of the velocity—The corrections to fulfill the condition
that the friction term is +v?/C*r, may be computed, to any desired
degree of refinement, by a somewhat laborious process explained in
detail in appendix IJ. As there shown, these corrections, designated
as 7, are proportional to the amplitude, B, of the primary current and
depend upon its angular lag, ¢, and its phase, af+ 6. The correction
factors, 2/B, as so computed for successive values of ¢, and for values
of at+ 8 from 0 to 180°, are shown in table X. For values of at+
between 180° and 360° the table is entered with at-+ 8—180° and the
algebraic sign of correction reversed. As will be seen from the table,
the corrections are small when ¢ is large, and the flow consequently
is nearly frictionless. They become zero when ¢=90°.

130

TABLE X.—Correction factor 1/B

o= 0 5° 10° 20° 30° 40° 50° 60° 70° 80°
at+B=0 0 —0.18 | —0.18 | —0.16 | —0.14 |—0.11 |—0.08 |—0.05 |—0.03 |—0.01
5° +. 18 —.14 —.16 —.15 =.13 9} =.10 «|| =.07 |>=.104 | — 1035 se Ss01
10° +. 21 0 —.09 =p WH =r = 308) 06) 2 OF 02 eae
15° +. 21 +. 12 0 — 0%) — 0 |) = 0 | = 0S || — 0 | S| =
20° +. 20 +. 16 +. 08 —.01 —.04 | —.04 | —.03 |) =7027 |) — 209 0
25° +. 18 +. 18 +. 138 +. 04 0 —.01 | —.01 0 0 0
30° +. 15 apo lla =. 14 +. 07 +.03 | +.01 0 +. 01 0 0
35° +. 12 +. 14 =. 14 +. 09 +.05 | +.03 | +.02 | +.01 | -+.01 0
40° SF ll) Spo Lil +. 12 +. 10 SOG) ol OFe st OSie ct O2 ictal Leen
45° +. 07 +. 08 +. 10 +. 09 spo || SO |) apoB | se O2 | se 0% | ss.
50° +.04 +. 06 +. 07 +. 08 =506 |) 005) |) 2103) nO) Nn O2 ene taOk
55° +. 02 +. 03 +. 04 +. 06 soe | SAOee PSB) SeeR} ese |) sell
60° =o (Hil 0 +. 02 +. 03 sro oe |) sa | 402 |) se.02 |) se. 0
65° —.02 —.02 —.01 +. 01 = O2 a 02) iO 2Ns | -t-O2ia etn On ton
70° —.05 — 704 —.03 —.01 spoil | SO | se Ol | ae O | seW |) se il
75° —. 06 —.05 —. 05 —.03 =o (Ul 0 0 0 +. 01 0
80° —.07 =, O7 —. 06 —. 04 =. 08} |} =p (dil —.01 0 0 0
85° —.08 =o (0 =o (07 —.06 = (028) 508} |) 5 | Sil | 0 0
90° —.08 —.08 —. 08 0 —.05 | —.04 | —.03 | —.02 | —.01 | —.01
95° —.08 —.08 —. 08 = Uy 309 |) = 05 || =U | 04 || = 02 || =
100° = (i = 07 —. 08 =, (07 —.06 | —.05 | —.04 | —.03 | —.02' >} —- 01
105° —. 06 —.06 =p Wi —.07 —.06 | —.05 | —.04 | —.03 | —.02 | —.01
110° —.05 —.05 —.06 —. 06 —.06 | —.05 | —.05 | —.03 | —.02) | —. 01
115° —.02 —. 04 —. 04 —.05 03 |) 30) |) 05) = 08 | = 0) Sn
120° (Mil —.02 —.03 =o (UH! —.05 | —.05 | —.04 | —.03 | —.02 | —.01
125° +. 02 0 —.01 —.02 —.03 | —.04 | —.04 | —.03 | —.02 | —.01
130° +. 04 +. 03 +. 02 0 —.02 | —.03 | =.03' | —.02 9) —202) |= 70%
135° +. 07 +. 05 +. 04 +. 02 0 —.01 | —.02 | —.02 |} —.01 0
140° +. 10 +. 08 aro We +. 04 +. 02 0 Sil | oO | =. Wil 0
145° +. 12 Sp ll +. 09 +. 06 sro | se.O% | ars Oil 0 0 0
150° +. 15 +. 13 =. 12 +. 09 +.06 | +.04 | +.02 | +.01 | +.01 | +.01
155° app Ils) +. 16 “pale +. 11 = 308) | tn05) 1 S035 0-02. sO ln en
160° +. 20 +. 18 +. 16 +. 13 S510 see |) =O | ae0R | = || =p, 0
165° +. 21 +. 20 +. 18 +. 15 a Fp ili! sralis |) sO i) S30 | oe || seo
170° +. 21 ap Zl ro ly) +. 16 +.13 | --.09 | —.07 | +.05 | —-.03) | --201
175° +. 18 +. 20 +. 20 Spo Uz Spelsy aces) orate apathy |] se) | Se5 il
180° 0 +. 18 +. 18 +. 16 +.14 | +.11 | +.08 | +.05 | +.03 | +.01

262. Hxramples.—The primary current in a section of the Cape Cod
Canal, at the time ¢ after a lunar transit, as derived in paragraph 249,
ee v=5.27 sin (mt+124°02’)
and its angular lag, ¢, is 10°.2.

The corrected velocity at say 3 lunar hours after a lunar transit
is to be found. When ¢ is reckoned in lunar hours m,=30°. Then, at
the given time, at-+ B=90°+124°02’=214°.03. Since this angle les
between 180° and 360°, table X is entered with ¢=10°.2 and

at-+ B=214.03°—180°=34°.03
The corresponding value of 7/B is, by interpolation, +0.14. Revers-

ing the sign, and multiplying by B=5.27, the correction is —0.74 feet
per second. The primary current at the given hour is

5.27 sin 214°.02’=-—2,95

and the corrected velocity is —3.69 feet per second.

263. The distortion of the primary current curve in this section of
the canal, derived by applying the correction at successive lunar hours,
is shown in figure 41. The distortion of the primary current curve in
a section of the Delaware River (par. 251 and fig. 39), is shown in
figure 42. In the latter case B is 1.96 feet per second, and ¢ 1s 36°.

5
- JANES ae euos
2, Fc me Ui UW
Meee Vee he
¢ CO
Ae ot
ee Seeeisich be iNee

USS SEE eee

Lunar Hee after See

Primary current——— Corrected

FIGURE 41.—Distortion of primary current when ¢=10°.2.

als
BoP E Me Nae
ida ei atin ae

Lunar Hour

Primary current ———
Corrected current

FIGURE 42.—Distortion of primary current when ¢=36°.

264. Correction for remaining approximations—The correction of
the primary current for the remaining approximations introduced in
its derivation may be computed by a variation of the procedure by
which the corrections shown in table X are derived. The computations
determine the corrections at selected time intervals through the tidal
cycle. In order that they may apply to repetitions of the cycle, the
intervals should be parts of the component hour of the simple har-
monic fluctuation of the head used for the determination of the
primary current, ordinarily the lunar hour of 1.035 mean solar hours.
Since the computations depend, in part, on the changes in the velocity
during these intervals, their accuracy is increased, but the labor mul-

132

tiplied, as the interval is decreased. Intervals of half a lunar hour
usually are sufficiently small to give acceptable determinations.
265. The corrected velocities must be such as to satisfy equation
(lene
he late —O

Let v be the velocity of the primary current, corrected by i, from
table X, on a given lunar half hour,

6, the further correction to 2,
69, the correction at the preceding half hour,
1, the length of the section.

266. For purposes of the computations, the surface head, h;, should
have a fluctuation which, although not necessarily a simple harmonic,
identically repeats itself every 12 lunar hours if the tides and the
surface head are wholly semidiurnal, or every 24 lunar hours if the
diurnal components of the head are so large as to require consideration.
This result may be accomplished by selecting tidal fluctuations at the
ends of the section which identically repeat themselves every 12 or 24
lunar hours. Under ordinary circumstances it is indeed apparent
that the tides on one day have but little effect upon the currents of the
next.

267. The expression for the acceleration head is, from equation (129)

ha= (U/g) 0(v+6)/ot
= (I/g) (Ov/ot-- 06/0t)

Since this relation remains approximately true when small finite incre-
ments are substituted for the differentials, it is permissible to place:

ha= (I/g) (Av/At+ A6/At)
= (1/gAt) (Av+ A6) (165)

In which At is the selected time interval, in mean solar seconds, and
Av and Aé are the increases in v and 6 corresponding thereto at the
given half hour.

268. It will be convenient to place:

L/gAt=b (166)
When the time interval is a half lunar hour:

At=% 1.035 X3,600 seconds
and:
b—0.00001671 | (167)

133

The values of Av are computed from the successive values of 2.
As the best approximation, Aé will be taken as the increase, 6— 6p, in
the preceding interval. Equation (165) then becomes:

ha=bAv+b(6—6)) (168)
269. The friction head, h;, is, from equation (130):
hy= +1 (0+6)?/Cr

in which C and r vary with the stage of the tide. A diagram may be
prepared showing the values of:

Gee (169)

corresponding to the stages of the tide.
The expression for the friction head may be written:

hy=+F+6)?
= + Fy +2 Fv + Fe (170):

Since 6 is a comparatively small correction, at least a first approxi-
mation may be derived by dropping its square and neglecting any
effect that it may have upon the algebraic sign of the corrected veloc-
ity, v-+6; giving:

hy= + Fv? +26(+0) F (171)

in which the positive sign is to be applied when v is positive, and the
negative sign when negative. Obviously, therefore, the factor (+)
is always positive. Representing the numerical value of v, on the
given half hour, as 7, equation (171) becomes:

h,=+ Fv? +25Fo (172)

270. The velocity head, h,, remains to be considered. The deriva-
tion of the tidal currents in a short section of channel was predicated
on the assumption that the section is so short that at any instant the
variation of the velocity between the ends of section is immaterial.
Under this assumption the velocity head would disappear. While the
derivation remains valid even though there be a sufficient difference
between the velocities at the ends of the section to produce some veloc-
ity head, yet in the ordinary case 1t is too small to be worth computing.
It may be included in the computations by determining the velocities
at the ends of the section at the successive intervals of time. For this
purpose the change in the discharge between the ends of the section
because of the storage and release of water with the rise and fall of the
tide, as developed in subsequent chapters, must be taken into consider-

134

ation as well as the cross section of the channel. In any case the
effect of the correction 6 upon the velocity head may be neglected.

271. Substituting in equation (131) the expressions derived for the
several heads:

h,th,+bAv+ Fv?+b(6—6,) +26Fo=0

and, placing:
h,+th,+bAavitk’?=—R (173)
1t becomes:
b(6—6)) +26Fv—R=0
whence:

5= (6 +R/b)/(1+2Fv/b) (174)

It should be observed that if » and Av were the correct velocity and
its increment for the given time, R would be zero. FR is then the
residual head which 6 is to remove.

272. The computation of 6 from equation (174) is most readily ex-
plained by applying it to the concrete example of the final adjustment
of the currents produced by the average tides in the section of the Cape
Cod Canal between stations 180-+30 and 225, as of September—October
1932. In the computation, these tides are referred to a datum 10 feet
below mean sea level, so that all tidal elevations are positive. At
mean sea level, elevation 10, the hydraulic radius of the section has
been taken as 22.7 feet, and C at 90 (par. 249). From the cross sec-
tions of the canal, the value of 7 at elevation 14 is found to be 24.6
feet, and at elevation 6 to be 20.9 feet. The corresponding values of
C will be taken as 91 and 89, respectively. Since /=4,470, the values
of F=l/C*r are:

Values of F
Tide (y) | Tr (6 F
14 24.6 91 0. 0219
10 PP Eve 90 . 0243
6 20.9 89 . 0270
|

A diagram prepared from this data gives the value of F for any
value of y between 6 and 14.
The coefficient 6, for intervals of a half lunar hour, is, from equation
GiGi):
b=0.0000167 x 4470 =0.0746

The lag of the primary current has been found to be 10°12’, and its
equation, with a lunar transit as the origin of time, to be (par. 249):

y=5.27 sin (mot+124°02’)

273. The computation of the residuals, R, may be made in the form
ulustrated in the following tabulation.

135

RESIDUALS
| ’
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)
. = so at — | | }
| |
t v v= Av Yo uN y iP hs bAv +R | R |
| |
|
0 4.32 | —0.38 | —0.45 11.95 11. 54 11.75 |, 0.0233 | —0. 41 —0. 03 +0. 44 0
5 3. 80 —. 52 —.63 11. 06 10. 76 10. 92 . 0238 —.30 —.05 +. 34 | +.01
1.0 3. 06 =. 7 —.90 10. 09 9. 94 10. 01 . 0243 =, Ie —.07 +. 23 | —. 01
.5 2.01 | —1.05 | —1.30 9.14 9.11 9.13 . 0249 —.03 —.10 +. 10! | +5. 03
2.0 .47 |} —1.54 | —2.04 |] 8.21 8. 35 8. 28 . 0255 +. 14 wo +. 01 0
.5 | —2.06 | —2.53 | —2.08 7.42 Tes! 7. 56 . 0259 +. 28 — 16 =, ili |) Sou
3:0) || —3.69 | —1.63 | —1.19 81 7. 21 7.01 . 0263 +. 40 —.09 —.36 | +.05
* * * * * * *
10.0 4.70 +. 26 +. 21 13. 73 13.18 13. 46 0222 —.55} -+.02 +.49 | +. 04
5 4.85 +. 15 +. 08 13. 63 13. 06 13. 35 0223 tl 0 +. 52 | --..05
11.0 4,85 0 —.08 13. 28 12. 72 13. 00 0225 —. 56 0 } -—+.53 | +.03
Bt) 4.70 =. 1s) =. 21 12.71 12. 20 12. 46 0228 = dL) =102 +. 50 | +. 03
i2.0 4,32 —. 38 —. 45 11.95 11. 54 11.75 0233 = 4H! —.03 +. 44 0
| j |

The time in lunar hours after a lunar transit is entered in column (1)
and the primary current corrected by 7, table X, in column (2).
Column (3) shows the increase in the velocity during the preceding
interval, the entry for 0 hour being repeated from the twelfth hour.
The values of Av, column (4) are the means of the entries for the given
and the following half hours, and are consequently the average of the
increments in v during the preceding and following intervals. The
tidal elevations, y, at the initial end of the section, station 180+30,
taken from the equation of the tide at this station, paragraph 240,
are entered in column (5), and those at station 225 in column (6).
The tidal elevation, y, in the section, column (7) is the average of the
entries in columns (5) and (6). The corresponding values of /’, from
the diagram, are entered in column (8).

The surface heads, h,, column (9) are the algebraic differences,
¥1—Yo, from columns (6) and (5). The entries in column (10) are the
products of Av, column (4) times the constant b=0.0746. The slide
rule affords satisfactory accuracy for these and the subsequent com-
putations. The entries in column (11) are obtained by multiplying
v2, from column (2) by F, column (8) and have the algebraic sign of v.
The residuals, R, column (12) are then the algebraic sum of the entries
in columns (9), (10), and (11), with the sign reversed; the sums of the
entries in columns (9), (10), (11), and (12) being zero.

The computation for hours 3.5 to 9.5 is not shown, but is carried
out by the same procedure. In the example chosen, the tides at
the ends of the section, and consequently the surface head, have in
fact a simple harmonic fluctuation, but the computation would be in
the same form if they had any other repeating fluctuation. The
velocity head is omitted. If computed, it would be entered in an
additional column and included in the derivation of FR.

136

274. The computation of the corrections is completed in the follow-
ing tabulation.

Corrections
(1) (2) (13) (14) (15) (16) (17) (18) (19)
t v R/b 1+2Fv/b | d0+R/b 6 60+R/b 6 vo!
0 +4, 32 0 3 56Q)Pa || Meee 0 Br YE Ga eC
3H i) SESE) 5,78) 3.42 +0. 13 +. 04 st, ili +.05 | +3.85
1.0 +3. 06 —.13 3. 00 —.09 —.03 —.08 —.03 +3. 03
§ | 22.01 +. 40 2. 34 SL 3 +. 16 erate 29) il7/
2.0 +. 47 0 1.32 +.16 =f TOF ll Sle bs ees eee +.59
5) —2.06 —.13 2. 48 —.01 QPP a nies || eee ae —2.06
30 | =a +. 67 3. 60 +. 67 SSOP |lh, cee Seta |e a —3. 50
3 | Saw +. 94 4.16 +1. 13 AES OT Aa Soe Saas eae Siti
A0 | =4,70 +. 80 4.35 +1.07 AES Obit Sevilla —4, 45
.5 | —4.85 +. 80 4,45 +1.05 STIESGY toate le PY yi. = —4 61
G0 | =A.05 +.80 4.49 +1. 04 Uy Phe ko MNS oo oe —4 61
a | Sao) +. 54 4.26 ea78 LENT Saltese Seay? lh crea —4. 52
GO | =coey +.54 3.95 dL Hea kstra| Pater te ee lode ce a —4,14
B | Ske 4b ig 3.53 SL, Bil 2 409)01|| SENS 8 | ei eC Al
iO | = 8.08 aL ig 2.99 4D Oise aoe pee —2.99
36 | 2.0 —.40 2. 28 = 88 —.13 OE A ce —2.14
8.0 SL 0 1.29 =, 13) SEO ee ae =
.5 | +2.06 JE oy 2.26 AL, 7 E108 4| hos 8 2 eee +2.14
9.0 | -+3.69 at, 113 3.23 se Dil SUA Meee Ie +3. 76
15 | (44.44 s6 D7 3. 65 +, 34 S400) a lens: J2e oe |S +4, 53
10.0 | +47 +. 54 3. 80 +. 63 EG (fae (eae eae +4, 87
5 | +485 +. 67 3.89 +, 84 zl EA eal le Uae coll on Se +5.07
11.0 | +4.85 +. 40 3.91 +, 62 S16) hips ee |e +5. 01
6 | ee +. 40 3. 87 +. 56 te aA, | Noe an Te ee eae +4, 84
| iq | +4oe 0 3. 69 +. 14 SER la || Meee PSS +4, 36
| |

The residuals from column (12), divided by 6=0.0746 are entered
in column (13). The divisor, 1+2F%/b, of equation (174) is derived
from the numerical value of v, column (2) and of F, column (8), and
entered in column (14). Since an initial value of 6) is not known,
the computation of 6 in columns (15) and (16) is started with an
initial correction of zero at 0 hour. At 0.5 hour, 6,+A/b=0+.13=
13. Applying the divisor, column (14) the first determination of 6
at 0.5 hour is 0.04. At 1.0 hour 6,+R/b=0.04—0.13=—0.09; and 6
at 1.0 hour is —0.03. A continuation of the process gives 6=0.04
at 12 hours. With this initial value the computation is repeated in
columns (17) and (18) until, at 1 hour, the value of 6 becomes that
previously found. “The adjusted velocities are then found, in column
(19) by applying the final values of 6 to the velocities in column (2).
'The velocities, before and after this final adjustment, are shown in
figure 43.

975. Discussion—The final corrections, 6, developed in the pre-
ceding example, are so small that the corrected velocities do not differ
substantially from those used in their determination, and a recompu-
tation is unnecessary. Since the surface head assumed in their
derivation has a simple harmonic fluctuation, these corrections are
due only to the variation of r and C with the rise and fall of the tide.
Such corrections depend upon the relative timing of the tide and cur-
rent and on the relation between the tidal range and channel depth.
For a given tidal range and channel depth they are the greatest when,

137

o_eSSaeae
Vice al a a P

eee aS ees
EERE EES E EERE
JSe CS Be 2a eee ae es
mi See) ane

Feet per sec.
O

-—5
CURRENT
Afterthinst .cOkreculions—— —
final correction
14

wy eo I ee ee
[REA VAS
MA ON
Se emmieisie” \s\ea Nae
pe dt ee
RA fee AH
CE Aceeee seo
sl tse: CAneos aman ail

Lunar nee after cea

FIGURE 43.—Final adjustment of computed current in Cape Code Canal.

as in the example chosen, the timing is such that the maximum cur-
rents occur at high and low tide. They become smaller as the angular
lag, ¢, of the primary current increases and the frictional resistance
consequently has a less effect upon the currents.

276. The residuals developed in the example chosen amount to but
a few hundredths of a foot. Much larger residuals, and consequently
much larger corrections to the velocity, would be produced if during
the tidal cycle the surface head varied from a simple harmonic fluctua-
tion by as much as a tenth of a foot. As will later be shown, consid-
erable distortions of the surface heads in successive sections of a long
tidal channel may be produced in the filling and emptying of the tidal
prism at different tidal stages, and other distortions are produced by
the diurnal components of the head. If the first computation devel-

158

ops comparatively large corrections to the velocity, the computations
should be repeated with the corrected velocities derived from the first
computation.

277. The relatively small distortions of the tides needed to produce
a large distortion of the surface head and currents is illustrated in
figure 44. The current there shown has the equation:

v=sin mt+sin 3mot
Neglecting the velocity head, the equation of motion becomes:
dy/Oxr-+ (m>/g) (cos mat+3 cos 3mozt) + (sin mot+sin 3m,t)?/C’r=0
Whence:

= (loy/oz)
—— (Im,/g) (cos myt+3 cos 38mg) + (1/C’r) (sin myt+sin 3mof)?.

PNERAdeee

SANE
Qa

we eoebeuee 4b
a=

eT [current | V7) |
o- O ee

SS 6 ie) a es = ee
HEAD

y

Lunar Hour

SS]

Tide at initial end —-——
at other end

Ficuri 44.—Comparative distortions of current and tides.

The surface head, A,, in a section 10,000 feet long, when r=20
and C=90, is plotted in the figure. Taking the tide at the initial end
of the section as a simple harmonic fluctuation with an amplitude of

139

2 feet, the tide at the other end would be distorted only to the extent
shown.

278. Comparison with measured current curves—The form of the
computed tidal current curves derived in the preceding paragraphs
may be compared with curves of measured velocities, and with those
derived by the method of cubature hereafter described. It should be
recollected that the computed curves show the mean velocity at a
cross section of the channel during a tidal cycle. A meter measure-
ment of the mean velocity in a tidal channel is quite a difficult under-
taking, as the velocity in each area of the cross section is changing
continuously, while its fluctuations are not identically repeated from
day to day. Available records often show the velocities only at a
single point in the cross section; but these indicate the characteristic
shape of the current curve. Current curves derived by the cubature
of an estuary show, on the other hand, the mean velocities at the
cross section.

279. The currents in the Cape Cod Canal afford a typical example
of the form of the current curve when the flow is markedly frictional.
The average measured midstream current velocities at 0.3 depth, at
station 225, after the time of a lunar transit, compiled from a series
made by the United States Engineer Department, September 28-
October 6, 1938; and the corresponding mean tide curve in the section
from station 180+30 to station 225, are shown in figure 45. The
velocity curve has, it will be seen, the characteristics of the computed
curve of mean velocities, shown in figure 43. As is to be expected,
the midstream velocities are about 25 percent in excess of the mean
velocities throughout the cross section.

280. The current in the estuary of the Delaware at the head o
Delaware Bay, as determined by a mean cubature made by the
United States Engineer Office in Philadelphia, shown in figure 49,
page 154, affords an example of a typical velocity curve when the
flow is of a less frictional character, and is not greatly modified by
overtides. This curve may be compared with the curve of computed
velocities shown in figure 42.

281. The marked effect of overtides on the currents is illustrated
by the velocity curve at Philadelphia 63 miles further up the Delaware
estuary, determined by the same cubature, and shown in figure 50,
page 155.

282. The even greater distortions of the current in some tidal chan-
nels is illustrated by the curve of measured channel velocities in
Seekonk River, R. I., shown in figure 46, page 141, taken from the
Manual of Current Observations, United States Coast and Geodetic
Survey.

192750—40——10

140

>
ao
+S
- oO
op
Oo Dn
p on o
iS} =
Oo
ww
re
os
Oo re
Se
oO
pb
L¢p)
3
fa
p
a 103
as
rit
3
¢
a
Zo
-4 © 100
Ens
a
3
»
rs
“i 07

Solar Hour after Moon's Transit
ta) 6 2
-+—— 1 4, 1} 4+, +--+, 4
fa) 6 \2

Lunar Hour after Moon's Transit

Average measured midstream velocity, 0.3 depth
Sta.225, Cape Cod Canal, Sept.28 - Oct.6, 1932.

FIGURE 45.

283. Summary.—The preceding formulas and examples show that
the deformation of the primary current because of the variation of
frictional resistance with the square of the velocity is not large unless
the currents are unusually strong and the channel is of moderate depth,
so that the flow is largely frictional. Its deformation because of the
varying channel depth depends on the relation between the timing of
the tide and the timing of the current, as well as on the ratio of the
tidal range to the mean depth in the channel, and usually is quite
small in deep channels. The deformation because of the effect of over-
tides may be quite large. If the ascertained variation in the surface

141

head in a short section of a tidal channel can be closely reproduced by
a simple harmonic fluctuation, and the proper coefficient of friction
selected, the computed primary current affords a fair representation
of the actual currents, but a closer approximation would be secured
by applying the corrections developed in the preceding paragraphs.

24 Flood
a
~~
Se
XE
Ebb.
2
[he Wa SY a MS EE a DT Vs ee ere
0 2 24

Solar hours

FIGURE 46.—Current in Seekonk River, R. I., showing effect of short-period constituents.

LIMITATIONS ON THE COMPUTATION OF CURRENT VELOCITIES FROM THE
OBSERVED HEADS IN A SHORT SECTION OF TIDAL CHANNEL

284. As is well recognized, the velocity in a natural channel cannot
be reliably determined from the observed head even when the flow is
steady. Ina short section of a tidal channel, the computation of the
velocities from the observed heads presents further complications.
These heads are the relatively small differences between the changing
tidal heights at gaging stations at the two ends of the selected section
of channel. Considerable accidental errors are inevitable in taking
off the tidal heights from the somewhat irregular curve produced by a
recording tide gage, and even greater errors in the timed readings of a
staff gage. When these departures happen to be in opposite directions
they produce errors which are large in proportion to the head. The
heads derived from the differences of observed hourly readings are apt,
therefore, to vary so erratically as to afford little basis for a determi-
nation of the velocities. The most workable procedure is to find the
harmonic fluctuations which most nearly represent the actual fiuctua-
tions of the tides at the ends of the section, and to derive therefrom
the corresponding harmonic fluctuation of the head. Obviously, more
consistent results may be secured from average tide curves than from
observations made during 1 day.

285. In a long tidal channel, the heads between the entrances
usually are so large that accidental errors in the observed tidal heights
at these entrances become of minor importance; but in such a channel
the currents may be due more to the storage and release of water in
the tidal prism than to the head between the entrances.

286. In short, a direct measurement of the actual velocities in a
channel, however crude, is more reliable than the most refined calcu-
lation from the varying head and an assumed coefficient of roughness.

142

The relation between the surface head and the velocity in a short sec-
tion of a channel, derived in this chapter, affords, however, a basis for
estimating the currents in a projected long canal, by a procedure which
is developed in detail in chapter VIII.

CURRENTS IN A SHORT SECTION OF CHANNEL WHEN THE
FRICTIONAL RESISTANCE IS NEGLIGIBLE

287. If a channel is so deep, and the current velocities are so small,
that the flow is essentially frictionless (par. 257), the currents pro-
duced in a short section of the channel by any fluctuation of the tides
at the ends of section have a simple relation to the amplitudes and
speeds of the harmonic components of these tides. Designating the
amplitudes of the several harmonic components of the tide at the
initial end of the section as M,’, S.’, ete., and at the other end as M,’’,
S,’’, ete., the equation for the tide at the initial end becomes:

Yo= M2’ cos (met-+ ay’) +S.’ cos (Soft ae’): - -
and at the other end:
Yi= Mb” cos (met a,”)-+S2” cos (Sef- a2”) => > >
The surface head through the section is then:

hs=Yi— Yo M2” cos (met-+a,”) — Mb)’ cos (mzt+ a’)
+5,” cos (Sof + ay”) —S,’ COs (Sof + Qs’)
+ete. (175)

288. Since the respective pairs of components of the same speed
unite into components of that speed, equation (175) reduces to one in
the form:

h,=H, cos (m t+ A,°) +H, cos (sot+ H2°)+: - -

In which the amplitudes, /7,, (7, and the phases 77,°, 72° of the com-
ponent surface heads could be computed by the process indicated in
paragraph 239.

When both the velocity head term and the friction term in equation
(112) are dropped, this equation becomes:

oy/Ox+ (1/g) 0v/ot=0 (176)
Whence:

p=—9 | (2y/d2) ot (177)

143

And in a section of channel so short that the change in slope in
the section is negligible:

oy/Ox=h,/l= (A, /l) cos (m.t+ H°) + (A2/l) cos (sot-+ Ae?) +: - -
=I, cos (mt+H,°)+J, cos (s.¢-+H2°)+° - > (178)

289. The slopes, J;, Jo, ete., in this equation quite evidently approach
definite limits as the length, /, of the section is reduced.
Substituting in equation (177) and integrating:

j= =o | (@ylanat
= — (/ig/me) sin (m2t-+ A,°) — (tog/se) sin (s2t+H2°)—---+K (179)

The constant of integration, K, is readily interpreted as an adven-
titious constant current through the channel, apart from the currents
due to tidal fluctuations, and may be disregarded.

If then the flow in a tidal channel is essentially frictionless, the
velocity of the current at any point in the channel is the resultant of
component velocities with the speeds of the tidal components.

290. The inference should not be drawn from equation (179) that
the amplitudes of the components of the velocity are proportional to
the ratios of the amplitudes of the tidal components to their respective
speeds; for the component heads and slopes, from which the velocities
are derived, are determined by the changes in the amplitudes and
phases of the tidal components at successive points along the channel,
and not by the magnitude of these amplitudes.

COMPONENT CURRENTS

291. As shown in paragraph 289, when the tidal flow is essentially
frictionless the current may be resolved into component currents,
fluctuating at the same speeds as those of the tidal components. If
the flow is not frictionless, each fluctuation of a tide of the semidiurnal
type has been shown to produce a primary current with a simple
harmonic fluctuation, to which minor corrections are to be applied.
The amplitude of the primary current must vary from day to day
with the variation in the amplitude of the resultant tide. The primary
current should then be resolvable into components of fixed amplitudes,
with the speeds of the tidal components. The corrections to the pri-
mary current, and its distortions due to overtides, are repeated almost
identically in each successive tidal fluctuation, and are then reproduced
by overcurrents whose speeds are antes al ignite of the speeds of the
principal tidal component.

144

292. Further minor current components are to be anticipated be-
cause of the variation of the friction term with the square of the
velocity; for, if the primary current components are:

Bsin (mt+ 8), B, sin (s+ ,), and so on, the friction term becomes:
F=-+[B sin (m.f+ 6) +B, sin (s.t+6,)+ ... P/C?r
— + (B?/C’r) sin? (m t+ B) + (B,/C?r) sin? (st+B,) ... (180)
++ (2BB,/C’r) sin (m.t+ B) sin (sot+ B;)

The terms in this expression for F which contain the squares of the
sines of functions of the speeds my, s:, etc., afford components of the
friction term with speeds of the corresponding harmonic components,
and their overtides. The terms which contain the products of the
sines of functions of these speeds may be replaced by the algebraic
sum of the proper trigonometric functions of the sums and difference
of the angles, and hence of the speeds. Components of the friction
term, and corresponding components of the current, with speeds which
are the sums and. differences of the speeds of the principal tidal com-
ponents, may therefore be anticipated. These may be termed
compound current components.

293. The currents set up by tides of the mixed or of the diurnal
types should equally well be resolvable into components with the
speeds of the harmonic tidal components, together with overcurrents
and compound current components. Furthermore, in the propagation
of the tide through a long channel, the overcurrents and compound
currents may create corresponding overtides and compound tides.

294. The mathematical relation between the components of the tide
and. the componenets of the current, when irictional resistance must
be taken into consideration, does not appear to offer a profitable field
for investigation; but, as explained in chapter X, the component cur-
rents may be determined by an harmonic analysis of the observed
currents in a channel.

Cuapter VI

CONTINUITY OF FLOW iN LONG TIDAL CHANNELS

Paragraphs
PERT Cer pen pote) eee ee MR TN nh eeyals 09 Aaah Re al oe 285
Pau rae UeCiecOmbilUltyes: Be Yea oy ee ee a hes Ss ee 296-299
DRAPE SRD GUEST CTO eae ee eae ey iver Oc 300
LED TIA DL Se ees Sea ae a unt ee a ee Nee Sobel eae Ce 301-302
Ean aiy Chiles = ie. stat eeteee ce. pts sg EES on aN es 303-306
BE rae ee eee PS Ba eee Re oe ee Le eS 307-310
eruoeoacischarce andy velocities. 2-22... 2) ae 311-314
LT CSU 2 a a a gk vet SN | 315.

DEFINITIONS

295. A few definitions may simplify the ensuing discussion:

The tidal prism of a channel is the prism between low water and
high water.

A long tidal channel is one of such length that the fillimg and empty-
ing of the successive sections of its tidal prism affects, more or less
profoundly, the tidal currents and the tidal heights through the
channel.

A connecting tidal channel connects two tidal seas. In a long con-
necting channel the tides and tidal currents through the channel are
caused both by the surface head between the tides at the entrances
and by the storage and release of water in its tidal prism. As a special
case, a connecting channel may join a tidal with a tideless body of °
water.. A natural connecting channel is usually termed a strait,
and a short connecting channel leading from the ocean to a tidal or
tideless bay or sound is termed an inlet.

A closed tidal channel leads inland from a tidal sea and terminates.
in a dead end. Its tides and currents are due solely to the filling
and emptying of its tidal prism, together with the discharge of any
flow which may enter the channel from the uplands.

A tidal canal is an artificially excavated tidal channel of regular
dimensions.

EQUATION OF CONTINUITY

296. Equation of continuity for steady flow.—Let:

X be the area of a cross section of a channel,

@ the quantity of water passing through the cross section in
a unit of time; designated as the discharge at the section.

v the mean velocity of the current at the section. Then
obviously:

OLX (181):
(145)

146

In steady flow, Q is by definition the same constant at all cross
sections, and equation (181) affords a complete expression of the con-
dition of continuity of flow.

297. Equation of continuity for tidal flow.—In tidal flow, water is
stored and released throughout a channel as the tide rises and falls,
and @ therefore varies from section to section as well as varying
at each section with the time.

Let So, figure 47, be a cross section of a tidal channel at a distance
x from the point chosen as the origin of distance, and Sj, an adjacent

FIGURE 47.

cross section at the elementary distance dz from Sj. At section So,
and at the time ¢, let:

z be the surface width of the channel,

X the area of the cross section,

D=X/z its mean depth,

y the elevation of the water surface above any assumed hori-

zontal plane of reference,

oy/Ot the rate at which y is increasing with the time,

v the mean velocity in the cross section,

Q the discharge.

The volume of water passing S) during the elementary time interval
dt is then Qdt. During the same interval the water surface between
So and S;, rises the distance (Oy/0t)dt. The volume of water passing
S; during the interval is then decreased by the contents of the prism
whose width is z, whose length dr and whose height is (dy/Ot)dt.
Designating the rate of decrease in discharge with the distance as
— 0/0x, the decrease in the discharge in the distance, dz, between
the sections, is —(0Q/0xr)dz, and the decrease in volume of water
passing section S,; in the time dt is —(0Q/0zx)drdt. Obviously, there-
fore:

— (0Q/0xr)drdt= zdx(Oy/Ot)dt
whence:
0Q/or-+ 2dy/dt=0 (182)

147

Equation (182) is the general equation of continuity in a tidal
channel.
298. Since Q=vX=vzD, equation (182) may be written:

O(veD)/d0x+ zoy/0t=0 (183)

If a channel is of both constant width and constant depth below
mean tide level, and the tidal fluctuation is so small with respect to
the depth that the variation in D may be neglected, equation (183)
becomes:

2Dov/dx+ zoy/ot=0
or:

Dov/dx-+ oy/ot=0 (184)

299. Distinction between mean depth and hydraulic radius.—Chan-
nels, whether natural or artificial, are usually so wide with respect to
their depth that, if the tide does not overflow the banks of the channel
proper, the mean depth, D, in the equation of continuity does not
differ materially from the hydraulic radius, 7, in the friction term of
the equation of motion. On the other hand, if the channel is bordered
by tide flats and sloughs, in which water is stored and released as the
tide rises and falls, but which carry no appreciable current, the value
of D may be much less than the value of 7. In other words, D is
computed from the gross width of the channel and r from the net
width after deducting areas which carry no substantial flow.

CUBATURE OF A CLOSED CHANNEL

300. Method of cubature —The currents in a closed tidal channel are
caused by the filling and emptying of the tidal prism, and by the fresh-
water discharge from any rivers and streams which may enterit. By
taking simultaneous readings of the height of the tide at a sufficient
number of stations between a given tidal station and the head of tide,
the changes in the volume of water in the tidal prism from hour to
hour, or at shorter intervals, may be computed, and the positive and
negative discharges at the station due to the filling and emptying of
the tidal prism ascertained therefrom. The total discharge is then
the algebraic sum of the tidal discharge and the measured or estimated
upland discharge. The mean velocity at the station at any given
time may be determined by dividing the total discharge by the area
of the cross section at the station at that time. This process is termed
the cubature of the channel. It is essentially the arithmetic integra-
tion of the general equation of continuity.

301. Basic data—The tidal stations established for a cubature
should be spaced at such distances that no material error is intro-
duced by taking the water surfaces between them as planes. This

148

condition ordinarily will be met by stations spaced some miles apart,
but suitably placed with respect to marked changes in the cross section
of the channel. The stations used in the cubature of the Delaware
River by the United States Engineer Office at Philadelphia are as
follows:

Distance from Distance from

head of tide in head of tide in

Station statute miles Station statute miles

1. Trenton municipal pier _ __ __ ORAS 3 ME Shortie Mathie ee ee 41. 67
20 Mrenton marine terminaias =o) 199) | SZ ae BS aildsyins 48, 45
3, Bordlemionas 2o.—-2.53-2--- =~ 5V AO) Sy Marcus h00 k= === 53. 83
AL IMC OCRO st aoe Se 6525) 4 shd¢ee Mocrs= 60. 42
Diet] OREM CEL eee eyo oa ee ee HOSS: New Castles ===. == 67. 52
Gay Burling tone = 2.2 ena 15. 32,16, Reedy Pomt: 2 === 75. 15
Hee DEVEL Vee oe eee es eee SAGO) Alive ckecdy lislandas =e 78. 88
Seorresd ale: sens a eee 23. 58) 18. Artificial Island _______—_= joe sehr
Qe ul) Slants pn ney eee A ed 29. 11|19. Woodland Beach__—__--___- 92. 48
1@, oilkeyooiie 2 eee 39-.20))/20) Ship Johni=. 2. 222s 97. 33

It may be necessary to establish as well tidal stations on any long
tidal tributaries which enter the channel. For the convenience of the
computations the tide of all stations should be referred or reduced to
the same horizontal datum, preferably taken so low that all tidal
heights are positive.

A reliable contour map is needed to show the tidal areas from low
water to high water and measurements must be made of the cross
sections of the channel at the stations where the velocities are to be
determined.

302. A tidal channel whose cubature is to be made usually is the
tidal portion of a river with a considerable drainage area. In the
United States, gaging stations with established rating curves have
been established above the head of tide on most rivers, and the upland
discharge of the main stream, and of any important tributaries which
enter the tidal section can be ascertained therefrom. If satisfactory
rating curves have not been established, meter measurements should
be made, at suitable stations above the head of tide, of the upland
inflow from the main stream and any important tributaries entering
the tidal section. The discharge from other drainage areas into the
tidal channel, including those below the gaging stations, is relatively
so small that it ordinarily can be derived with sufficient accuracy by
estimating, from general data, the run-off per square mile.

303. Selection of representative tides for cubature-—The process of
cubature would become an overwhelming task if repeated through the
tides occurring during a considerable number of days. If the tides
are of the semidiurnal type, with no great variation in range during
the month, the cubature of the tides on a single day, chosen almost
at random, will develop the characteristic fluctuations in the discharge
and in the velocity at stations along the channel. The effect of the

149

diurnal inequality may be ascertained by extending the cubature
through two semidiurnal tidal cycles; 1. e., through a period of 25
hours. If the difference between spring and neap tides is large, a
cubature might be made of a representative tide of each kind. If the
tides are of the mixed or diurnal types, a cubature of a representative
tropic tide and of a representative equatorial tide would be necessary
to determine the characteristic currents produced by each.

304. Average tide curves.—A cubature based on average tide curves
gives a better general picture of the discharges and currents in a tidal
channel than one based on the tides during a single day. Cubatures
prepared from average tide curves before and after a major change has
been made in a channel afford a conclusive determination of the effect
of the change upon the tidal discharge and currents. Average curves
of tides of the semidiurnal type may be prepared by averaging the
tidal heights, taken from the graphic record of an automatic tide gage,
at hourly or half-hourly intervals for the 12 hours beginning with the
time of each lunar transit. The observations should extend over a
period of 15 or 29 days, or a multiple of the latter. A consideration
of the principles of harmonic analysis, explained in chapter II, indi-
cates that an average curve so prepared is substantially that of the M2
component of the tide and its overtides.

Average curves of spring and neap tides may similarly be prepared
by averaging the recorded tidal heights at hourly or half-hourly inter-
vals after the lunar transit immediately preceding the times of spring
and neap tides respectively; and average-curves of tropic or equatorial
tides by averaging the heights at the same intervals for a period of 25
hours after the lunar transits next preceding the times of such tides.
Obviously, a long continuous record of the tides at each of the stations
must be available to prepare good averages of tides which occur but
twice a month.

305. Composite curves of mean tidal fluctuations.—The range of an
average curve of all semidiurnal tides, prepared by the process out-
lined in the preceding paragraph, is less than the actual mean tidal
range during the period. For the mean cubatures of the Delaware
River made by the United States Engineer Office at Philadelphia, tide
curves were prepared by computing, by the ordinary methods, the ele-
vations and lunitidal intervals of mean low and high water, and con-
necting them with a composite curve derived from 10 recorded tide
curves whose range, duration of rise and fall, and half-tide level were
nearly the same as the range, duration of rise and fall and half-tide
level of the mean tide. The composite curve is prepared by adjust-
ing, proportionally, the duration and height of the rise and of the
fall of each of the recorded tides to the mean duration and mean rise
and fall, and averaging the results. For this purpose the periods
from low water to high water and from high water to low water on

150

each of the recorded curves are divided into, say, 10 equal intervals,
the recorded rise or fall during each interval is multiplied by the
ratio of the total mean rise or fall to the total recorded rise or fall, and
the results added successively to the computed elevation of low water.
The tides at the proportional intervals are then averaged, and plotted
at the corresponding mean intervals. The tides at semihourly inter-
vals after a lunar transit then may be taken off the plotted curve.
The result is a tide whose high water and low water are at the times
and elevations of mean high and low water, and whose semihourly
rates of rise or fall are the composite of those of the selected tides.
This composite tide curve has a total period, from high water to high
water, or from low water to low water, of half of a mean lunar day,
12.42 mean solar hours.

306. Similar composite curves of spring or neap tides could be pre-
pared by adjusting a number of tides near the time of spring or neap
tides to the computed times and elevations of mean low and mean
high water of spring tides; and composite curves of tides of the mixed
type by similarly adjusting suitable recorded tides to the times and
elevations of mean lower low, higher low, lower high, and higher high
waters. It may be observed that the sum of the durations of the rise
and fall of spring and neap tides differs slightly from the mean lunar
half day or day.

307. Computations —Designating the successive tidal stations along
the channel, beginning at or near the head of tide, as station 0, station
1, station 2- - - station JN, let:

Yor Yt) Yoo + * * Yn be the heights of the tide at these stations at
the time ft, this time usually being on the hour and half hour.

At, the time interval used in the computations, usually \% hour,
or 1,800 seconds.

Yo’, Yr’, Yo’ + + + Yn’, the tidal heights at the time ¢+ Af.

U,, U2, U; - - - U,, the mean area of the water surface between
stations 0 and 1, | and 2, etc., during the time interval
between ¢ and ¢-+ At.

Ay;, Ay2, Ay3 - - + Ayn, the mean rise in the water surface
between the successive stations during the same interval.

AV,, AV2, AV; - - » AV,, the algebraic increase in the volume
of water between the successive tidal stations during the
same interval.

Then evidently

A Wo U,Ay,, AV= U,Ay2, a suey tr, AV = (OLIN

If the stations are sufficiently close together, the mean rise in the
water surface between any two stations during the time interval Af
may be taken as the increase in the mean elevation of the tides at
the two stations during that period so that:

151

AV,= UI (y’o+y'1)/2— (Yoty:)/21,
AV2= Uf (y’:+y’2)/2— (yit+ye)/2], cte. (185)

The total increase in the volume of water in the tidal prism from
the head of tide to any station, N, is then:

SAVE AVA Aetna = AV (186)

This summation obviously should include the increase in volume in
any long tidal tributary above station N which is separately cubatured.
Taking the origin of distances at the entrance to the channel, the
tidal discharge and the velocity at a given station are positive during
the tidal flood currents, when the total volume of tide water between
the station and the head of tide is increasing, and negative during the
ebb, when this volume is decreasing.
The mean tidal discharge, Q,, at station N, during the time interval
At, is then:
Q,=ZAV/At (187)

The fresh water discharge, Q;, at the station is similarly the sum of
the fresh-water discharges entering the channel above that station.
This discharge may be regarded as constant during the period of cuba-
ture. Since it is an outward discharge, it is intrinsically negative.
The total discharge. Q, is then

Q=0.-Q,; (188)

Designating the mean area of the cross section at station N during
the interval At as X, the mean velocity during this interval is:

v=Q/X (189)

308. The mean areas U,, U2, etc., of the water surface between the
tidal stations during the successive intervals may be derived by tak-
ing off with a planimeter, from a map of the waterway, the areas at
successive stages of the tide, and constructing a diagram from which
the area at any elevation may be read. Ordinarily, it is sufficient to
take off from the map the areas at high and low water and to join
them on the diagram with a straight line. The area at each semi-
hourly interval is read from the diagram at the mean elevation of
the mean of the tides at the ends of the section.

The areas between the stations should include any tidal tributaries
which enter the section, and should extend to the head of tide in these
tributaries, unless tidal volumes in the tributary are cubatured from
stations thercon. They should include also the effective storage area
in any tidal marshes adjacent to the channel.

152

The cross section areas, X, at the tidal stations at which discharges
and velocities are to be computed, similarly may be read from a
diagram, constructed by taking off with a planimeter the area of the cross
section at high and low water, and at intermediate elevations if necessary.

309. Form for computations —The computations proceed from the
head oi tide downstream. A convenient form, developed for the
cubature of the Delaware River, is shown in figure 48. To briefly
illustrate the process, the computations for six half-hourly intervals
only in the first two reaches below the head of tide and in the lowest
reach, are entered on the same sheet. In the actual computation a
separate sheet is used for each successive reach between the tide
stations. The computation shown for the first reach is abbreviated
as explained in paragraph 310.

The times and the heights of the tides at the upper and lower sta-
tions, selected for the cubature as explained in paragraphs 303 to 306
are entered in columns (1), (2), and (3), and the mean oi columns (2)
and (3) is entered in column (4). Column (5) designates the interval
to which the entries in the succeeding columns apply. Column (6)
is the mean of the given and preceding entries in column (4) and is
therefore the mean elevation, during the interval, of the mean tides in
the section. Column (7) is the surface area in the reach at the cleva-
tion shown in column (6). Column (8) is the algebraic increase in
the entries in column (4) during the interval. The product of columns
(7) and (8) is the value of AV for the interval (equation 185); entered in
column (9). The increases, during the interval, in the tidal volumes
of any separately cubatured tidal tributaries which enter the reach are
inserted in columns (10) and (11). The total tidal volume, column
(12), is the sum of columns (9) to (11). The total increase in volume
during the interval in the upstream reaches, as previously computed
for these reaches, is entered in column (13). The addition of the
increase in the reach, column (12), gives the total increase, ZAV, at
the lower station (column 14). The division by At=1,800 seconds,
gives the mean discharge during the period, column (15), and the
addition of the fresh-water inflow (with the negative sign) give the
total discharge, column (16). The mean elevation of the tide at the
lower station, column (17), is the mean of the given and preceding
entries in column (3). The corresponding area of the cross section at
this station is entered in column (18); and the mean current velocity
during the interval, column (19), obtained by dividing the entries in
column (16) by those in column (18).

310. Because of the steady increase in the width of nearly all natural
tidal channels from the head oi tide to the entrance, the increases in
the tidal volume between the stations near the head of the estuary are
relatively very small. The upstream station may therefore be placed
below the head of tide, and the successive values of AV at this

153

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SIILIDONTAA ANY S3AbYUVHISIG

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154

station determined by multiplying the area U, between this station
and the head of tide, by the half-hourly increases in the tide at the
station, as shown in tabulated computations in figure 48.

311. Graph of discharges and velocities —The fluctuations in the dis-
charge and in the velocity at the successive tidal stations during the
tidal cycle, and the relative importance of the tidal and fresh-water
discharges, are made apparent by plotting the tidal heights, discharges

ee
[ol econ | ee
(Ae

Seca

L£

1 ' = + a

a0 a ae ere

° (9) ° O
te} o

f=) oO

° 9 ie) tS) s

VELOCITY IN FEET PE,

Di stHarce 1n 1/0005 of SECOND FEET

x
qa
to)
fe)
6
z

MEAN SoLraR HoOurR

At
wre

N
fo)
o

a
S
(=)

S|
ae
N
8
Q
t
Wy
me

Ceoss SECTION IN 10005 OF 5a.F7

500
FIGURE 49.—Discharges and velocities at Ship John Light, Delaware Bay, from mean cubature.

and. velocities through the tidal cycle. The discharges and velocities
at Ship John Light, near the head of Delaware Bay, and at Phila-
delphia, 64 miles up the estuary of Delaware River, as derived from
the mean cubature of this channel, are shown in figures 49 and 50.

312. The tidal discharges computed in the cubature of a channel
(column 15 of fig. 48) are the mean discharges during successive half-
hour intervals. These are shown by the stepped lines in figures 49
and 50. The curve of instantaneous tidal discharges is then so drawn
that the area under the curve during each time interval is the same as
the area of the rectangle of mean discharge for the interval.

The fresh-water discharge is plotted as a horizontal line above the
zero line of tidal discharges. Since the fresh-water discharge is flowing

155

outward, and therefore intrinsically negative in sign, the difference
between the ordinates of the instantaneous tidal and the fresh-water
discharge is the total discharge at the instant. These total discharges
are then the ordinates measured from the line of the fresh-water
discharge.

The area of the cross section at the Ship John at the heights shown
by the tide curve is plotted on the diagram for the station. By taking

+|50

oh
°
co)

+
G
o

le)

'
6)
°

at
(s)
co)

VeLocsry in FEET PER SECOND

-150

D1sCHarRGE 1 1000's 0F SECOND FEET

REE
(SABES

Bier ie abe VE aS ake bo
fo
a NN :
CESS e aaa eee

FIGURE 50.—Discharges and velocities in Delaware River at Philadelphia, from mean cubature.

FEET ABOVE DaATU/7
&

off from the diagram the total discharge and the cross sectional area at
a given instant, a refined value of the current velocity at that instant
may be computed.

313. An examination of these curves shows that at Ship John Light
the average duration of the rise of the tide is about 1 hour less than
the duration of the fall; and that the duration of the ebb current is
nearly 45 minutes longer than the duration of the flood. At Phila-
delphia, the average duration of the rise of the tide is about 2 hours
less than the average duration of the fall; and the duration of the ebb

192750—40-—11

156

exceeds the duration of the flood by more than 2% hours. A study of
the figures shows that the differences in the durations of the flood and
ebb currents is to be ascribed principally to the larger areas of the
cross sections of the river during the flood; because of which the tidal
prism is filled in a shorter period. The fresh-water discharge evi-
dently is insufficient to have any large effect upon the tidal flow
except in the upper reaches of the river.

314. Other characteristics of the flow in the estuary, such as the
relative timing of the tides and currents, the average and maximum
discharges and currents, are quantitatively and definitely brought
out by the diagrams at the successive stations. The total volumes
of the inflow and outflow at the stations during the tidal cycle are
readily derived by measuring the areas between the instantaneous
discharges and the line of fresh-water discharge.

315. Conclusion —The cubature of a tidal channel affords complete
and reliable data on the discharge and mean velocities at successive
stations along the channel. It is perhaps the only means by which
a satisfactory determination of the discharge in the wide sections in
the lower part of an estuary may be secured. On the other hand the
cubature of a long tidal channel is a costly undertaking. It affords
no information on the distribution of the velocities in a cross section
of the channel, or of the distribution of the flow through the channels
on either side of islands and through other secondary channels.
Direct measurements of the current velocities in the ship channel of
an estuary are of far greater value to navigators than most refined
computations of the mean velocities throughout the entire cross
section, and are more readily made. The proper design of training
works also may depend principally upon the distribution of the
velocities in the cross section. For these reasons, extensive cubatures
of tidal channels have not often been made. Nevertheless the com-
plete and convincing data afforded by a detailed cubature of the tides
in a channel is of such value in the planning of works dependent upon
the discharges and velocities that its cost is fully justified when major
works of this character are under consideration. “Thus the applica-
tion of the principles of cubature to the estuaries of the Sacramento
and San Joaquin Rivers in California afforded information essential to
the study of a proposal to construct, at great cost, a barrier dam to
prevent the intrusion of salt water into the lower reaches of these
rivers. The cubature of the Delaware River has corrected misconcep-
tions of the influence of fresh-water discharge upon its tides and
currents, and has contributed to the measures by which the large
expense of maintaining the ship channel in the river has been greatly
reduced.

In summary, a cubature of an estuary affords much desirable infor-
mation, but is not warranted unless the information is worth its cost.

Cuapter VII

| FRICTIONLESS FLOW IN A LONG CANAL OF
UNIFORM DIMENSIONS

Paragraphs

Equations of a component of the tide and current ina canalif flow were
ITHPTR@ UBIO 1d ASS, LG SR ea ee eee ep Ace eee 316-324
ANDDIICATCIN TO). Comins Cine l= = a oe ee ee eee 325-329
Imnstamcanecoussproniles and wave lengths] 222 = === 2-2 2s Pe 326-327
Tides and currents at middle of a connecting canal_____-__----------- 330-331
SS RLDL CRSA eae es eS a ee ee a ee ree oe Se eee: OE ae 332-333
ROM SMEBIOMUN MENG CSE ee see sree oe a, el oS Lee oe 334
ROS CImE REIN Seg metre ols Ae ne a eB Le a ee ee ee es 335-337
Progressive, retrogressive, and stationary waves -_-_-------------~---- 338-345
GiqiemMensthicwest oes 28 wk ee ee ee ee ee eee Sa ee 346-347
i CEI MEATC OSCONCAD EA oe 52 = Syn ne tee See Sen ee ee 348
Shallowayater coMmponentSs=s2. 2-28 2 lo 6 a ee see ee ee eee 349
SOUTER BSS os cis aE ae Fa ae ey pete PE 350

316. Frictional resistance to flow must, in fact, be considerable in
the deepest artificial channel that can be conceived of, if the currents
are sufficient to be of any consequence; but the inclusion of frictional
resistance imposes insuperable limitations on a general analysis of
the flow in a long tidal canal. An analysis of the tides and currents
that would be created by frictionless flow in a long canal of uniform
cross section, while not affording a quantitive determination of the
tides and currents in an actual canal, develops certain general char-
acteristics of the flow in such canals, and affords a background for the
procedure, explained in the next chapter, for computing the actual
tides and currents. In this analysis of frictionless flow, the currents
are considered to be so moderate that the velocity head term of the
general equation of motion also may be dropped; and the channel so
deep with respect to the tidal range that the variations in the mean
depth of the channel, as the tide rises and falls, may be disregarded.

317. The tide at any station in long tidal canal, whether connecting
or closed, may be presumed to be the resultant of semidiurnal and
diurnal harmonic components of the speeds established in chapter IT.
The height of the tide above mean sea level at the time ?, at a station
distant x from the origin of distances, is then:

y=M2 Cae (mot+ a;) +S, cos (Sef-+a2)+: ° : (190)
(157)

158

The amplitudes, Mz, S:, etc., and the phases a;, a2, ete., of the
several components may vary with the distance z, of the station from
the origin. This origin is conveniently taken at either entrance to a
connecting canal, or at the single entrance to a closed canal. The end
so chosen will be termed the initial end.

Expanding the cosines in equation (190):

y—M, cos a cos mat— Mp sin ay Sin Mgt+S; COS az COS Sof—
Ss sin (eo) sin Sot +: < he (191)
Equation (191) may be written:

y= X, cos mt+ Y; sin mot-+X2 cos s,t+ V2 sin sof - - (192)

In which X,, Y,, X5, Yo, etc., are functions of z.

318. When the flow is frictionless, the current is likewise the result-
ant of component currents having the speeds of the harmonic com-
ponents of the tide (par. 289).

The velocity at the time ¢ is then given by an equation in the form:

v=V, cos mt+Z, sin m:f+ V2 cos st-+Z, sin sof: - - (193)

in which V,, Z, V2, Zs, etc., are similarly functions of z.

The form of the functions, Y, Y, Z, and V necessary to satisfy the
equation of fluid motion for frictionless flow, and the equation of con-
tinuity, is then to be determined.

319. When the frictional and velocity head terms are omitted the
equation of motion is (equation 176):

oy/Or+ (1/g) 0v/0t=0

And in a channel of uniform width and depth, with a tidal fluctua-
tion small in comparison with the depth, the equation of continuity is
(equation 184):

dy/Ot+ Dodv/dor=0

Substituting the differential coefficients derived from equations
(192) and (193), the equation of motion becomes:

(OX,/dxr)cos mot+ (O0Y;/Ozx)sin mot+ (OX2/dzx)cos sof
+ (O0Y,/0zx)sin st+ - > > —(m:Vj/g) sin mt+ (m2Z,/g)cos mot
— (s)V2/g)sin set (S:Z2/g)cos st— - - - =0

or:

(ONX,/0r+m,Z,/g)cos mzt+ (0 Y;/0z—m;V;/g)sin mot
+ (0X2/Oxr-+-s2Z2/g) cos sof-+ (0 Y2/0x—s_V2/g)sin st * + - =0 (194)

159

And the equation of continuity becomes:

—m,X; sin mot+m,Y;cos mot—s,X2 sin sot+s,¥>, cos st+ -- }
+ D(OV,/dzr)cos mat+ D(0Z,/dxr)sin myt+ D(OV2/dxr)cos sot
+D(0Zs/Oxr)sin sot - - + =0
or:
(DOoV,/Or+m:Y;) cos mot+ (DOdZ,/or—m,zX,) sin mot

+ (DOoV,/dx-+s, Y2)cos sot + (DOZ,/O0xr—s,X2)sIn Sof + ett at &——(() (195)

Equations (194) and (195) are satisfied by all values of f, if:
oX,/d0r-+m,Z,/g=0 0.X,/0r-+s.Z,/g=0, etc.
OY,/0r—m,V,/g=0 0 Y,/0r—s.V2/g=0, etc.
DoV,/o0rz+m,Y,=0 DOoV,/0x+s2¥2=0, ete.
DoZ,/o0r—m2X,=0 DOoZ,/O0x—s,X,=0, etc.

320. Expressions for the components of the tide —An examination of
these equations shows that the variable coefficients for each of the
tidal and current components are related by the equations:

OX /dr+aZ/g=0 (196)
OY /orx—aV/g=0 (197)
DoV/ox+aY=0 (198)
DoZ/or—aX=0 (199)

in which a@ is the speed of the component.
From equations (196) and (197)

Z=— (g/a)0X/0x V=(g/a)0Y/0x (200)
whence:
OZ/0x= — (g/a) 0?X/Ox? OVi0r— (Gi @oP (0x7 (201)
Substituting these expressions in equations (198) and (199):
0? Y /02?+- (a@?/gD) Y=0 (202)
NX/02?+ (a/gD)X=0 (203)

Evidently the solution of equation (203) will afford also the solution
of equation (202).

Placing for convenience, gV=c’, and multiplying the termsin equa-
tion (203) by 20.X/d2 this equation becomes:

2(0.X/Ox) (07.X?/Ox?) +2 (a?/c?) XOX/Or=0 (204)
The integration of which gives the equation:
(0.X/0x)?+ (a?/c?) X?= K? (205)

in which K? is a constant of integration. .

160
From equation (205):

OX/+/K2— (aX/c)?= dx (206)

The integration of which gives:
sin [(aX/c)/K]=az/c+FR’, (207)
in which K’ is a second constant of integration.
From equation (207):
X= (cK/a) sin (ax/e+K’) (208)
This expression for X may be placed in the form:
X=WM cos (az/c)+WN sin (az/c) (209)

in which M and WN are constants.
’ Since the differential equation (202) for Y is the same as that for_X,
the expression for Y is similarly:

Y=P cos (az/c)+Q sin (az/c), _ (210)

in which P and Q are constants whose values are independent of those
of M and N.

The height of a component of the tide in the canal at a station dis-
tant x from the origin of distances, is then given by an equation in
the form:

y=[M cos (az/c)+N sin (az/c)] cos at
+[P cos (az/c)+Q sin (az/c)] sin at (211)

321. Expressions for components of the current—The component of
the current due to the same component of the tide is, from equation
(193):

v=V cos at+Z sin at (212)

From equations (200) and (210):

V=(g/a) 0Y /Ox= (g/a)[— (a/c) P sin (az/c)+ (a/c)Q cos (az/c)]
= (g/c)[Q cos (az/c)—P sin (az/c)] (213)

And from equations (200) and (211):

Z=— (g/a)0X/dx=— (g/a){[— (a/c) M sin (az/e) + (a/e) N cos (az/e)]
=—(qg/c)[N cos (az/c)—M sin (az/c)] (214)

161

The component of the current at a point distant x from the origin is
then:

v=(g/c)[Q cos (ax/c)—P sin (az/c)] cos at

—(g/e)LN cos (ax/c)—M sin (az/c)] sin at. (215)

The constants VM, N, P, and Q in the equation of the current are the
same as those in the expression for the corresponding component of
the tide.

322. Determination of the. constants—The constants in equations
(211) and (215) may be determined from the amplitudes and phases
of the corresponding component of the tide at the two ends of the
canal.

Let L be the length of the canal, Ay the amplitude and a the phase
of the component at the initial end, and A, and ,a, the amplitude and
phase of this component at the other end.

The equation of the component tide at the initial end is then:

Yo= Ay cos (at+ ap) = Ay cos at cos a—Ay sin at sin a (216)
and at the other end:
y= A, cos (at-+a,) =A, cos at cos a,— A, sin at sin a (217)
At the initial end, x=0, and equation (211) becomes:
Yo= M cos at+P sin at (218)
Since this equation must be identical with equation (216)
M=A), cos a (219)
P=— Ay sin a (220)
At the other end of the canal, r=Z and equation (211) becomes:

4¥:=[M cos (aL/c) +N sin (aL/c)] cos at
+[P cos (a L/e)+Q sin(aL/c)] sin at

Therefore:
M cos (aL/c)+N sin (aL/c) =A, cos ay (221)
P cos (aL/e) +Q sin (aL/c) =— A, sin ay (222)
It will be convenient to place:

aL/c=y (223)

162

It may be noted that y (gamma) is an angle which is measured in
radians, if a@ is expressed in radians per second, or in degrees, if a is
expressed in degrees per second:

Substituting the expressions for M/Z and P, from equations (219) and
(220) in equations (221) and (222), and solving for N and Q

N= (A, cos a,;— Ap COS a cos y)/sin + (224)
Q=— (A, sin a;— Ap sin ay cos y)/sin (225)

323. Component tides.—Substituting the expressions for the con-
stants found in the last paragraph, equation (211) becomes:
Y= [Apo cos ap cos (az/c)
+ (A; cos a,— Ap COs ay Cos y) sin (axz/c)/sin y] cos at
—|[Ap sin a) cos (ax/c)
+ [(A; sin ay— Ay sin ap Cos y) sin (ax/e)/sin y)] sin at
={Ap COS ay [sin y cos (ax/c)—cos y sin (az/c)]
+A, cos a sin (az/e)}cos at/sin y
—{Apo sin a)[sin y cos (az/c)—cos y sin (az/c)]
-- A, sin aq sin (az/c)} sin at/sin >
=[A) cos ay sin (y—az/c) + A, cos a sin (az/c)| cos at/sin
—[Ap sin ap sin (y—az/c)+ A, sin a, sin (az/c)| sin at/sin +
=[(A)y cos ap cos at— Ay sin ap sin at) sin (y—az/c)
-++ (A, cos a, cos at— A, sin a, sin at) sin (ax/e)]/sin
= A) cos (at+ ao) sin (y—az/c)/sin
+A, cos (at+a,) sin (az/c)/sin y (226)

Since, from equation (223):
ax/e= (a/L)y (227)
equation (226) also may be written:

y= Ap cos (at+ ao) sin (1—a/L)y/sin y
+A, cos (at-+a;) sin (2/L) y/sin vy (228)

324. Component currents —The equation of the corresponding com-
ponent of the current, obtained by substituting in equation (215) the
same expressions for M/, N, P, and Q, similarly reduces to:

v= (g/c) Ap sin (at+ ao) cos (y—az/c)/siny
— (g/c) A, sin (at-+a,) cos (az/c)/sin y (229)

And this equation may be further transformed into:

v= (g/c) Ap sin (at+ a9) cos (1—2/L) y/sin
— (g/c) A; sin (at+a,) cos (2/L)y/sin (230)

163

325. Computation of tides and currents produced by frictionless flow
in a connecting canal.—The tides and currents in a connecting canal
are determined by the known amplitudes Ay and A,, and initial phases
a and ay, of the several components at the two entrances to the canal.
As shown in paragraph 239, equation (228) may be reduced to the
form: —

y=A cos (at+a)

by placing:
A cos a= Ay Cos a sin (L—2/L)y/sin y+ A; cos a sin (x/L)y/sin y (231)
A sin a= Ap sin op sin (1—2/L) y/sin y+ A, sin a sin (2/L)y/sin y (232)

_ The value of y in degrees, for any component of the tide, is given
by equation (223):
y=aL/e=aL|\gD

in which ZL is the length of the canal, D its mean depth, and a the
speed of the component, in degrees per second. Thus the value of a
for the M, component is 28°.9841/3600=0°.00805.

The initial phase, a, and the amplitude A of each component of the
tide at a point distant « from the origin of distances may be deter-
mined from the values of A cos a and A sin a, equations (231) and
(232).

The equation of the @ component of the current in the canal at a
point distant « from the origin, equation (23C), may be reduced to
the form:

v=B sin (at+ B)

by a similar procedure.

326. Instantaneous profiles and wave lengths —The longitudinal sec-
tion of the water surface in a long tidal channel at any instant is a
curve designated as the instantaneous profile. The instantaneous
profile of a component of the tide at any time, fo, in a long connecting
canal of uniform cross section with frictionless flow, is derived at once
by placing t=¢) in equation (226).

This equation then takes the form:

y=C sin (axr/e—y)+C’ sin (az/ec) (233)
in which

C=— Ay cos (dtp ap)/sin y, and C’=A, cos (at)+a,)/sin y

164
This equation is readily transformed into:
y=W sin (az/c+w) (234)

The graph of this equation, if sufficiently extended, is the sinusoidal
curve shown in figure 51.

FIGURE 51.—Instantaneous Profile.

The distance, } (lambda) from crest to crest of the profile is desig-
nated its wave length. The crests of the sinusoidal curve representing
equation (234) are at the poits at which:

at%/c+w=7/2, arx,/c+w=5n7/2, etc.
So that:
\=2)—2)=5me/2a—1e/2a=21e/a=2rygD/a (235)

In which a is the speed of the component, in radians per second, D
the mean depth of the canal, and g the acceleration of gravity. The
wave length of the M; component of the tide, in a channel whose
mean depth is 30 feet, is, for example

204/309
= 1,389,000 feet=263 mil
0.00014053 % a

Evidently, the length of a canal is usually but a small fraction of
the wave length of its principal tidal components.

The form of equation (229) shows that the graph of the instan-
taneous component velocities through a long canal of uniform dimen-
sions with frictionless flow is a similar sinusoidal curve with the same
wave length as the instantaneous profile.

327. Relation between y and \.—From equations (223) and (235)

y=aL/c=2rL/d (236)

It may be noted that if the length, Z, of a connecting canal is one-
half the wave length, , of a component of the tide, y=a and
sin y=0. As subsequently discussed in paragraph 347, the tides and
currents in a canal of this length would become infinite if there were
no frictional resistance to flow.

165

328. Example of frictionless tides and currents in a long connecting
canal.—The characteristics of frictionless flow in a connecting canal
whose length is less than one-half of the wave length of the tidal
components, may be exemplified by the tides and currents that would
be produced in a canal 200,000 feet (37.8 miles) in length, of uniform
width, and with a mean depth of 30 feet, by the M, component of the
tide, if its range at the initial entrance is 8 feet, and at the other en-
trance 4 feet, high water at the latter being 2 lunar hours, or 60°,
earlier than at the former. Taking the origin of time at an instant
of high water at the initial entrance, the given data are:

Ap=4 feet; a=0; A\=2 feet; a:=60°;
L=200,000 feet; D=30 feet; c=WgD=31.06;
a@=m,=0°.00805 per second; y=m2L/e=51°.83=51°50’.

The computation of the amplitudes and phases of the tide and cur-
rent at the entrances to the canal (r=0 and x=Z) and at its quarter
and mid points (r=\L, 4%L, and %Z) is summarized in the following
tabulation:

# 0 44D YeL 345 L

(a1) py ne 51°50’ 38°52’. 5 25°55’ 12°57’. 5
(GID) 7a ne eee 0 12°57’. 5 25°55’ 38°52’. 5 51°50’
PAUSIN (sa 0 0. 494 0. 963 1. 383 1. 732
PAU COSI ana wee 4 3. 478 2. 780 1,9) 1. 000
(972 525 0 8°05’ 19°06’ 35°30! 60°
il a eee 4 3. 513 2. 942 2. 382 2. 000
PBESII Beene —2, 281 —2. 223 —2. 052 S11, 778 —1. 410
IB COS} Se 1. 938 2. 818 3. 554 4. 109 4, 454

eteaeat a or Shatin — 49°39’ —38°15/ —30°0’ — 23°22! —17°34!
PES ape eae 2. 993 3. 588 4.104 4.476 4, 672

The equations of the tides and currents then are:
At the initial end:

y=4 cos mof

v=2.99 sin (m,t—49°39’)
At the first quarter point:

y=3.51 cos (mt+8°5’)

v=3.59 sin (m,t—38°15’)
At the middle:

y=2.94 cos (m.t+19°6’)

v=4.10 sin (m,t—30°)
At the third quarter point:

=2.38 cos (mt+35°30’)
v=4.48 sin (myt— 23°22’)

166

At the further end:

y=2 cos (m,é+60°)
v=4.67 sin (m,t—17°34’)

Since the speed of any competent, in degrees per component hour
(par. 84) is 360°/12=30°, and the component hour of the M, com-
ponent is the mean lunar hour of 1.035 mean solar hours, the tides and
currents represented by these equations are most conveniently
plotted in terms of lunar hours, by placing m.t=0, 30°, 60°, ete. The
tides and currents at the entrances and at the middle of the canal, and
the instantaneous profiles at the successive lunar hours, designated as
0, I, II, iI, etc., are plotted in figure 52.

It will be noted that the amplitudes and phases of the tides and
currents go through a progressive, but not uniform, variation from
one end of the canal to the other, and that the amplitude of the
current increases as that of the tide decreases.

329. Progression of high and low waters, and of the strength and turn
of the current, through a connecting canal—The times of high and low
water, of the strengths of the positive and negative currents, and of
the turn of the current, at points along the canal may be determined
immediately from the phases of the tides and currents at these points.
Thus in the example set forth in paragraph 328, in which the origin of
time was taken at a high water at the initial entrance to the canal,
the current at this entrance turns when v=0 and mfi—49°.6=0. As
the speed of the component is 28°.98 per solar hour, the turn of the
current occurs 49°.6/28°.98=1.71 solar hours after high water; and
the strength of the positive current 90°/28°.98=3.11 hours after the
turn, or 4.82 hours after high water. Similarly, at the first quarter
point the phase of the tide is 8°.1, and high water occurs when
t= —8°.1/28°.98, or 17 minutes before high water at the entrance; so
that if high water at the entrance is at noon, high water at the quarter
point is at 11.43 a.m. <A plot of these times at successive points along
the canal (fig. 53, page 168) shows how high and low water and the
strength of and turn of the current, progress through it.

Obviously, high water and low water must travel through a connect-
ing canal at such rates as to reach the far end at the time of high and
low waters at that entrance, the total time of travel being fixed by the
difference in the timing of the tides at the two entrances; but the rate
of travel is not in genera! uniform. The progress of the strength and
turn of the current through the canal is determined, on the other
hand, by the rate of storage and release of water in the successive
sections of the canal, and is dependent on the depth of the canal. No
fixed relation exists therefore in the general case between the progress
of high water and of the strength of the current through a connecting
canal.

167

Feet per Seo

Lunar Hours

4 @)
Kal
oG
| u-x 1X-X1
4 0-WIIT
pee UWI-Lk I-VI
2 V-v nev
EB mvs
V-V
= A aval
} 190,000 200,000 feel,

Jnstantaneous Profiles

FIGURE 52.—Frictionless tides and currents in connecting canal.

330. Tides and currents at the middle of a connecting canal.—At the
middle of a canal, c=%L and equation (228) becomes:

Ym=Ay cos (at+ao) sin %y/sin y+A; cos (at+ a1) sin 4y/sin ¥
Since sin y=2 sin 4 y cos 4 y this equation reduces to:

Ym='s[Ap cos (at+ ao) +A; cos (at+a)]/cos hy (237)

ts
° 100,000 206,000 feet
Distance from Initial Entrance.

FIGURE 53.—Progression of tide and current through connecting canal.

The height of the tide at the middle of the canal is then the mean
of the heights at the entrances, increased by the factor 1/cos ky.

The velocity at the middle of the canal is, similarly, from equation
(230):

Um=(g/c)[Ao sin (at+a)— A, sin (at+a)] cos y/sin +
=34(g/c)[Ay sin (at+-a9) —A; sin (at+a)]/sin by (238)

The total surface head between the two entrances to the canal is:
h,=A, cos (at+a;)— Ap cos (at+ a)

Tf the effect of the tidal storage were neglected, the water surface
would have the uniform slope of:

oy/Or—=[A, cos (at-+a,)—Ap cos (at+ap)]/L (239)

And from equation (177) the velocity through the canal would be:
— —9 | (y/r) Ot=(g/aL)[Ap sin (at+ ay) —A, sin (at-+a)] (240)

The ratio of the velocity at the middle of the canal, when tidal
storage is considered, to the velocity through the canal without tidal
storage, is then, from equations (238) and (240):

,— 2(g/e)/sin by _
Vm|[Vi= gjaL

In equation (241), y is measured in radians. Its relation to the
wave length, \, of the component is given by equation (236):

y=2rL/X=L/(A/277)

= (aL/c)/2 sin ¥y=y/2 sin by (241)

331. The nature of the ratios, 1/eos %y and y/2 sin %y can perhaps
be shown more clearly on a diagram.

169

Let A B CF figure 54, be a circle
of radius \/27, and hence with a cir-
cumference of length equal to ; and ©
let A B C be an arc of length L.
The subtended central angle AOC is
then L/(\/27)=y radians, and the
length of the chord AC is 2(A/27). sin }y.
The ratio v/v, 1s therefore the ratio
of the arc ABC to the chord AC;
and 1/cos sy is the ratio of OC to OD.
It is apparent therefore, that if the
length of a canal is not too large a
part of the wave length of the tidal
components, frictionless tides and cur- ;
rents at its middle do not differ greatly from the values which they
would have if the effect of tidal storage were neglected.

FIGURE 54.

SPECIAL CASES

332. A canal connecting a tidal with a tideless sea.—If the sea at one
end of the canal is tideless, all of the components there have a zero
amplitude and equations (228) and (230) become:

y=A) cos (at+a) sin (1—2/L)y/sin y (242)
v=(g/c) Ap sin (at+ a9) cos (1—2/L)y/sin (243) ©

It is apparent that, when the equations reduce to this form, y be-
comes a Maximum and v becomes zero, for all values of x, when at--a,.=
0, or when t=—a,/a; and that y=0 and v is a maximum, when t=
—a,/at+7/a. Both high water and low water occur therefore at the
same respective instants throughout the canal, and the current turns
at the same instants.

If, for example, the canal described in paragraph 328 entered a
tideless sea at its further end, the tides and currents at the entrances
and at the middle of the canal, and the instantaneous profiles at suc-
cessive lunar hours, would take the forms shown in figure 55, page 170,
were the flow frictionless.

333. When high water occurs at the same time at both entrances, or
when the tides at these entrances are exactly opposite —If the phases of
the tides at both entrances are the same, a;=ap, and equations (228)
and (230) reduce to:

y=cos (at+a)[Ay sin (1—2/L)y+A, sin (x/L)y]/sm y (244)
v=(g/c) sin (at-++ay)[Ap cos (1—2/L)y— A, cos (x/L)y\/sin y (245)

In this case, also, high water and low water each occur at the same
instants throughout the canal, and the currents turn throughout the

canal at these instants. It is readily shown that the same conditions
result if the phases of the tides at the ends of the canal differ by 180°.

Feet per see.

Lunar Hours

4 Q
i
~~
S
oe
) I
=
o Iv
ae
Gs v
os VW
== 1 a or Ge aah cal ieee ae nL
fa) 100,000 200,000 feet

lustantaneous Profiles

FIGURE 55.—Frictionless tides and currents in canal connecting a tidal with a tideless sea.

334. Meeting the tides —If the tides at the two entrances are identi-
cal in range and in timing, equations (244) and (245) further reduce to:

y=Ay cos (at+a)[sin (1—2/L)y+sin (a/L)-y]/sin > (246)
v=(g/c) Ap sin (at+a)[cos (1—a2/L)y—cos (a/L)y]/sin y (247)
and at the middle of the canal:
v= (g/c) Ap sin (at-+ a) (cos ’y—cos Ky) /sin y=0

al

Obviously, under such conditions, the total surface head through
the canal is always zero, and the currents are due solely to the filling
and emptying of the canal from the two entrances. At the middle of
the canal the currents disappear, and the tides are said to meet.

FRICTIONLESS TIDES AND CURRENTS IN A CLOSED CANAL

335. The amplitude and initial phase of each component of the tide
at the head of a closed canal is determined by the condition that the
currents are there zero. Taking the open end of the canal as the initial
entrance, the equation of a component of the tide at this entrance is
Yo= Ap cos (at+ ay) and at the other end, at the head of the canal, y,=
A, cos (at-++a;). At the head of the canal xz=Z, the length of the canal,
and the velocity of the corresponding component of the current is,
from equation (230):

v= (g/c) Ay sin (at+ap)/sin y— (g/c) A; sin (at-++a,) cos y/sin y=0

Whence:
A, sin (at+ Bn) — Al sin (at+ a) /Cos a (248)

Since this equation is identically true for all values of ft:
Ag —Ay/ Cosi (249)
Ql = ay (250)

Substituting these equivalents in equation (228), the equation of a
component of the tide at any point in a closed canal becomes:

y= Ap cos (at+ ao) sin(1—2/L)y/sin +

+ Ay cos (at+ap) sin (x/L)y/siny cos

== Ay cos (at+ ay) [(sin y cos (z/L) y—cos yx sin (x/L)y) cos

+sin (z/L)-y]/sin y cos

= Ay cos (at+ay)[sin y cos (x/L)y cos ¥
+(1—cos? y) sin (2/L)y|/sin y cos

= Ay cos (at-+-a)[cos (x/L)y cos y+sin (2/L)y sin y]/cos

=A) cos (at-+ao) cos (l—2/L)y/cos y. (251)

The substitution of the same values of A; and a, in equation (230)
gives the equation of the corresponding component of the current,
which similarly reduces to:

v=—(g/c) Ay sin (at+-ao) sin (1—2z/L)y/cos y (252)

336. The form of equations (251) and (252) shows that, if the flow
were frictionless, high water and low water each would occur simul-
taneously throughout a closed canal, and the current would turn at
these instants at every point in the canal. The maximum currents

192750—40—12

172

would occur at mean tide. The tides and currents at the
entrance, the middle, and the end of a closed canal 30 feet in mean
depth and 200,000 feet in length, which would be produced by a
simple harmonic fluctuation of the tide at the entrance with the speed
of the M, component, and a range of 8 feet, were the flow frictionless,
are shown in figure 56, with the instantaneous profiles through the
canal. ,

337. Relation of amplitudes of the components of the current to those
of the components of the tide in a closed canal—From equation (251),
the amplitude of a component of the tide at a point distant x from the
entrance is:

A=A) cos (1—2/L)y/cos (253)

and from equation (252) the amplitude of the corresponding compo-
nent of the current is:

B=(g/c) Ay sin (1—2/L)y/cos y (254)

so that:
B=(g/c)A tan (1—2/L)y (255)

Designating the amplitude of any other component of the current
at the same point as B,, the amplitude of the corresponding component
of the tide as A;, and the value of y for this component as 7, then:

i=(g/e) A; tan (1—2/L) 1

and:
B,/B=(A,/A) tan (1—2/L)y,/tan (—2z/L)y (256)

If the length of the canal is a small part of the wave lengths of the
components, the angles (1—z/Z)y and (1—2/L)y; are small, and are
approximately proportional to their tangents.

Then, approximately:

B,/B= (Aj/A) (1/7) (257)
Since y=aL/c and y,=a,L/c, equation (257) becomes:
B,/B= A,a,/Aa (258)

It follows, therefore, that unless a closed canal is quite long, the
amplitudes of the components of the current produced at a given
point by frictionless flow, are nearly proportional to the products of
the amplitudes and speeds of the corresponding components of the
tide at that point. These speeds, it may be observed, may be ex-
pressed in any units.

x=0
SY
ra)
9
7)
ae)
a
re Current
te

Lunar Hours

Tide in feet

)
(op)

WA
VI
FSS ye a ae ee aT |
cs) 190,000 200,000 feet
Instantaneous Profiles

FIGURE 56.—F rictionless tides and currents in closed canal.

174

The diurnal components of the current in a closed canal have there-
for a much smaller ratio to the semidiurnal current components than
the diurnal components of the tide have to the semidiurnal tidal com-
ponents. It is easy to see, indeed, that the filling and emptying of the
tidal prism of the canal by the diurnal components is at but half the
rate of the filling and emptying by the semidiurnal components. In
a connecting canal, on the contrary, the ratio of the diurnal to the
semidiurnal components of the current may be increased because of
the proportionally large acceleration heads set up by the semidiurnal
components.

PROGRESSIVE, RETROGRESSIVE, AND STATIONARY WAVES

338. The progressi -A special condition of frictionless tidal
flow arises when aay am ana of a component of the tide is the same
at both entrances of a connecting canal, and the phases of the compo-
nent at the two entrances differ by the snclle vy. Taking first the case
in which high water at the further entrance is later than at the initial
entrance, and placing in equation (226), A;=Ao, and a;=a,—y, this
equation becomes:

y= Ay cos (at+ ao) sin (y—az/e)/sin
+ Ay cos (at+a y—vy) sin (az/c)/sin y

Expanding by the formula:
cos A sin B=% sin (A+8B)—% sin (A—B) (259)

y=%A)[sin (at+a+y—aa/c)—sin (at+a—y+az/c)
+sin (at+a)—y+az/c) —sin (at+-a—y—az/e)|/sin y
—¥A)|sin (at+a)—az/e+y)—sin (at+a,—az/e—y)]/sm 7
— Ay cos (at-+ a )—azx/c) sin y/sin (260)
=A, cos (at—ax/c+ ao)

The expression for the current is most readily derived by applying
equation (177):

= —9 | (y/dn)01

From equation (260):

Oy/Or=(Aya/e) sin (at—azx/c+ av)
Whence:

r=—9 | (Atle) sin (at—azx/e+a9) Ot=(g/c)Ao cos (at—az/e+-a9) (261)

175

339. From equation (260) it is seen that the component tide has,
under these conditions, a constant amplitude throughout the canal.
At a point distant v7 from the origin its high water occurs when
at —axJe+-a,=0; or when t=2/¢—a)/a.

The time of high water therefore increases with the distance x at
the uniform rate of c=VgD feet per second. Successive instantaneous
profiles when sufficiently extended, have the relation shown in figure
She

eI Te oh

At dnE ag ane

FIGURE 57.—Successive instantaneous profiles of progressive wave.

Evidently, in this case, a wave progresses through the canal with a
speed, <i gD, which depends only on the depth, D, of the canal and is
independent of the speed of the tidal fluctuations and of the wave
length of the component, and independent also of the maximum cur-
rents in the canal.

The current velocities (equation 261) similarly have the constant

amplitude, Aogle= Avy g/D throughout the canal. At every station
along the canal the strength of the current occurs at high water.

340. Example of a frictionless progressive wave.—In a canal 200,000
feet in length, with a mean depth of 30 feet, the value of y tor the Mz
component of the tide has been found to be 51°50’. The tidal flow
through a canal of these dimensions will then have the form of a pro-
gressive wave if the tides at the entrances have a simple harmonic
fluctuation with the speed of the M, component, the same amplitude,
and the phase of the tide at the initial entrances 1s 51°50’ larger than
that at the other entrance. The tide at the farther entrance is then
51°.83/28°.98=1.79 solar hours, or 51°.83/30°=1.73 lunar-hours later
than at the initial end. Ii tidal range at the entrances is 8 feet, the
equations of the tides and currents in the canal are, when the origin
of time is at the time of high water at the initial entrance:

y=4 cos (mf—0°.000259z)
v=4.14 cos (mst—0°.000259z)

The currents and tides at the entrances and at the midpoint of the
canal, and the instantaneous profiles at successive lunar hours, are
shown in figure 58, page 176.

176

Feet per sec.

Feet

Lunar Hour

A
»
i}
iS}
Pats)
£
Vv
=
K- -4

° 100,000 200,000 feet

Instantaneous Profiles

FIGURE 58.—Tides and currents of frictionless progressive wave.

341. It readily may be shown that a progressive wave would be
propagated through an endless deep canal of uniform section, from any
fluctuation of the water surface at the entrace, whether harmonic or
otherwise, if the flow were frictionless. Such a wave would also be
propagated, under the same conditions, through a canal of finite
length, if the energy transmitted could be wholly absorbed at the far
end. As will be shown later, the tidal flow in an estuary of the usual

177

shape, in which the cross section diminishes at such a rate as to counter-
balance the friction losses, also takes the form of a progressive wave,
but with the current out of phase with the tide. The progressive
wave is commonly regarded, therefore, as a normal form of tidal
motion in a long channel; but tidal flow does not take the form of a
progressive wave in all cases.

342. The retrogressive wave. —If the amplitude of a tidal component
is the same at both ends of the canal, and its phase at the far end
exceeds by the angle y the phase at the initial end, so that high
water is later at the initial end, equations (260) and (261) become:

y=Ay cos (at+ar/e+ a) (262)
v= — (g/c) Ay cos (at-+az/c+ ay) (263)

A wave then retrogresses through the canal at the rate of + gD
feet per second.

343. Resolution of frictionless tides in a connecting canal into pro-
gressive and retrogressive waves.—The application of equation (259)
to the equation of any component of the tide in the form derived in
equation (226):

y=Ay cos (at-+ao) sin (y—az/c)/sin y+A, cos (at-+ a) sin (azx/c)/sin Y
gives:

y='%Ay sin (at+ao+y—az/c)/sm y—%Ap sin (at+a,—y+az/c)/sin v
+%A, sin (at+a,;+az/c)/sin y—%A, sin (at+a,—az/c)/sin y

The first and fourth terms of this equation may be combined into a
term in the form:

W, cos (at—azr/e+u)
and the second and third into one in the form:

W, cos (at+ax/e+ wu»)
giving:
y= W, cos (at—az/e+w,)+W, cos (at+at/e+u,). (264)

From the form of equation (264) it is seen that if the flow were
frictionless, the water surface in a connecting canal produced by a
component of the tide at the entrances would be the resultant of two
waves, one progressive and the other retrogressive. Since the speed,
gD, of the waves produced by each component of the tide depends
only on the depth in the channel, the resultant of all of the compo-
nents of the tide is similarly resolvable into two compound waves
traveling in opposite directions through the canal. The combination
of these two component waves produces, in general, a wave which

178

travels through the canal with changing amplitude and varying
speed. The currents may be correspondingly resolved.

344. The resolution of the frictionless tides in a connecting canal
200,000 feet in length and 30 feet in mean depth, into component
waves, when the equation of the tide at the initial and farther entrances
are respectively:

y=4 cos mot y=2 cos (mt+60°)
gives the equation:
y=1.30 cos (mat—myx/e—46°10’) +3.24 cos (met+myr/e+16°47’)

It may be seen, therefore, that the amplitudes and phases of the
progressive and retrogressive waves in a long connecting canal generally
differ widely from the amplitudes and phases of the tides at the en-
_trances, and have no simple relation thereto.

345. The stationary wave.—In a closed canal, and in certain special
cases In a connecting canal, as has been seen, the phase of the tide and
of the current produced by frictionless flow is the same at all points,
so that high and low water and the strength and turn of the current,
each occurs simultaneously throughout the canal. When the tide in
a closed canal, derived in equation (251), is resolved into progressive
and retrogressive waves by the process indicated in paragraph 343, its
equation becomes:

y='%Ap> cos (at—az/c+a+y)/cos y
+A, cos (at+azr/e+a,—vy)/cos (265)

These two component waves have the same amplitude, A)/2 cos y.
The retrogressive wave may be regarded as the reflection of the pro-
eressive wave from the end of the canal. The resultant of the
progressive and retrogressive (or reflected) waves of the same ampli-
tude is a wave which neither advances or retreats, but remains
stationary. It will be noted that when a stationary wave is produced
by frictionless flow, the current turns at high and low water; while if
a simple progressive wave is produced, the strength of the current at
each station along the canal occurs at high and low water, and the
current turns at midtide.

Frictional resistance must modify the conditions of flow in a long
closed canal, since the lag of the current increases as the currents
decrease toward the head of the canal. From another viewpoint, the
absorption of energy by friction reduces the amplitude of the reflected
wave. The tides at the head of a closed channel are therefore always
later than those at the entrance, and a completely stationary wave is
never found. It is nearly realized in such deep channels as the fiords
of Alaska. Thus in the Portland Canal, a fiord from 600 to 1,000 feet

Wy

in depth, and of quite regular section, the published high water inter-
vals show that high water at Eagle, at the head of the fiord, is but 2
minutes later than at Halibut Bay, 55 miles down the channel and
not far from the entrance. In closed channels of more usual depths,
the frictional resistance is considerable if the length of the channel
is sufficient to set up any appreciable currents, and the tidal fluctu-
ations may more nearly resemble a wave of the progressive type.

CRITICAL LENGTHS OF A CANAL WERE THE FLOW FRICTIONLESS; NODES
IN A CLOSED CANAL

346. Critical lengths of a closed canal—The formulas for the tides
and currents which would be produced in a closed canal by a com-
ponent of the tide at the entrance, were the flow frictionless (equations
(251) and (252)) shows that these would reach infinite proportions if
the length of the canal were such that y=7/2, 37/2, 57/2, etc., since
cos y would then become zero. Since y=27 L/\ (equation 236) these
critical lengths occur when L=)/4, 3\/4, 5X/4, etc.; 1. e., when the
length, Z, of the canal is one-quarter, three-quarters, etc., of the
wave length, \, of the component.

It is apparent that if a closed canal is not of great length, the cur-
rents set up by the fillmg and emptying of the tidal prism are
moderate, and the slopes produced by a small increase in the tidal
ranges in the canal are sufficient to check the momentum of the mov-
ing water. As the length of a canal increases, the increase in the tidal
ranges in the canal further accentuates the currents, and if these
were not restrained by frictional resistance, they would reach infinite
proportions if the canal had the critical length of one quarter of the
wave length of the component. If the length of the canal exceeds
this critical length, the currents at the head of the canal are in the
opposite direction to those at the entrance, and the momentum of
the water is correspondingly controlled. The currents then are finite
until the length of the canal reaches thesecond critical length ;andso on.

347. Critical lengths of a connecting canal——Unless the amplitudes
and timing of the component of the tide at the entrances to a con-
necting canal are such as to produce a simple progressive (or retro-
eressive) wave, a critical length for the component is reached when
sin y=0, and hence when L=15X, A, 14d, etc. The condition of flow
in each half of the canal at the first of these critical lengths is like that
in a closed canal of one quarter of the wave length of the component.
The water entering through both entrances would pile up in the
canal without limit, were there no frictional resistance. When the
length of the canal is equal to the wave length of the component, the
positive and negative currents exactly balance each other, and the
net work done in the acceleration and deceleration of the current
becomes zero, so that frictional resistance would alone limit the flow.

180

Obviously, frictional resistance must control the currents when a
connecting or a closed canal approaches its critical lengths.

348. Nodes in a closed canal.—The equation of a component of the
tide at a point in a closed canal distant x from the entrance is, equation
@oald:

y= Ay cos (at+-ay) cos (1—2/L)y/cos

The tide, y, becomes zero, for all values of ¢, at the points at which
(1—a2/L)y=n/2, 32/2, 57/2, etc. At these points c=L—7L/2y,
L—3rL/2y, L—5rL/2y, etc., and hence s=Z—)/4, L—3)/4, L-5)/4,
etc.

If the flow were frictionless and the canal long enough, each com-
ponent of the tide would then disappear at points one-quarter, three-
‘quarters, etc., of its wave length from the head of the canal. At
these points the component current would reach a maximum ampli-
tude of (g/c)Ao/cosy. These points are termed nodal points.

Similarly a component current in a closed canal would become zero
at the points at which sin (1—2/L)y=0, and hence at which:

r=L, 1—%d, L—X, ete.

And at these points the component of the tide would have a maxi-
mum amplitude of A,/cos y.

It is perhaps needless to point out that true nodal points have no
counterpart in actual channels.

349. Shallow water components of frictionless tides and currents.—
The variation of the mean depth, D, in the equation of continuity
(equation 184) with the rise and fall of the tide has been neglected in
the preceding analysis of the tides and currents in a canal without
frictional resistance. This variation must in fact produce distortions
of the simple harmonic fluctuations of the components of the tides
and currents that have been deduced. It may be shown that these
distortions, like those due to the form of the friction term, may be
reproduced by overcurrents and compound currents, and overtides
and compound tides. As illustrated in the cubature of the Delaware
River, in chapter VI, this variation in the depth may produce marked
distortions of the current in a long and comparatively shallow channel;
but a mathematical analysis of the distortion with frictionless flow
does not serve much useful purpose.

SEICHES

350. An accidental tilting of the surface of a deep lake or enclosed
sea, such as may be produced by wind, or a variation in the barometric
pressure, or by any other cause, often is followed by periodic oscilla-

181

tions of the surface before it returns to normal level. These oscilla-
tions are called seiches. Since the currents set up by seiches are
never strong, the characteristics of a seiche in a long canal of uniform
dimensions, closed at both ends, may be derived from the equations
for frictionless flow in such a canal. The frictionless tides and cur-
rents in a canal open at the initial end and closed at the other have
been derived in equations (251) and (252). If the canal is closed at
both ends, the currents at the initial end are zero. Placing r=0 in
equation (252) the equation of the current at the initial end becomes:

v=—(g/c) Ap sin (at+ ay) tan y

This current is zero if the speed of the oscillations, a, is such that
tan y=0, or if:
y=aLl/c=x

whence:
a=re/L

The period, 7, of these oscillations is therefore, from equation (28),
paragraph 49:

T=2n/a=2L/e=2L/VgD (266)

Thus the period of free oscillation in a canal 5,000 feet in length,
and 30 feet in mean depth is 10,000/+/30g=322 seconds=5}4 minutes.

Jf then the entrance were closed, an oscillation of this period, once
started, would continue, like the oscillations of a pendulum, until
damped out by friction.

At a point distant x from the initial end of the canal, and at the
time ¢, the height of the water surface above its normal level, is found
by placing y=7 in equation (251), and is:

y= Ap cos (at-+ ao) cos (rx /L) (267)
and the current is, from equation (252):
v=—(g/c) Ay sin (at+ ao) sin (rr/L) (268)

At the middle of the canal, x=, so that rz/L=7/2; and y=0
The middle of the canal is then the node of the oscillation. The
currents there reach the maximum amplitude of gAo/c.

Since the current at the initial entrance also is zero when the oscil-
lations have such a period that y=2r7, 32, 47, etc., similar oscillations
with periods of one-half, one-third, one-fourth, etc., of the first may
be set up in a canal closed at both ends. These have two, three,
four, etc., nodal points respectively.

182

The corresponding oscillations of a deep lake or arm of the sea are
more complicated, but those observed at any locality often have one
or more fairly well defined periods. In a deep narrow lake with a
regular shore line, the observed period of the oscillations usually
agrees with the computed period for a canal of equivalent dimensions,
closed at both ends. Even when the conformation of the bed and
shores of the lake is not so regular, the observed periods may agree
with those computed for an equivalent canal extending to a selected
poit on an opposite shore; but as the selection of the point must be
based on the known period, the analogy does not serve much useful
purpose.

Seiches are damped out by the frictional resistance of the currents
which they produce. For an oscillation of a given amplitude these
currents decrease as the depth of the lake increases. Seiches there-
fore are most marked in deep bodies of water. At many localities on
the Great Lakes, seiches of a foot or more occur with such frequency
that an allowance is made for them in the design of navigation channels.

Seiches in tidal waters are superimposed upon, and more or less
obscured by, the fluctuations of the tide; but the tide gage records at
some stations occasionally are marked by saw-toothed irregularities
produced by seiches. Their characteristics may be ascertained by
taking off their departures from a smooth tide curve. In San Fran-
cisco Bay, seiches produced by variations in the barometric pressure
and winds have a period of about 45 minutes, and may have a range
of as much as a third of a foot. It has been observed that earthquake
waves reaching the bay from distant points set up oscillations of the
same period.

At some of the ports on the Pacific coast in California, the currents
set up by seiches often cause a surging of large vessels at wharves,
sometimes with sufficient violence to break the mooring lines. The
surge 1s experienced chiefly at the wharves in the less enclosed parts
of the harbors. The reason for the prevalence of a troublesome surge
in this region is not clear. It is possible that the characteristic
periods of the seiches may agree with the period in which a vessel, as
customarily moored to a wharf, comes and. goes with the stretching
and slackening of its lines. The usual remedy is to make fast to the
wharf with short, taut lines.

Cuapter VIII

COMPUTATION OF TIDES AND CURRENTS IN LONG CANALS
WHEN THE FLOW IS NOT FRICTIONLESS

Paragraphs
Cannectine canals: general consideration._..~......2-.. 2222 2 4- ._- 351-352
Selection of representative entrance tides____________-__-_----_____- 353-358
emMerNeeNIDTANCe ICES... == 46 a be SL eee ae ee 359-360
amputation of primary tides and currents__..__.---_..__-____-__=_= 361-374
Hicsiexample canal oh umirorml cross Section_=-2-_2—))--— e 375-384
Lo LS EYISSNOSM GL/-TRESUI Itsy ee eae teehee Ee ey ee 385-391
Second example, canal whose cross section is not uniform_____________ 392-403
Adjustment for distortions of primary tides and currents_____________ 404-419
rmmmniacons Tor Closed Canals. 2-2. 2+ 2. 4 -= ce ee ee 420-432
terri uTRC NIETO UGS Hae = Fe) een te eee a) ert ee ee 432-434

CONNECTING CANALS

351. General considerations—Perhaps the most important applica-
tion of the principles of tidal hydraulics is in the computation of the
currents which may be expected in a projected long open connecting
canal because of the tides at the entrances; and in the concurrent de-
termination of the elevation of low and high water through the canal.
As has been pointed out, no artificial canal can be so deep that the
tidal flow approaches a frictionless condition, if the currents are of
any consequence whatever. Frictional resistance to flow has there-
fore an important effect upon the tides and currents. The variation
in the depth and width of the water prism as the tide rises and falls
also may have a sufficient effect to warrant consideration, and the
entrance, recovery and velocity heads may be more than negligible.

352. Accuracy required —The usual purpose of such computations is
to ascertain whether the currents will be strong enough to affect ad-
versely the use of the canal for navigation, or to cause serious erosion
of the bed and banks. The computations also show the depths to
which the successive sections of the channel must be excavated to
afford the designed depth for navigation at the selected low-water
datum. These purposes are fulfilled if the maximum current ordin-
arily to be expected in any part of the canal is reliably determined to
- say the nearest half of a foot per second, and the elevation of low water
in the successive sections to the nearest half foot; but good workman-
ship in the calculations usually requires that they check to the nearest
tenth of a foot per second, and tenth of a foot of elevation respectively.

(183)

184

It is well to recollect that the results rest on the selection of the coeffi-
cient of friction, and that this depends upon the undeterminable
irregularities of the channel as actually excavated. Furthermore, the
actual currents will vary from day to day with the varying range and
even the varying form of the tides at the entrances. These variations
may be intensified by winds and storms, which may pile up the water
at one entrance and draw it away from the other. <A precise deter-
mination of the currents and tides produced by a particular fluctuation
of the entrance tides is of academic interest only. The computations
call therefore for reliability rather than precision.

353. Selection of representative entrance tides —The computation of
the tides and currents in a canal is far too laborious to warrant repe-
tition for each successive tidal fluctuation at the entrances, even if
any useful purpose would be served thereby. Representative tidal
fluctuations at each entrance should therefore be selected from a study
of the actual tidal fluctuations during a month or more.

If the tides at both entrances are of the semidiurnal type, with no
large variations between springs and neaps, and are not much de-
formed by overtides, the representative tide at each entrance may be
taken as a simple harmonic fluctuation with the speed of the M, com-
ponent and the amplitude of the mean semirange of the tide. The
most convenient origin of time in this case is at a high water at the
initial entrance. The initial phase of the tide at this entrance is then
zero. The initial phase at the other entrance may be obtained from
the difference between the average lunitidal intervals at the two en-

trances, corrected, if necessary for the differences in longitude (par. 10). _

The corrected difference, in solar hours, multiplied by the speed of
the M; component, 28°98 per hour, gives the initial phase of the tide
at the far entrance. This phase is positive if the tide at the far en-
trance is the earlier, and negative if it is later than at the initial en-
trance. Published data on the lunitidal intervals at stations near the
ends of the canal may be based on such a limited number of observa-
tions as to have no great weight. If a reliable determination of the
lunitidal intervals is not available, the recorded differences in the
times at high water at the two entrances, and in the times of low
water, Over a period of 29 days, or a multiple thereof, should be
averaged. A material discrepancy between the average difference in
the times of high water at the two entrances to the canal and the
average difference in the times of low water, indicates that overtides
are of sufficient importance to warrant consideration.

354. If the tides at the entrances are of the same general type as
those considered in the preceding paragraph, but the daily tide curves
at one or both entrances are so distorted by overtides that they cannot
be represented satisfactorily by a simple harmonic fluctuation, either
average or composite tide curves may be prepared by the methods

aT

185

outlined in paragraphs 304 and 305. The tidal heights on these
curves should, however, be taken off at lunar hourly intervals, begin-
ning generally either at a lunar transit or at the time of high water at
the initial entrance to the canal.

355. If the tides at an entrance have a marked variation between
springs and neaps, it may be advisable to determine the tides and
currents in the canal at mean spring tides, or at ordinary spring tides
(par. 181); and perhaps at the corresponding neap tides as well. If
the daily tide curves show no marked distortions, the representative
spring and neap tides may be taken, without introducing errors which
will affect the results materially, as simple harmonic fluctuations with
the speed of the M, component, and amplitudes of one-half of the
spring and neap ranges respectively; otherwise average or composite
curves of spring and neap tides may be prepared.

356. Probably the most satisfactory method for dealing with an
entrance tide of the mixed type is the preparation of a composite
curve whose timing and elevations conform to the times and elevations
of mean lower low, higher low, lower high and higher high waters, as
outlined in paragraph 306; or a representative tropic tide could be
selected from the records.

357. If the entrance tides are of the semidiurnal type, the currents
produced by the repetition of a single properly selected average, or
spring, or neap semidiurnal fluctuation will afford an adequate indica-
tion of the average, spring or neap currents and tides to be expected
in the canal. To facilitate the computations, the representative
entrance tides of this type should be adjusted as necessary to afford
a smooth curve when identically repeated in successive periods of 12
lunar hours. When either or both of the entrance tides are of the
mixed or diurnal types, the representative tides similarly should follow
curves which are identically repeated every 24 mean lunar hours.

358. The representative tides at the two entrances to a canal should
be referred to the same horizontal datum. If either end of the canal
takes off from the upper part of a generally shallow bay or river
estuary, the mean level of the tides at the two entrances may not be
at the same elevation. The same situation may arise if the canal
connects two oceans or seas in which the water density and mean
meteorological conditions are not the same. Any uncertainty may
be removed by connecting the tide stations at the proposed entrances
by a line of precise levels. Daily variations due to winds, freshets,
and other meteorological causes may produce a constant component
of the head between the entrances, of sufficient magnitude to have a
marked effect upon the currents. It may, therefore, be advisable to
select one or more typical concurrent storm tides at the two entrances,
which would produce large differences in the head through the canal,

186

and determine the currents due to these, as well as those produced by
the representative normal tidal fluctuations.

359. Primary currents and tides in the canal.—If the selected repre-
sentative entrance tides are not simple harmonic fluctuations of the
- same speed, they may be approximated more or less closely by such
fluctuations. These may be termed the primary entrance tides. The
primary currents and tides in the canal which would be produced by
the primary entrance tides are first computed. These computations
are based on the depth and width of the canal at mean tide, and omit
the effect of the minor components oi the friction term (par. 226). The
primary currents and tides may then be adjusted to develop the
deformations produced by the minor components of the friction term,
by the variation in the width, depth, and area of the cross section of
the water prism as the ride rises and falls, and by the entrance, recovery,
and velocity heads; and to develop also the variations because of a
departure of the representative entrance tides from the simple har-
monic fluctuations from which the primary currents and tides were
derived.

The primary currents and tides usually afford a fair representation
of the currents and tides to be expected in the canal because of the
ordinary tidal fluctuations; but their adjustment, although a laborious
procedure, may be warranted to give a more complete and assured
picture of the anticipated tidal flow. The effect of storm tides can
be ascertained only by going through the latter process.

360. Determination of primary entrance tides.—The primary tides
most nearly conforming to the selected representative tides ordinarily
will have the speed of the M, component, whose component hour is
the mean lunar hour of 1.035 mean solar hours. By taking off from
the representative tide curves the heights on 24 successive lunar hours
after any assumed origin of time, the amplitude A, and the initial
phase, a, of the primary tide at this origin of time, may be computed
from equations (56), (57), (47), and (48) developed in Chapter IJ, viz:

12¢,= (hp cos O+h; cos 30°+hz cos 60°+. . .+hs3 cos 330°) (56)
12s,= (ho sin 0+, sin 30°+h, sin 60°+. . .+/o3 sin 330°) (57)
tan (=8)/c2 A=s,/sm ¢=c,/cos ¢ (47) (48)

In which ho, fy, etc., are the tidal heights at the successive lunar hours,
and ¢=—a.
The abbreviation of the computations is explained in paragraph 94.
A consideration of the derivation of these equations shows that if
the representative entrance tides are taken as a fluctuation which is

187

identically repeated every twelve lunar (or other component) hours,
the values of c, and s, may be derived from the equations

6¢:= (ho—he) cos 0+ (hi—hz) cos 30°+. . .+(As—hi) cos 150° (56.4)
652= (Ao—he) sin 0+ (i—hz) sin 30°+. . .+(hs—Au) sin 150° = (57A)

The mean tide elevation of the primary entrance tides should be the
mean of the elevations computed from the representative tides at the
two entrances.

COMPUTATION OF PRIMARY TIDES AND CURRENTS IN
A CONNECTING CANAL

361. A general mathematical analysis of the tides and currents in
a long canal appears impossible if the friction term in the general
equation of motion is taken as a reversing function which varies with
the square of the current velocity. A general solution when the
frictional resistance is assumed to vary with the first power of the
velocity is given by Maurice Lévy in ‘“Lecons sur la Théorie des
Marées” (Gauthier-Villars, 1898); and its application to the Cape
Cod Canal is presented in a paper by William Barclay Parsons con-
tained in the Transactions of the American Society of Civil Engineers,
volume LXXXII (1918), pages 1-157. The equations developed by
this analysis are lengthy and unwieldy, and the established coefficients
of frictional flow are not applicable thereto. Another method for
determining the tides and currents is presented by Col. Earl I. Brown,
Corps of Engineers, United States Army, in a paper on the Trans-
actions of the American Society of Civil Engineers, volume 96 (1932),
pages 753 et seq. The solution therein presented proposes that the
currents at high water be determined from the mean depth of the
canal at high water, and the currents at low water from the mean
depth at low water.

362. A better method is to compute the primary currents and
tides from established frictional coefficients, by a process of successive
approximations, on a line of procedure somewhat similar to that
applied in computations of steady flow. The canal is divided into
subsections so short that the variation in the velocity of the current
because of channel storage is not material in any subsection. The
primary currents in the subsections, and the primary tides at the
ends of the subsections, must satisfy the two conditions:

(a) The fluctuations of the current in each subsection must
conform to the fluctuations of the surface head set ue by the
tides at the ends of the subsection.

(b) The currents in the subsections must also conform to the
storage and release of water from subsection to subsection

because of the rise and fall of the tides.
192750—40-—_13

188

The computations are started by determining the tides which
would be produced if the instantaneous profiles were straight lines.
The primary current in the subsection at the middle of the canal is
then computed from the surface head established by these tides, and
the currents in the other subsections determined therefrom by com-
puting the increments in the velocity due to tidal storage between the
successive subsections. The surface heads corresponding to these
currents are next computed and adjusted to the total head through
the canal. These subsection heads establish corrected primary tides,
and corrected instantaneous profiles, more nearly conforming to the
true profiles than the straight lines initially assumed. The primary
current in the middle subsection is then recomputed from the adjusted
surface head in this subsection, and corresponding currents in the
other sections from the storage and release of water with the corrected
tidal fluctuations. The process is repeated until the further corrections
become negligible.

363. It may be noted that the computations are started by taking
the instantaneous profiles as straight lines, and the current at the
middle of the canal as unaffected by channel storage. As shown in
paragraph 331 both of these conditions would be approximately
realized in as long a canal as is likely to be undertaken, if frictional
resistance were neglected. An examination of the recorded instan-
taneous profiles in actual canals shows that they do not, in fact,
depart widely from straight lines. The procedure will be found
applicable to any case likely to be encountered.

364. Coordinate components of the primary tides and currents.—
The equation of the primary tide at any station on the canal is in
the form:

y=A cos (at+a)

The speed, a, at all stations is that selected for
at the primary entrance tides, but the initial phase,
a, differs from station to station. To carry out
the computations outlined in the preceding
paragraphs, these tides are resolved into com-
ponents with common initial phases. As is
apparent from figure 59, the primary tide at a
given point may be resolved into two com-
ponents; the Y component with an amplitude
Ficure 59—Coordinate com- of A cos a and an initial phase of 0°, and the
ponents of the tide. 3 : 5
X component with an amplitude of A sin a,
and an initial phase of 90°.
The equation of the primary surface head in a subsection of the
canal is similarly

h,=H cos (at+H°)

189

and the surface heads may likewise be resolved into Y components
whose amplitudes are H cos H° and whose initial phases are 0°; and
X components whose amplitudes are H sin H°, and whose initial
phases are 90°.

Designating the amplitudes of the Y and X components of the
tide at the initial entrance as Ay cos ap and Apo sin ay respectively, the
amplitudes of the Y and X components of the tide at any point in:
the canal ‘are:

A cos a=A,y cos a +2H cos H° (269)

A sin a=A) sin m+2H sin H° (270)

In which 2H cos H° and YH sin H® are respectively the sums of the
amplitudes of the Y and X components of the heads in the successive
subsections of the canal between the initial entrance and the given
point.

365. The primary tides and heads may. then be computed in terms
of the amplitudes of their coordinate components. After the values
of the components have been satisfactorily established, the amplitude
and phase of the resultant tide is readily determined from the
equations:

tan a=A sin a/A cos a, A=A sin a/sm a=A cos a/cos a (271)

The quadrant in which a lies is fixed by the algebraic signs of
Asinaand A cosa. A schedule of the values of a corresponding to
the value, (a), taken from a table of tangents, is set down for con-
venient reference.

Asine | Acose

The primary current at a given point in the canal:
v=B sin (at+ B)

similarly may be resolved into two components, one with the ampli-
tude of B sin 6 and the initial phase of 0°, and the other with the
amplitude of B cos 6, whose amplitude differs by 90° from the first.

366. Coordinate amplitudes of the tides for first computation The
computations are started with the tides which would be produced if
the instantaneous profiles were straight lines.

190
The equations of the entrance tides are:
p= Alp cos (at+ Qo) iA cos (at+ a4).

If the instantaneous profiles were straight lines the equation of the
tide at a point in the canal distant x from the initial end would be:

A cos (at+a)=Yyot (2/L) Yi—Yo) = Ao Cos (at+ a)
+ (2/L)[A; cos (at+- ayo) — Ap cos (at+ ay)]. (272)

The coordinate amplitudes of the tide at the point z are then:
A sin a=Ap sin ay+ (2/L) (A, sin a,— Ap sin a) (273)
A cos a= Ay cos apt (x/L) (A; cos a,— Ap COS a). (274)

367. First computation of primary currents in subsections.—The
primary current in each subsection may be taken as that at the middle
of the subsection, and the midpoints of the successive subsections will
be designated the velocity stations. The amplitude, H, and the initial
phase, H°, of the head, and the amplitude, S, of the slope in the middle
subsection of the canal are computed, as described in paragraph 239,
from the components of the tide at the ends of the subsection, derived
from equations (273) and (274). The amplitude, Bo, the initial phase,
GB, and the resulting coordinate amplitudes, By sin B) and By cos Bo, of
the primary current at the velocity station at the middle of the sub-
section are then computed by the process set forth in paragraphs 246
and 248.

368. The corresponding primary currents at the other velocity sta-
tions are determined by the general equation of continuity (equation
S2)e .

0Q/or+ zdy/dt=0.

Since differential equations remain approximately true when small
finite increments are substituted for the differentials, equation (182)
may be written

AQ/Az+ zoy/ot=0
or
AQ=— zAroy/dt. (275)

In this equation AQ is the algebraic increase, at any instant,in the
discharge from one velocity station to the next, z the mean width of
water surface between the stations at mean tide and Az the distance
between the sections. It should be noted that Az may be large when
expressed in feet while being small in relation to the change which it
produces in the discharge.

191

Designating the area, zAz, of the water surface, at mean tide,
between successive velocity stations as U, equation (275) becomes:

AQ=— Udy/ot. (276)

369. Strictly speaking, the rate of rise of water surface, Oy/Ot, should
be computed at the center of gravity of the water surface between the
‘two velocity stations, but its location need not be determined with
mathematical precision. If the canal has a constant width at mean
tide, the storage stations, at which oy/Ot is to be computed, are mid-
way between the velocity stations; and if the subsections are also all
of the same length, these storage stations are at the ends of the sub-
sections. If the width of the canal is tolerably constant, the storage
stations also may be taken at points half way between the velocity
stations; otherwise the location of the center of gravity of the water
surface may be roughly estimated and the storage stations selected
accordingly.

The equation of the primary tide at a storage station is in the form:

y=A cos (at+a).
Differentiating:
oy/Ot=—aA sin (at+a).

Substituting this value in equation (276):
AQ=aUA sin (at+a). (277)

370. Designating respectively the area of the cross section of the
velocity station at the middle of the canal, at mean tide, as 14); the area
at any other velocity station as M; the discharges at these stations
at a given instant as Q) and Q; the amplitudes of the currents as Bo
and B; and the initial phases of the currents as 8) and 6; then;

Qo=Mv=B,.M sin (at+ Bo)
and
Q=Q+2AQ=B,M sin (at+ B)+=aUA sin (at+a). (278)

Since G@= WB sin (at+ 8), equation (278) becomes, after dividing
both members by ,

_Bsin (at+ 8) = (MG/M) By sin (at+ Bo) +(o/M)=(aU/M,)A sin (at+a).

The coordinate amplitudes of the current at the velocity station are
then:
B sin B=(M,/M)B,y sin By) + (M,/M)>(aU/M,)A sna (279)

B cos B=(M,/M)B) cos B+ (M,/M)=(aU/M,)A cos a. (280)

192

371. Placing for brevity:
M,/M=m (281)

aU/M,=1 (282)

equations (279) and (280) may be sniiion.
(B/m) sin B=By sin &+2Z/IA sin a (283)
(B/m) cos B=B) cos {y+21A cos a. (284)

In equations (283) and (284), m is the ratio of the areas, at mean
tide, of the cross section at the middle velocity station to that at the
given station, A sin a and A cos a are the coordinate amplitudes of the
tides at the intervening storage stations, and the values of J are deter-
mined from the surface areas, U, at mean tide, between the successive
intervening velocity stations and the speed, a, of the primary tidal
fluctuations, whose usual value is 0.0001405 radians per second.

372. If the canal has a uniform width and cross section, and con-
sequently a uniform mean depth, D, at mean tide; m=1, and
T=azAxr/M,=aAz/D._ If, further, all of the sections are of the same
length, the value of J is the same for all.

The computations indicated in equations (283) and (284) establish
the first values of the amplitudes B, and the initial phases, 8, of the
currents in the subsections of the canal.

373. Heads corresponding to computed velocities in subséclpreenne
relations between the amplitude, B, of the primary current in a short
section of channel of length, J; its angular lag, ¢, and the amplitude, S,
of the slope in the section, have been developed 1 in chapter V. From
equation (153) in that bay tpn.

tan ¢= (37/8) (a/g)C?r/B. (285)
It is convenient to place:
p= (32/8) @/g) Cr (286)

in which a is the speed of the fluctuation, in radians per second,
g=32.16, and C and r the Chezy coefficient and hydraulic radius in
the section at mean tide.

If the speed of the primary tidal and current fluctuations has its
usual value of 0.0001405 radians per second, equation (286) becomes:

p=0.000,005,148 C?r. (287)

The logarithm of the coefficient is 4.71160—10.
From equations (285) and (286):

tan ¢=p/B. | (288)

193
Again, from equation (151):

S sin ¢=aB/g
whence:

H=IS=l(a/g)B/sin ¢. (289)

If a=0.0001405 and g=32.162, the value of a/g is 0.000,00437 and
its logarithm is 4.64042—10.

The relation between the initial phase, H°, of the head and the
initial phase, 8, of the primary current is, from equation (150):

5 Sa-O=R 00". (290)

The values of H and /7°, derived from equations (288), (289), and
(290) give the values of the coordinate amplitudes, H cos H° and
H sin H°, of the heads in the subsections corresponding to the first
computation of the currents.

374. Corrected tides and velocities—The computed coordinate ampli-
tudes of the heads in the subsections are so adjusted that their sums
are equal to the differences between the coordinate amplitudes of the
tides at the entrance. The corrected coordinate amplitudes of the
tides at the ends of the subsections produced by the adjusted coor-
dinate amplitudes of the heads are computed from equations (269)
and (270). The coordinate amplitudes, By sin By) and By cos Bo,
of current in the middle subsection are next recomputed from the
adjusted head in the subsection, but the recomputation may be
somewhat abbreviated. Let /7 and H’ be the initial and adjusted
values of the amplitude of the head, and S=H/I and S’=H’/l the
corresponding values of the amplitude of slope in the subsection.
The ratio of the corrected value of P’=1.0854 C./rS’ (par 245) to
the value, P=1.0854 Cy7rS, initially computed, is:

P'/P= 8" /JS=1H'/VH

So that:
log P’=log P+ (log H’—log H) (291)
and similarly:

log (P’/S’)=log (P/S)—%dog H’—log HA). (292)

The corrected values of ¢, By and 8; are determined from the corrected
values of P’/S’ and P’ as explained in paragraph 246.

The currents in the other subsections are then recomputed from
equations (283) and (284). In applying these equations, the coordi-
nate amplitudes of the tide at any storage station which does not
coincide with the end of a subsection are interpolated between the
corrected values at the ends of the subsection in which it lies, on the
assumption that the instantaneous profiles in each subsection are sub-

194

stantially straight lines. The corresponding heads are then recom-
puted and the process repeated until a satisfactory concordance of the
currents and heads is attained.

375. First excample-—The computations of the primary tides and
currents in a canal of uniform dimensions will be illustrated by apply-
ing the procedure outlined in the preceding paragraphs to a canal
200,000 feet (37.8 miles) in length, and of uniform cross section, with
a bottom depth of 40 feet at mean tide, a bottom width of 250 feet, and
side slopes of 1 on 3% (fig. 60).

<— — — 2go'— — — >
FIGURE 60.—Cross section of assumed canal.

The representative tide at the initial end of the canal has an ampli-
tude of 4 feet and the speed of the M2, component (28°.98 per mean
solar hour, or 30° per mean lunar hour). The representative tide at
the other entrance has an amplitude of 2 feet, and. the-same speed.
Its high water occurs 2 lunar hours, or 60°, before that at the
initial entrance. Taking the origin of time at high water at the
initial entrance, the equation of the tide at this entrance is then
y=4 cos mf and at the other entrance, y=2 cos (mt+60°).

The area of the cross section of the water prism at mean tide is
15,000 square feet and the surface width is 500 feet, giving a mean
depth of 30 feet at mean tide. The hydraulic radius at mean tide is
also taken as 30 feet, as the refinement of computing the wetted
perimeter is superfluous in view of the uncertainty in the Chezy
coefficient. The Chezy coefficient at mean tide is taken as 120.

376. Division into subsections—The canal will be divided into a
middle subsection and two subsections on either side, total of 5 sub-
sections, each 40,000 feet in length, as shown in figure 61.

I \ | I

@) 20 40 60 8o 100 =: 18O 140 160 (80 00

FIGURE 61.—Division of canal ir to subsections.

A canal of uniform dimensions should always be divided into an
odd number of sections each of the same length; but shorter sub-
sections are required in a shallower canal. The ends and midpoints
of the subsections are conveniently indicated by station numbering,
as shown in the figure, the stations being taken as 1,000 feet in length.
In the present example the velocity station at the middle of the
canal is at station 100, and the velocity stations of the other sub-
sections at stations 20, 60, 140, and 180. The storage stations

195

coincide with the ends of the subsections, at stations 40, 80, 120
and 160.

377. Initial component tides at the storage stations.—These are com-
puted from equations (273) and (274):

Ay=4 ajp=0 AG a,=60°

Yay Sit) oe == An cos a,— 10

Ap sin a=0 Ay cos a=4.0

ae =a3.0

Initial component tides
(1) (2) (3) (4) (5) (6)
Station | 2/L YEE) GID, Nat COGS reieta Cerenll D ana eayr

0 0 0 0 0 4.0
40 2 346 =f 346 3.4
80 4 693 =F) 693 2.8
120 6 1.039 21.8 1. 039 2.2
160 ‘8 1. 386 2.4 1. 386 1.6
200 1.0 1. 732 a0 1, 732 1.0

378. Coordinate currents in middle subsection—The head in the

middle subsection is computed from the component tides at stations
80 and 120.

A sin H°=A sin ajyy—A SIN ag=1.039—0.693=0.346
EL Cos Heli Vl cos Qy99— A sin Ago = 2.2 —2.8= — 0.600

tan H° =0.346/(—0.600) = —0.577 Ho N80 7—30-— 502
H=0.6/cos 30°=0.693
Then S=H/I—0.693/40,000—0.00001732.

L—s,

And from the given data:

Cg I20 rS=0.0005196

The coordinate components of the current in the middle subsection
are computed as described in paragraphs 246 and 248:
log rS=6.71567—10 logycos ¢=9.93987—10

log-¥7S=8.35783—10 log P= .47260

log 1.0854= .03559 loess ye —
log C=2.07918 Bo

41247
= 2.580

loot 4 7260
log S=5.23855—10
log P/S=5.23405
From table IX: ¢=40°.7=40°40’

From equation (162):
plel Gi 0"
—150°—40°40’—90°
—= 92207
By cos Byp=2.440
By sin Byp=0.854

196

379. Primary currents in other subsections —The amplitudes and
phases of the currents which would be produced at the other velocity
stations under the initial assumption are computed from equations
(283) and (284). These computations are tabulated in figure 62,
facing page 198. The coordinate components of the tide at the
entrances and at the storage stations, found in paragraph 377, are
entered in columns (2) and (38). Since the canal has a constant
cross section.

I=aAz/D=0.000,140,5 X40,000/30=0.187.

This value is entered in column (4). A much larger value of J would
indicate that the canal should be divided into shorter subsections.
The resulting coordinate velocity increments, JA sin a and IA cos a,
are entered in columns (5) and (6). For these and for the subsequent
computations, the slide rule affords satisfactory accuracy.

The coordinate amplitudes, B) sin 6) and By cos Bo, of the current
at station 100, determined in paragraph 378, are entered opposite this
station in columns (7) and (8). The values of (B/m) sin 6 and
(B/m) cos B at station 60 are found by subtracting, algebraically, the
coordinate velocity increments JA sin a and JA cos a at station 80 from
the coordinate amplitudes of the currents at station 100 (since the
summation is in the negative direction); and the values at station 20
by subtracting the increments at station 40 from the values found at
station 60. The values of (B/m) sin 6 and (B/m) cos 8 at station
140 are similarly found by adding, algebraically, the coordinate
increments at station 120 to the coordinate amplitudes at station 100;
and at station 180 by again adding the increments at station 140.
The totals of columns (5) and (6) may be checked against the differ-
ences between the last and first lines of columns (7) and (8) respec-
tively.

The values of tan 8, from columns (7) and (8) are entered in column
(9), and the corresponding values of 6 in column (10). Since, in the
present example, the area of the cross section of the canal is the same
at all stations, m=1 and B/m=B. The entries in columns (11) and
(12) therefore are omitted, and the amplitudes, B, of the current at the
velocity stations entered in column (13) from the relation:

B=(B/m) sin-B/sin B= (B/m) cos B/cos B.

The values of 6 and B at station 100 derived in columns (10) and
(13) should check with those found in paragraph 378.

380. Surface heads corresponding to computed currents in subsec-
tions—The computation is continued in columns (14) to (24) of
figure 62. The value of p, column (14) for all subsections is, from
equation (287):

p=0.000,005,148 X 120? X30=2.224.

197

Dividing by B, the values of tan ¢ (equation 288) are entered in column
(15) and the corresponding values of ¢ in column (16). The com-
puted value of ¢ at station 100 should check with that derived in
paragraph 378.

The value of /a/g, column (17), for all subsections, is (par. 373):

la/g=0.000,004,37 40,000 —0.175.

The values of Bla/g are entered in column (18) and those of
H= (Bla/q)/sin ¢, from equation (289), in column (19). Tne values of
H°, derived from the computed values of 6 and ¢, are entered in
column (20), and the corresponding computed coordinate amplitudes
H sin H° and H cos H° in columns (21) and (22). The computed
values at station 100 should check with those found in paragraph
378. The initially computed values at this station should be entered
in these columns, even if minor inaccuracies have produced slight
differences in this check computation. A material difference would
indicate the need for reviewing the entire work.

381. Adjustment of coordinate heads.—As is to be expected, the sums
of the computed cvordinate amplitudes of the surface heads in the
subsections, in columns (21) and (22) differ by small residuals from
the total coordinate amplitudes of the heads between the entrances
shown in columns (2) and (3). The computed values are adjusted
in columns (23) and (24), by dividing the residuals as equally as may
be between them, so that the sums agree with the actual coordinate
amplitudes of the head between the entrances.

382. Recomputation of tides, currents, and heads.——The corrected
coordinate amplitudes of the tides at the storage stations, which in
this case are at the ends of the subsections, are next recomputed in
columns (2) and (3) of figure 62, by successively adding the adjusted
values of H sin H° and H cos H°, found from the first computation,
to the coordinate amplitudes of the tide at the initial entrance. The
coordinate amplitudes of the tide at the further entrance, station 200,
afford a check on the results. It will be seen that the corrected coor-
dinate amplitudes of the tides at the storage stations differ materially
from those used in the first computation. The currents and the
resultant heads are therefore recomputed as shown in figure 62 from
the corrected data. In the present example, the adjusted coordinate
amplitudes of the head in the middle subsection, station 100, in col-
umns (23) and (24) of the initial computation, differ so little from
- their original values, in columns (21) and (22) that a recomputation
of the coordinate amplitudes of the currents in the middle subsection
is unnecessary.

198

383. Final computation—The values of H cos H° derived from the
second computation are practically the same as those determined from
the first. While the recomputed values of H sin H° also are in suffi-
cient concordance to be reasonably acceptable, a third computation,
as shown in figure 62, affords a desirable check. In this final compu-
tation the primary currents at the entrances to the canal are derived
by correcting those at the adjacent velocity stations for the inter-
vening storage. The primary current at the initial entrance, station 0,
is derived from that at station 20 by subtracting the increment of the
velocity due to the fluctuation of current at station 10. The values
of A sin a and A cos a at station 10 are obtained by interpolation
between their values at station 0 and station 40. For the half sub-
section, 0 to 20, Ar=20,000 and 7=0.0935. The primary current at
the other entrance, station 200, is similarly derived from that at

station 180, by correcting for the storage due to the tidal fluctuations
at station 190.

384. Summary of computations.—The results of the computations
are tabulated below. The values of A sin a and A cosa, derived from
the final computations of the subsection heads, are shown in columns
(2) and (3), and the resulting amplitudes A, and initial phases a@ of
the tides at the ends of the subsections in columns (6) and (5). Those
at the middle of the canal are inserted by interpolation. The ampli-
tudes, B, and initial phases, 6, of the primary currents are the final
values found in figure 62.

Summary

(1) (2) (3) (4) (5) (6) (7) (8)

Station Asina | A cosa tan a a A B B
0 0 4. 000 0 0 4. 00 1.14 47°20’
20) aati ay Bil pee cee S| ited all Rae ae 1.41 36°10
40 —.014 sees —. 003 0°10’ 3.7 Se Uh
GO EM eee eh ee Fe ey AlN eee 2.01 24°20/
80 +. 134 3. 246 +. 043 +2°30/ 3. 24 =s22 ieee
100 306 | 2.944 . 104 6° 0! 2. 93 2.58 19°20’
120 478 2. 642 . 181 10°20’ 2. 68 32 eee
LAH hal) gee ee et A ae eh oes ae 3.08 17°50’
160 1. 022 1. 899 . 540 28°20 2, Ld ee
TRSKO MPa HG es ASE | ae ee 1) | cee Sete || [oe = es = 3.48 19° 0’
200 32 1. 000 576 SOS (OY 2. 00 3. 65 20°30

This summary affords the data fer writing the equations of the
primary currents and tides at the stations. Thus the equation of the
primary current at station 0 is:

v=1.14 sin (m.t+ 47°20’),
and of the primary tide at station 40:

Y=. 10) COsm mist — O10):

ides and Currents.

Primary T

sr

24

Computed.

o

19 23
H. Adjusted.
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192750—40

FIGURE 62.

atc
4 J i
Mal add
‘ cal
iy ee
fie
aoe
‘
'
a
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i
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j
at
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h

*

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tana cnaan wa poem herman

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net ah

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Fi

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ia
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a a
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:.
ks
r YS -BRO 7 OY,
* ; Be hat Caer sR T
re 3 |
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eeeeree. (rene
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199

The primary tides and currents at the entrances and at the middle of
the canal are shown in figure 73, page 218.

385. Shape of instantaneous profiles.—The characteristic shapes of
the instantaneous profiles developed by the successive approxima-
tions is readily shown by plotting the values of A sin a and A cos @
successively derived in figure 62, since A cos a is the tidal height at a
station at 0 hour and A sin a=A cos (—90°+<a) is that at —3 lunar
hours. The instantaneous profiles at 0 hour are shown by the lines:
marked 0—0 in figure 63. The differences between the first and secondi

{so
FIGURE 63.—Instantaneous profiles at 0 and III hour.

recomputations are too small to be distinguishable. The successive
determinations at —3 lunar hours are shown by the lines marked
JII—III on the figure.

386. Sufficiency of subdivision.—The question may arise whether
subsections 40,000 feet in length, in a canal having a mean depth of
30 feet, are sufficiently short to afford a reliable determination of the
currents. An independent computation, based on a subdivision into
9 subsections, each 22,222 feet in length, affords the following com-
parative figures in the equations of the currents at the entrances:

Computed from 9 subsections Computed from 5 subsections
PANU eTIGGAN CO sess ee oe US 2Siny (Qt-|-4ijcl 00) ee eee v=1.14 sin (at+47°20’).
Mtiouherentrance.-9-.). 45-5 22 D=3260) Sim (Qt-+-20°40)) eae v=3.57 sin (at+20°50’).

It is apparent, therefore, that the results are but little improved
by using the shorter sections.

200

387. Effect of frictional resistance —The effect of frictional resistance
upon the primary tides and currents in the canal selected for the first
example is shown by a comparison between the results of the preced-
ing computations and of those derived in Par. 328, Chapter VH, for
frictionless flow in the same canal with the same entrance tides. The
origins of time and distance are the same in the two cases. Angles are
written to the nearest 10 minutes of arc.

Frictionless flow Friction considered
VELOCITIES
NF hawiell Gh he RNC v=2.99 sin (m2f—49°40’)___________. v=1.14 sin (mof+47°20’).
PANG arr d Gee eee sk nee ee v=4.10 sin (mot—30°)______________ v=2.58 sin (mof+19°20’).
ACUGunermenthancesssa= == =e v=4.67 sin (Mmof—17°30’)___-________ v=3.65 sin (m2f+20°30’).
TIDE
/ip MCL OF GEMEN ee y=2.94 cos (mot-+19°10’) ___________ y=2.96 cos (msf-+6°).

As is to be expected, frictional resistance reduces considerably the
amplitude of the currents. Its effect upon the timing of the currents
is even more marked. When frictional resistance is neglected, the
strength of the positive current at the initial entrance was found to
be (90°+49°40’)/28°.98=4.82 solar hours after high water, while
when frictional resistance is considered, the strength of the positive
current is (90°—47°20’)/28°.98=1.47 hours after high water.
Frictional resistance advances therefore the strength of the current
at the initial entrance by 3% hours. At the other entrance, it ad-
vances the current by 1% hours. ;

In the present example, the amplitude of the tide at the middle of
the channel is but little affected by frictional resistance, but the rela-
tive phases of the tide at this station show that the time of high water
is altered by nearly half an hour.

388. Relation between channel storage and the primary currents and
discharges at the entrances to a cana/.—The primary discharge at an
entrance to the canal is derived immediately by multiplying the cur-
rent velocity by the area of the cross section at mean tide. Obviously,
the difference in the discharges at the two entrances at any moment
must be equal to the rate of storage or release of water in the tidal
prism of the canal at that moment.

In the present example, the area of the cross section at both
entrances is 15,000 square feet. The entrance velocities and dis-
charges and the rates of filling and emptying of the canal during
successive tidal cycles are shown diagrammatically in figure 64.

389. The discharge at the initial entrance, station 0, is an inflow
into the canal when positive, and an outflow when negative; while
at the further entrance, station 200, it is an outflow when positive
and an inflow when negative. During the time interval marked AB

201

on the diagram the discharge is in the positive direction at both
entrances, and the outflow at station 200 largely exceeds the inflow
at station 0. The tidal prism in the canal is therefore emptying
through the further entrance. During the interval BC it is emptying
through both entrances; and during the brief period CA’ through

= 7
e3 sage
Tee ei
SS
716
o ,
a8
e20-°2--0--
s
See
: r)
m4
te) g
0-34 =
Ges
= ae
ee (=) ter

16)
)
D
im

Lunar Hours
FIGURE 64.—Discharges and storage, canal 200,000 feet long.

the initial entrance. The filling of the tidal prism during the interval
A’ A’’ follows a similar sequence.

It is apparent from the figure that, in this case, the discharges and
velocities at the further end of the canal are caused principally by the
filbng and emptying of the tidal prism. Because the filling and empty-
ing is largely through this entrance, the currents are there the strong-
est, although the tidal range is the least.

390. The strength and timing of the currents in a connecting canal
depend upon the relative timing of the tides at the entrances, as well
as upon the amplitudes of these tides. Thus if in the example devel-
oped in the preceding paragraphs, high water at the further entrance
is taken 2 lunar hours (60°) after, instead of 2 lunar hours before, that
at the initial entrance, the equations of the current are found to be:

At the initial entrance: v=3.04 sin (at+108°30’).
At the further entrance: v=2.30 sin (at+52°20’).

The change in the timing of the tide produces therefore the strongest
currents at the initial entrance, where the tidal range is the greatest,
instead of at the further entrance, where the range is the least. A
diagram of these velocities, and consequent discharges and the storage
and release of water in the tidal prism of the canal, is shown in figure 65,
page 202.

In this case the interval, AB, during which the prism is emptying
because of the greater outflow through the further entrance, station
200, is shorter than the interval CA’, during which the outflow

202

through the initial entrance is the greater; and the mterval BC,
during which the prism is emptying through both entrances, is rela-
tively long. The tidal prism fills and empties through both entrances,
but somewhat more water enters and leaves through the initial
entrance than through the other.

Oo 77)
rs) at
ng 37570
a

Ss. nike
Qo es
AS

Ss s)
g —
+ O-] -
2 =
=
at ©
Ne 3
o AS
0-31 %
S 2
= A

Luhar Hours
FIGURE 65.—Discharges and storage, with different timing of entrance tides.

391. The longer a canal the less is the net slope set up by given
tides at the entrances, and the more are the currents determined by
the storage and release of water in the canal prism. It may be
expected that in a very long canal the tidal prism will fill and empty
from both ends during most of the tidal cycle, and that the currents
will be stronger at the two entrances than in the interior of the canal.
Thus a computation of the primary currents in a canal 360,000 feet
(68.5) miles in length, and 30 feet in mean depth at mean tide, with
the same entrance tides as in the first example, 1. e.,

You= COs 4Mot Y,;=2 cos (mt+60°)

and with the same coefficient of roughness, shows the strength of the
current decreasing from 2.3 feet per second at the initial entrance to
1.0 foot per second at a point 100,000 feet from that entrance; thence
increasing to 3.8 feet per second at the further entrance. The equa-
tions of the entrance tides are:

Initial entrance: v=2.3 sin (m.t+149°).
Further entrance: v=3.8 sin (m,t+10°50’).

The diagram of the entrance velocities, discharges, and channel
storage, figure 66, shows that in this case the entrance currents are
due principally to the filling and emptying of the tidal prism through
both ends of the canal.

203

It further may be expected that in an even longer canal the currents
would become so small at some point near the middle of the canal that
the tides could be said to meet.

eee
= 3
S o
7)
a alte)
2 16
x ae .@)
c =
HO ce
21
oO es
o-3-'o
7 aL 50
> A

fe) 12 is
Lunar Hours

FIGURE 66.—Discharges and storage, canal 360,000 feet long.

392. Second example.—The computations of the tides and currents
in a connecting canal whose width and cross section is not uniform
throughout may be illustrated by applying them to the Chesapeake
and Delaware Canal, connecting the estuary of the Delaware River
with the head of Chesapeake Bay, after it had been converted into a
sea level canal for barge traffic but before it was enlarged to accom-
modate ocean shipping. The canal then had a horizontal bottom at a
project depth of 12 feet below Delaware River low water datum, a
designed bottom width of 150 feet from the 12 foot depth contour in
the Delaware (station 3+ 400) to station 10+300, a distance of 9,900
feet; and of 90 feet thence to Back Creek, a tributary of Chesapeake
Bay, at station 77-++000; a distance of 66,700 feet. Between -Reedy
Point Bridge, station 9+780, and Biddles Point, station 19+600,
the canal was bordered by wide marshes having a large tidal storage.
A second outlet to the Delaware, with a depth of 6 feet below datum,
and a bottom width of 50 feet, entered the canal near Biddles Point.
Some tidal storage outside of the prism proper extended to the deep
cut beginning at about station 50, but thence almost to station 77
the only tidal storage was in the prism of the canal. The upper part
of Back Creek afforded a tidal basin inside of the bridge at station 77.
The canal was designed with side slopes oi one on two, but as
excavated the cross section generally was somewhat in excess of that
projected.

393. Representative entrance tides selected —To afford a comparison
between the computed and measured currents, the representative
entrance tides will be taken as the recorded tides at stations 5-000

192750—40—14

204

and 77 +000 on November 27-28, 1928 when tide and current measure-
ments were made at these stations, at Biddles Point (station 19-+600)
and at Summit Bridge (station 51-+-200). The day selected is one on
which the tides had little diurnal variation. The origin of time is
taken at 7 a.m. on November 27, when the record of the observations
begins.

394. Primary entrance tides—The amplitude, initial phase, and
mean elevation of the primary tide at station 5+000, are computed
in the following tabulation from the recorded tidal heights at this
station, by the method explained in paragraph 360.

Primary entrance tide, station 5-+-000

GQ) eiaumnarihours==s5 nd 1 2 3 4 5 6 7 8 9 10 11
(2) Mime se == 2288 7. 00 8. 04 9.07 | 10.11 | 11.14 | 12.18 |18.21 |14. 25 |15. 28 |16.31 | 17.35 | 18.38
@Ovhide === .6 2.4 3:9 5.0 5.5 5.0 3.5 2.3 1.2 nt) —.2 —.7
(4) Lunar hour__-_| 12 13 14 15 16 17 i} | Tey |i oy | on Op} || 98
(5) im eee 19.42 | 20.45 | 21.49 | 22.52 | 23.56 | 0.59 | 1.63 | 2.66 | 3.70 | 4.738 5.77 | 6.80
(Gewide: 23s s—-= 12 2.7 4.1 Ayal 5. 5 4.7 3.2 2.2 ils al =} —.2 +.3
(@) (@RH@)s-s-2-5- 1.8 tis al 8.0 10.1 11.0 9.7 6.7 4.5 7733 .8 | —.4 | —.4
(S)eheitovnaee se 6.7 4.5 2.3 8 —.4 —.4

(°) (0) eel (:) —4.9 +.6 | +5.7 | +9.3 |+11.4 |+10.1

The mean solar time in lines (2) and (5) corresponding to the lunar
hours in lines (1) and (4) is derived by successively adding the length
of the lunar hour, 1.035, to the initial time at zero hour. The repre-
sentative tidal heights, taken in this case from a plot of the recorded
tidal heights, on mean solar time, are entered in lines (3) and (6) and
summed in line (7). The last six entries are subtracted from the first
six in lines (8) and (9). The values of s) and ¢, and of Ay and ¢ are
then computed from the equations:

12s.=hp sin 0+h, sin 30°+. . .h; sin 150°.
12c,=hy cos 0O+hz cos 30°+. . .hs cos 150°.

Products Products
Angle Sin h SS Cos
+ = =F =
0 0 —4.9 0 1.00 4.90 tan ¢= bar se 85.
30° . 500 +.6 0. 30 . 866 0. 52 €=180 © —§1°40’ =118°20',
60° . 866 +5.7 4.94 500 2. 85 Ao=s82/sin [=2.79.,
90° 1. 000 +9.3 9. 30 0 Ho=Zh/24=2.48,
120° . 866 +11.4 9. 87 —. 500 5. 70
150° . 500 +10. 1 5. 05 —. 866 8. 69
L
12s2=29. 46 3.37 19.29
Se= 2.45 3. 37

205

The sum of the tidal heights in line (7) divided by 24 gives the mean
tide elevation, Hy, above the tidal datum, which in this case is Dela-
ware River low water datum. The equation of the primary entrance
tide at station 5+-000 is then:

y=H)+ A) cos (at—f£).
=2.48+-2.79 cos (at—118°20’).
395. The equation of the primary tide at station 77 +000, derived by
the same procedure, is:
y=2.79-+1.45 cos (at—73°50’).
The computed primary tides and the recorded tidal heights at the

two stations are plotted in figure 67. They show a satisfactory
concordance.

PES)

o oO
°

STA. 5+000

STANDARD TIME

NOV. 27 NOV. 28
7 3 1s at 3 6
° 3 3 12 is (8 2) 24
LUNAR HOUR
PRIMARY. TIDE ————— RECORDED. SIDE. o>

FIGURE 67.—Primary and observed entrance tides, Chesapeake and Delaware Canal,

206

396. Mean sea level of primary tides —The mean sea level at the two
entrances as found in the preceding paragraph differs by 0.31 feet.
Long period observations show a similar difference. For the com-
putation of the primary currents and tides the elevation of mean sea
level is taken as the mean of the elevations at the entrances, or 2.63
feet above Delaware River datum. The depth of the bottom of the
canal is then 14.63 feet throughout. .

397. Division into subsections.—A subdivision based on the variation
in the cross section and in the storage areas is shown in figure 68;
station 5+000 being selected as the initial entrance.

oe aK ma
ae ol ce 91 cae al l 3
ce) ol S| | O| | Q| \ 9!
bl %. » R 98 1 fe) 4 +|
+ 2 +| a + + #
Bestar we es i gee ee

Back.Cr

Delaware River.

FIGURE 68.—Diagram of Chesapeake and Delaware Canal.

The subsections and their constants are:

1 2 3 4 5 6 7 8 9 10

Subsection (sta- Velocity a no

tion to station) Length / siipsrisiara Area M |m=Mo/M|Widthz| r C ) la/g

5+000 | 10-+-000 5, 000 7+500 3080 0. 78 260 11.8 g4 0. 537 | 0.022
10-+000 | 29-++000 19,000 | 19-+-500 2400 1: 00 190 12.6 95 . 585 . O83
29+-000 | 45-+-000 16,000 | 37-++000 2400S || aaa eee 190 12.6 95 . 585 . 070
45+000 | 61-+-000 16,000 | 53-+-000 2400 1.00 170 14.1 97 . 683 . 070
61-+-000 | 77-+-000 16,000 | 69-+-000 2400 1. 00 170 14.1 97 . 683 . 070

72, 000

The velocity stations, column (3), are the midpoints of the swbsec-
tions. Station 37, nearest the middle of canal, is taken as the base
station for velocities. The cross section areas, M/, column (4), and
the surface widths of the canal, z, column (6), are at mean tide, 2.63
feet above datum, and are taken from a sheet of measured cross sec-
tions. Since the area at the base station is 2400, the values of m,
column (5), are 2400/M. The hydraulic radius, 7, column (7), is taken
as M/z, and the Chezy coefficients C, column (8), are from the Bazin
formula, C=87/(0.552+‘m’’/+/r), with “m’=1.30. The “m” in this
formula has no relation to the ratio m in the tabulation. The cor-
responding values of p (equation 286) and of la/g (par. 373) are shown
in columns (9) and (10).

398. The surface areas, U, at mean tide, between the velocity sta-
tions, and between the entrances and the adjacent velocity stations,

207

derived from a topographic survey made in 1928; and the consequent
values of J=aU/M)=0.000, 1405 U/2400 are shown in the following
tabulation.

Surface
From To area, U Storage
z : (1.000’s of IE F
station station square station
feet)
5+000 7+500 325 0.019 6+250
7-++500 19++500 9, 907 . 580 13-++500
19-+-500 37-+-C00 6, 900 . 404 28-+-250
37-+000 53-++000 5, 300 .310 45+-000
53-+000 69-++-000 2, 720 . 159 61++000
69-+000 77+000 1, 847 . 108 73-+-+-000

The storage stations are midway between the velocity stations.

399. Initial computation of primary tides and currents.—The compu-
tation of the tides and currents, by the procedure previously explained,
is tabulated in figure 69 facing page 208. In columns (2) and (3) the
coordinate amplitudes, A sin a and A cos a, of the tides at stations 5
and 77 are computed from the values of A and a@ at these stations,
found in paragraphs 394 and 395. In the initial computation the
values at the other stations are interpolated from equations (273) and
(274). The values of By sin B) and By cos By at station 37, columns
(7) and (8), are determined from the head between stations 29 and 45.
From columns (2) and (3) ,in this subsection:

Een $P=—i1 = (— D1) =
Hs cos H° = —0.37— (— 0.75) =+0.38
Ee 322200 el — OAS) S— 0.4 5/16,000— 000050281
loge P=0.28785, los P/S=4.83914
@=17°10’, f=F°—e—90°=—74°50’
Bo— 190, By sin 6j——".83, Bo cos 65 —0:50

The initial computation is completed as explained in the first ex-
ample, with due regard to the algebraic sign of the items. The minor
discharge through the branch outlets from Biddles Point to the Dela-
ware is neglected.

400. Recomputation of primary tides and currents ——In the recom-
putation, also shown in figure 69, the primary currents at the entrances,
and the primary tides and currents at Biddles Point (station 19+ 500),
and at Summit Bridge (station 51+200) are included. The under-
lined coordinate amplitudes of the tides in columns (2) and (3) are
at the ends of the subsections, and are derived from the adjusted
coordinate amplitudes of the heads found in the initial computation.
Those at the other stations shown are interpolated. As in the pre-
vious example, no correction of the values of By sin By) and By cos Bo

208

at the base velocity station (station 37)is required. The coordinate
increments to the current due to the flow through the branch channel
extending from near Biddles Point to the Delaware are computed
from the head in this channel. As this correction is small, the tides
at the Delaware entrance to this branch may be taken as the same
as those at the entrance to the main canal, at station 5. From the
tabulated data, in this branch:

Hf sin H°=—2.29—(—2.45)=+0.16
H cos H°=—1.30—(—1.33) = +0.03
ane ==" a2 0% HONG

The length of the branch channel is 9,000 feet. Taking the bottom
depth as 6 feet below datum, or 8.63 feet below mean tide, the bot-
tom width as 50 feet, and the side slopes as 1 on 2, the area of the
cross section is 580 square feet, the surface width is 85 feet and the
hydraulic radius, 7, is 6.8 feet. The corresponding value of C is 82.
From this data:

p=13°40’ B=0.965 B=— 24°20’

m=2400/580=4.14
Whence
(B/m) sin B=—0.10 (B/m) cos B=+0.21

These increments evidently are to be subtracted, together with the
intervening storage, from the values of (B/m) sin 6 and (B/m) cos 6
at station 19-+500 to give the values at station 7+500.

The values of (B/m) sin B and (B/m) cos B at station 51+200,
enclosed in parentheses, are interpolated between those at the adja-
cent velocity stations. These values are disregarded in the summa-
tions by which the entries in columns (7) and (8) are derived.

The recomputed amplitudes and phases of the currents at the
velocity stations are so close to those derived from the first computa-
tion that a recomputation of the heads is not made.

401. Summary of results —The amplitudes and phases of the com-
puted primary tides and currents, at the stations at which the actual
tides and currents were observed, are shown in the following tabu-
lation.

Computed primary tides and currents, Chesapeake and Delaware Canal

Station | Asina | Acosa| tana a A B B
5-++-000 —2.45 = 33) | see —118°20’ 2.79 ise} +19°30’
19-++500 —2. 29 —1.31 1.755 —119°40’ 2. 65 1.39 —43°10’
51+200 —1. 84 —. 66 2.78 —109°50’ 1.95 2. 39 —83°40’
1.45 2.90 —86°0’

77-+-000 S11, 81) sro4i0) |s2osscse- —73°50!
| |

! 2

3

rs

Sta. |AsinalAcosa| I

5 +4+000|~2

= 1,33

S Ss Zz

TAsina|l Acos: B inp cose ane

8

Primary Tides and Currents.

=)

10

6

|| 3

°

7

iS ié

pa | & | lag

°

17

f=}

19

21 22

23 24

Bla/g cayping

H° || computed.

Adjusted.

Le Hsin H'|Hcos H|Hsin H®|HcosH”

oe

=

+0,5"

B

EX)

=343

=[ 4

4

FIGuRE 69.

192750—40 (Face p.

208)

e

re or

oe

209

402. Primary discharges and storage -—Multiplying the area of the
cross section at the entrances by the amplitude of the primary cur-
rents, it is found that, with the given tidal fluctuations, the primary
discharge at Reedy Point reaches a maximum of 4,096 c. f. s.; and at
the Chesapeake City entrance, 6,960 c. f. s. Evidently, the larger
part of the filling and emptying of the canal is through the latter
entrance.

403. Hiffect of the flow through the canal upon the primary entrance
tides —At Reedy Point the canal opens into the wide estuary of the
Delaware, and the flow in and out of the canal obviously is insuffi-
cient to produce a measurable effect upon entrance tides. At the
other entrance, at Chesapeake City, the discharge is into the com-
paratively restricted channel in Back Creek, whose area of cross
section, in its upper part, is given as but 3,700 square feet. The
discharge through the canal therefore, is sufficient to effect the cur-
rents and tides in this approach to the canal. The computation of
the tides and currents in the canal has been made from the actual
recorded elevations at Chesapeake City, after the canal was opened.
If equally good records were available at the mouth of Back Creek,
‘and physical data were at hand to determine the constants for the
successive reaches in that approach, the computations profitably
could have been extended to include this approach as a part of the
canal prism.

DISTORTIONS OF THE PRIMARY CURRENTS

404. The distortions of the primary current in a short section of a
tidal channel have been developed in paragraphs 260 to 276 ot chapter
V. In a long tidal canal, further distortions are introduced by the
variation, with the rise and fall of the tide, in the area of the water
surface between successive velocity stations, and in the area of the
cross section of the water prism at these stations. The corrected
velocities at any stations at which a determination is desired may be
computed by a procedure which now will be described.

405. Intervals —The corrected velocities are computed at selected
intervals of time which, like those chosen for deriving the corrections
in a short section of a tidal channel, should be parts of the component
hour of the primary tides and currents. This component hour usually
is the lunar hour of 1.035 mean solar hours. Intervals of one-half a
lunar hour, or 1,863 mean solar seconds, ordinarily are sufficiently
small to afford reliable results.

406. Procedure.—A first adjustment of the currents, which usually
is sufficient for all practical purposes, may be accomplished through
the following procedure:

(a) The primary tides at the ends of the subsections of the canal
are adjusted to the selected representative tides at the entrances, if

210

these depart from the simple harmonic fluctuations of the primary
entrance tides. The simplest adjustment, and one which appears as
tenable as any other, is to assign the departures of the total head
between the entrances to the primary heads in the subsections in
proportion to the length of the subsection.

(b) The primary current in the subsection in which the amplitude,
B, is the greatest 1s corrected to conform to the adjusted tides at the
ends of the subsection, and for its other deformations, by the pro-
cedure described in chapter V. The currents in this subsection may
be expected to have the largest influence upon the currents through
the canal.

(c) The discharges at the velocity station of this base subsection
are determined by multiplying the corrected velocity by the area of
the cross section at this station at the given time.

(d) The currents at other stations are computed from those at
this base station by a cubature of the adjusted tides through a modi-
fication of the process developed in chapter VI.

407. Example——To illustrate the procedure, the corrected currents
in the Chesapeake and Delaware Canal will be computed from the
primary tides and currents derived in the preceding paragraphs, and
entrance tides conforming to the observed tides. To curtail the
computations, the small diurnal variation of the entrance tides is
omitted and the tidal cycle is completed in 12 lunar hours. The
representative tide at each entrance at zero hour (7 a. m., November
27, 1928) is taken as the mean of the recorded tides at 0 and 12 lunar
hours; that at 0.5 lunar hour as the mean of the tides at 0.5 and 12.5
lunar hours; and so on. A minor adjustment at 11.5 hours produces
fairly smooth repeating tide curves with a period of 12 lunar hours.
The heights, above Delaware River datum, of the entrance tides so
derived are as follows:

Selected entrance tides

a ae
Tunat | station 5 | Station77| Tuma | station 5 | Station 77
0 0.90 3.35 6 3.35 2. 50
B 1.75 3. 55 cB 2. 80 2.10
1 2. 55 3. 85 7 2,95 1.75
5 3. 30 4.05 5 1.65 1.45
2 4. 00 4.15 8 115 1.35
5 4. 60 4. 20 5 "80 1.35
3 5. 05 4.10 9 "40 1.35
es 5.35 400 B “10 1.50
4 5. 50 3. 80 10 — 20 1.80
A 5. 30 3. 52 6 — 35 2.07
5 4. 85 3.15 u — 35 2. 40
4 4. 20 2.90 +10 2. 87

408. Adjustment of primary tides at ends of subsections —The primary
heads in the subsections are given by the equation:

h,=H cos (at+H°)

211

in which H and H° may be taken as the unadjusted values shown in
columns (19) and (20) of figure 69, and at increases by 15° at each
successive half lunar hour. Thus the surface head in the subsection
between stations 61 and 77 at zero hour is:

0.770 cos 22°30’=+0.71
and at 0.5 lunar hour it is:

0.770 cos (15°-+ 22°30’) =+0.61
and so on.
The tides at 0 hour are then adjusted to the selected entrance tides
as follows:

(1) (2) (3) (4) (5) (6)
- Primary . Adjusted | Adjusted
Station Head Factor Correction iron tide
ee eee aire [vey eS meme! (pet oe te A eee | ee ee 0. 90
—0. 04 10/72 +0. 05 SBQR017 hoo ae
i () ee eaten NS ee ca Se A ve ee Bt nt SE) [fe ts | Le han ee -91
+.11 19/72 18 = eo REE Baza
DAD) pei at ee eee ee a eee 2 Vee See ee ee ee 1. 20
+. 40 16/72 15 Sn | ee ee
A te a oS |e as Ree ale Byte eG oe PS a eI ES ake Sh eS Se io 7)
+. 58 16/72 15 a ET (Yall |e es
Geen Se |= ee he ak eee ee ee Sa 2. 48
+. 71 16/72 16 Suey bei eee =
TO Se RE | MES Ste Tee ee Se ee es ce ee Lees Pee 3.35
Swain 1.76 72/72 69 2y45) Messen ee
Motalahea amen tran Conti d Stes meee Seen en ene ew ee ee 3. 35—0. 90= ze 45
Primary tides 5S ENE oe a ee ee SS ee ee eee 76

FRO UAICOLLE CLIO Nes eee ee Sa ee ee oe eee ee . 69

The primary heads in the subsections, column (2), total 1.76 feet,
while the head between the selected entrance tides is 2.45 feet, giving
a correction of 0.69 feet. The correction factors, column (8), are the
lengths of the sections divided by the total length of the canal. The
corrections derived by applying these factors to the total correction
are shown in column (4), an odd hundredth being assigned to the
subsection with the largest head. The adjusted tides, column (6),
are found by successively adding the adjusted heads, column (5), to
the tide at station 5, the initial entrance.

The tides at ends of the subsections at subsequent intervals of time
are adjusted by a repetition of the process. The corrections generally
are much smaller than those which, in this case, happen to occur at 0
hour.

409. The heights above Delaware River datum, of the adjusted tide
- at any other stations on the canal may be found, at half lunar hour
intervals, by linear interpolation between the adjusted tides at the
ends of the subsections. The adjusted and observed tides at Biddles
Point (station 19+500) and at Summit Bridge (station 51-+200) are

212

plotted in figure 70. The computed tides are seen to be in satisfactory
concordance with the observed tides.

Feet

Feet

8
Standard Time

Adjusted tides
Recorded tides 0 9 Oo
FIGURE 70.—Adjusted and observed tides, Chesapeake and Delaware Canal, November 27-28, 1928.

410. Adjusted velocities and discharges at base velocity station——The
primary currents are the largest in the subsection between stations 61
and 77. The corrected velocities at station 69, the velocity station of
this subsection, are determined from the adjusted tides at stations 61
and 77, by the procedure described in paragraphs 260 to 276 of chapter
V. The value of the hydraulic radius, 7, as derived from the given
cross section of this part of the canal, varies from 12.3 at 0 tide to 16.9
when the tidal height is 6.0 feet. The corresponding values of C, from
the Bazin formula, with the coefficient used in the determination of
the primary tides and currents, are 94.3 and 100, respectively. The

213

computations which are not here repeated, give the corrected velocities
shown in column (3) of the following tabulation, from which the dis-
charges are computed.

Discharges at station 69

(1) (2) (3) (4) (5) (6)
Cor-
Lunar Primary | rected Tide BE Q
hour current | current
0 —2.73 —2.81 2.92 2, 410 —6, 770
5 —2. 59 —2. 59 3. 20 2, 450 —6, 350
1 —2. 26 —2.31 3. 58 2, 520 —5, 820
5 —1.80 —2.02 3. 87 2, 570 —5, 200
2 —1.20 —1.61 4.08 2, 600 —4, 190
5 —.52 —1.03 4,23 2, 640 —2, 720
3 +.19 —.22 4. 24 2, 640 — 58
5 +. 87 +1. 05 4. 22 2, 630 +2, 760
4 +1. 56 +1. 94 4.10 2, 610 +5, 060

The tidal heights at station 69, column (4), are the means of the
adjusted heights at stations 61 and 77. From a sheet of typical cross
sections, the area of the water prism at station 69 is found to be 1,900
square feet at 0 tide and 2,944 square feet at a 6-foot tide. The areas,
X, of the cross section at the tidal heights shown in column (4) are
taken off a straight line diagram and entered in column (5). These
areas, multiplied by the velocities in column (3), give the discharges,
Q, in cubic feet per second at half lunar hour intervals, shown in column
(6). These, and subsequent computations, are by slide rule. The
discharges through the rest of the 12 hour tidal cycle are computed in
the same way.

411. Discharges and velocities at other stations—The adjusted dis-
charges and velocities at any other station on the canal are determined
from the discharge at station 69 by the cubature of the adjusted tides
through successive intervening reaches. As shown by equation (276),
paragraph 368, the increase, AQ, in the discharge between two succes-
sive velocity stations is:

AQ=— Uody/ot
in which U is the area of the water surface between the stations and
Oy/dt is the rate at which the average tidal height between the stations
1S increasing with the time. For sufficiently small increments of
time, At, this equation may be written:

Q=— UAy/At

in which U is the area of the water surface at a given time, At is the
selected time interval, and Ay may be taken as the mean increase in
the average tide between the stations during the preceding and follow-
ing intervals. In the present case, At is the half lunar hour of 1,863
seconds.

214

412. The cubature between stations 69 and Summit Bridge, station
51-+200, takes the following form:

Cubature—stations 69 to 51+ 200

Tides Increments Q
Tunars| aman ee = ay, Area
hour Sta- Sta- ah ee a Sia. Sta- xX i
eas tion ean | Prior Ay see tion
tion 69 51-4200 tion 69 51-+200
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)
0 2. 92 2.05 2.48 +0. 54 | +0. 46 1,795 | —830 | —6,770 | —5,940 | 2,250 | —2. 64
aii) 3. 20 2. 51 2. 85 +. 37 +. 42 1,815 | —760 | —6,350 | —5, 590 | 2,330 | —2.39
1 3. 58 3. 04 3. ol +. 46 +.42 | 1,840 | —770 | —5,820 | —5,050 | 2,430 | —2.00
5) 3. 87 3. 52 3. 69 +. 38 +.35 1,855 | —650 | —5, 200 | —4,550 | 2,520 | —1.80
2 4.08 3. 93 4. 00 +. 31 +. 29 1,870 | —540 | —4,190 | —3,650 | 2,580 | —1.41
50) 4.23 4.29 4. 26 +. 26 +.19 1,880 | —360 | —2,720 | —2,360 | 2, 640 —.90
3 4.24 4. 51 4.37 +.11 +.09 1,890 | —170 — 580 —410 | 2,680 —.15
.5 4, 22 4. 66 4.44 +. 07 +. 02 1, 890 —40 | +2, 760 | +2, 800 | 2,710 | +1.03
4 4.10 4. 69 4. 40 —.04 —.14 1,890 | +270 | +5,060 | +4, 790 | 2,720 | +1. 76
-5 3. 85 4. 50 4.17 —.23

The tides at station 69, column (2), are those shown in column (5) of
the preceding tabulation. Those at station 51-+200, column (3), are
derived by linear interpolation between the adjusted tides at stations
45 and 61. The mean tide in the reach from station 51+200 to
station 69 is shown in column (4). The increase in the mean tide in
the preceding interval is entered in column (5), the entry at 0 hour
being repeated from that for 12 hours (not shown). The mean of the
entries on the half hour and on the succeeding half hour, in column (5),
gives the mean rise during the preceding and following intervals, and
is the value of Ay, column (6). The area, U, of the water surface
between stations 69 and 51+200, from topographical maps of the
canal, is 3,128,000 square feet at zero tide, and 3,662,000 square feet
at a 6.0 foot tide. Dividing by At=1863, the value of U/At at 0 tide
is 1,679, and at 6-foot tide, 1,966. The values of U/At at the mean
tidal heights shown in column (4) are taken off a straight line diagram
and entered in column (7). The values of AQ, column (8), are then
the products of the entries in columns (6) and (7), with the sign
reversed. The discharges at station 69, previously found, are entered
in column (9). Since the cubature is in the negative direction, the
values of AQ are subtracted therefrom, algebraically, to give the
discharges at station 51+ 200, column (10). The typical cross sections
of the canal show the same section at station 51+ 200 as at station 69,
and the areas X of the cross section at the latter station, at the tidal
heights shown in column (3) are taken from the diagram previously
prepared. The quotient of the entries in columns (10) and (11) then

215

gives the velocities at station 51+200, in column (12). The compu-
tation for the rest of the 12-hour cycle is in the same form.

413. The velocities at the entrance to the canal at Chesapeake City
Bridge, station 77, are similarly derived by cubature from station 69.
The discharges at station 37 are derived from those at 51-+200, and
thence successively the discharges and velocities at station 19+ 500
(Biddles Point) and at station 5 (Reedy Point). The flow through
the branch canal which makes off from near Biddles Point is too small
to warrant the labor involved in including it in the adjustment. The
surface areas, U’, and the areas of the cross sections, used for these
computations, are as follows:

Surface areas, U (square feet)

Reach 0 tide 6.0. foot tide
Stagione 09s toni(—— phism a= 1, 406, 000 1, 646, 000
Back Creek. .-.____ 583, 000 4, 294, 000
AICO eas a no Re ee ee 1, 989, 000 5, 940, 000

Stations:

HET 2 OORCO LOOMS Te Sethe 2 oe 3, 128, 000 3, 662, 000
SevOnola 200 ee a a ee 3, 505, 000 7, 699, 000
NOE HOOhtO: i= ke ee ae 4, 294, 000 9, 447, 000
HevOmlOat =) OQCE eames 2s a8 se 8, 208, 000 13, 568, 000

Cross sections, X (square feet)

Stations: 0 tide 6.0-foot tide
> [reo 2) WO Weegee Fo Ses 1, 900 2, 944
MOE) Qe ee raat ts ene lH 1, 936 3, 065
i pe wets, ee NINE eer AOE a 2, 428 4, 060

414. Comparison with observed velocities —The computed primary
currents and adjusted currents, at Reedy Point (station 5) and at
Biddles Point (station 19+500) and the mean velocities from meter
measurements at these stations on November 27-28, 1928, are shown
in figure 71, page 216. It may be seen that the adjustments produce
large distortions of the primary velocity curves at these stations.
While the recorded velocities are somewhat erratic, the adjusted
velocities conform fairly well to the observations. The computed
and observed currents at Summit Bridge (station 51+200) and at
Chesapeake City Bridge (station 77) are shown in figure 72, page 217.
The recorded velocities at these stations are much more consistent,
and with some minor variations the adjusted currents resulting from
the computations are in close accordance therewith. Although the
constants and data used in the computations were selected without
regard to the observed currents, the agreement is perhaps closer than
could ordinarily be expected.

216

415. Second approximation.—The corrections applied to the veloci-
ties in the subsections of the canal must change, to some extent, the
distribution of the total surface head between the subsections. A
second adjustment may be made by computing the surface heads
produced by the corrected velocities in each subsection of the canal, at

Lunar Hours
O 6 12 [s} 23

2 eS ES Eee)

Feet per sec.
oO

Feet per sec.
oO

Nov.28

75 12 18 fe) 67
Standard Time

Computed primary current ——-—
Adjusted current

Recorded currents o 0 o

FIGURE 71.—Computed and observed currents, Chesapeake and Delaware Canal, November 27-28, 1928

the adopted half lunar hour intervals. These heads are determined
from the equation:

hst+h,+hat+h;=0

by computing the values of hf», ha, and h; from the corrected velocities.
The surface heads so computed may then be adjusted to the heads
established by the selected entrance tides, corrected tidal heights at
the ends of the subsections derived therefrom, and a second determina-
tion of the velocities made from the corrected tides. The procedure
is too laborious to be warranted in any ordinary case. The results of
its application to the tides and currents in the canal chosen for the first
example, in which the entrance tides were taken with a simple harmonic

20G

fluctuation, are shown in figure 73, page 218; and the instantaneous
profiles derived therefrom in figure 74, page 219. In these figures
the tides are referred to a datum 10 feet below mean sea level. It
may be noted that the changes produced in the tides by the adjust-

Lunar Hours

Feet per sec.

Summit Bridge— Sta. 514200

per sec.

Feet

18
Standard Time

Computed primary current ——--
Adjusted current
Recorded currents 000

FIGURE 72.—Computed and observed currents, Chesapeake and Delaware Canal, November 27-28, 1928

ment are very small. The currents derived from the second adjust-
ment do not differ materially from those derived in the first. The
weaker currents at the initial entrance show a considerable distortion.
A plot of the distorted discharges at the two entrances, not here
- shown, develops no material departures from the relation between
the storage and discharges due the primary currents in the canal,
discussed in paragraph 388.

416. Preponderance of flow in a connecting canal.—The flow in the
two directions through a connecting canal ordinarily is at different
stages of the tide. Because of the consequent difference in the mean

218

areas of the cross section of the water prism while the flow is in the
opposite directions, and because of the distortions of the currents, the
total volume of flow in one direction may be expected to differ from
that in the other. This preponderance of flow may be estimated by
computing the arithmetical mean of the adjusted discharges at
the given intervals.

417. In the canal selected in
the first example the algebraic
mean of the finally computed
discharges at the initial en-
trance is —275 c. f. s. and at
the further entrance it is —158
ce. f. s. A closer adjustment
would be necessary to remove .
the discrepancy between these
two figures. The maximum
discharges at these entrances
are 22,220 c. f. s. and 50,700
ce. f. s., respectively. These
figures show that in this case
the total volume of flow is
nearly the same in both direc-
tions, but indicate a slight
preponderance of flow toward
the initial entrance, where the
tidal range is the greater.

418. In the Chesapeake and
Delaware sea-level barge canal,
taken as the second example,
the adjusted discharges pro-
duced by entrance tides on the
day selected show a prepon-

Ficure 73.—Primary and adjusted currents in frst derance of flow averaging —441

example sane

ce. f. s. at the initial entrance,
station 5, and of —376 c. f. s. at the further entrance, station 77.
These figures indicate an average net discharge during the day of
about 400 ec. f. s. through the canal in the negative direction, from
Chesapeake Bay into Delaware River. This preponderance of flow
may be attributed to the higher mean tide elevation in the head of
Chesapeake Bay.

419. It is not difficult to see that in a comparatively short canal,
with a wide difference in the tidal range at the two entrances, the tidal
elevations and the surface heads through the canal are dominated by
the tide at the entrance having the larger tidal range; and because of
the greater cross section and less resistance to flow at the higher tidal

Feet per sec.

Feet

Lunar Hour

Primary Tides and Currents —---—
Adjusted

219

stages at that entrance, more water will flow through the canal from
that entrance than will flow back at low tide when the direction of the
flow is reversed; provided at least that no adverse constant component

of the head is produced
4, 0

by a difference in the 1-X!

elevation of mean tide ~ pee Ne
at the two entrances. IX-xI
The more _ intricate raed pO rae
tides and currents in « te
a longer canal and re WV

differences in the ele-

vation of mean tide ae

at the entrances may ee
Sta O 40 80 120 160 200

produce a preponder-

Fi 2 FIGURE 74.—Instantaneous profiles (adjusted).

ance of flow which is

not necessarily from the entrance having the larger tidal range.
CLOSED CANALS

420. A computation of the currents and tides in a projected closed
canal seldom is necessary, as usually it may be taken for granted that
the currents in such a canal will not be troublesome; but should the
occasion arise, the primary tides and currents may be computed by
a procedure paralleling that applied in the preceding paragraphs to
connecting canals. Aside from a practical application, the develop-
ment of the effect of frictional resistance upon the primary tides and
currents in a long closed canal of uniform dimensions will cast some
light upon the characteristics of tidal flow in closed channels in general.

421. Computation for closed canals of moderate length.—If a projected
canal is so short that the instantaneous profiles will not depart widely
from horizontal lines, the computations may be started by determining
the currents that would be produced in successive subsections of the
canal if the primary tides in each subsection had the same amplitude
and phase as at the entrance. The surface heads in the subsections
are then computed, corrected tides derived therefrom, the currents
recomputed, and the computations repeated until further corrections
become negligible.

422. Since the discharge at the head of the canal is zero, the dis-
charge, Q, at a velocity station at the middle of any subsection is,
from equation (278):

Q= MB sin (at+ 8) =SaUA sin (at+a) (293)
in which M is the area of the cross section at the velocity station, B
the amplitude and @ the initial phase of the primary current at the

station; and 2aUA sin (at-+a@) is the summation, from the head of the
192750—40- 15

220

canal to the velocity station, of the products of the surface areas,
U, between the successive velocity stations, at mean tide, and
aA sin (at+a), the rate of increase of the tide at the storage stations
midway between the velocity stations.

Designating the area of any typical cross section of the canal as
M), and placing, as in equations (281) and (282):

M,/M=m
aU/M)=I
equation (293) may be written:
B sin (at+ 8)=m2Z/A sin (at+ a)
whence:
(B/m) sin B=Z/JA sin a (294)
(B/m) cos B=ZIA cos a. (295) |

If the canal is of uniform dimensions, and the subsections of equal
length, Mo>=M, 1/m=1, and J=aAz/D.

The component currents in the subsections are computed from
equations (294) and (295), and the resulting surface heads from
equations (288), (289), and (290).

423. Hxample-—The computations may be illustrated by applying
them to a closed canal of uniform cross section, 60,000 feet (11 miles)
in length, with a mean depth of 16 feet at midtide, when the tidal

fluctuation in the entrance has a

range of 6 feet, and the speed of

the Ms, component, 0.0001405

bon SOO eo =o . 18 © radians per second. The origin

FIGURE poem mney state stations (stations of distances is at the head of the

canal, and the origin of time at

a high water at the entrance. The canal will be divided into three
subsections, each 20,000 feet in length, as shown in figure 75.

Station 0 is at the head of the canal. The velocity stations, at the
middle of the subsections, are at stations 10, 30, and 50. The storage
stations are at stations 5, 20, and 40. As the currents near the head of
the canal are extremely small, the surface head in the quarter section
between stations 0 and 5 is always negligible, and the components of
the tide at station 5 may be taken as those at station 0.

424, Coefficients —An appropriate value of Chezy coefficient, C, is
100. The storage coefficient, J, at all storage stations, exeee for the
half section at the head of the canal, is:

T=aAz/D=0.0001405 X 20,000/16=0.176.
For the half section at the head of the canal, J=0.088.

Since the canal is of uniform dimensions, m=1. The coefficients
for the determination of the subsection heads (par. 373) are:

221

p=0.000,005,148 C?r=0.823
la/g=0.000,004,37 < 20,000=0.0874.

425. First computation of the primary currents and heads —The com-
putations are started by taking the equation of the tide throughout
the canal as:

Y—3s COS Ct.

At all storage stations therefore:
A cos a=3 A sin a=—0.

The computations are conveniently made in the form previously
used for connecting canals, and are shown on figure 76, facing page
222. The value of (B/m) cos 6 at station 10, is the storage increment
for the half subsection, 0 to 10; that at station 30 is obtained by
adding the storage increment between stations 10 and 30, and so on.
The subsection velocities and heads are then computed, but since the
tide at the entrance to a closed canal is alone fixed, the computed
coordinate heads are not subject to adjustment.

426. Recomputation of currents and heads.—The currents and heads
are next recomputed as shown in figure 76 from the tides established
by the heads determined in the initial computation. The component
tides, A sin a and A cos a, at stations 40, 20, and 0 are obtained
by successively subtracting, algebraically, the component heads,
H sm H° and H cos H°, found in the first computation, from the
component tides at station 60.

In the final computation the current at the entrance, station 60, is
determined by adding to the component currents at station 50, the
storage increments from stations 50 to 60. The component tides at
the storage station, station 55, are interpolated.

427. Results of computation —The amplitudes and initial phases of
the tides at the ends of the subsections, derived from the final com-
putation, are:

Station: At =
(5) eee ey Se Ei eS eS eS 3. 00 0
A () Ee ene ee ernie eh he 1 DO NO yl eee ee & NAB ae 33, 11% — 3°50’
AO) es cot Re Ie Ce een ee ge 3. 19 — 5°10’
(Ree ease See ae ie Re eh eet eas Ad Ly 3. Wil — 5°20’

The tidal range therefore increases from 6.0 feet at the entrance to
6.42 feet at the head of the canal. High water at the head of the
canal is 5°.33/28°.98=0.18 hours=11 minutes later than at the
entrance. The strength of the current at all sections is nearly at
midtide, and decreases from 1.66 feet per second at the entrance to
zero at the head of the canal. The currents are so weak that the
tides and currents approach the condition of frictionless flow.

428. Computations for a longer canal.—The procedure which has
been described is applicable only to a comparatively short canal. As

222

the length increases, the successive approximations converge more
slowly, and after a certain length is reached, run completely wild. To
compute the primary tides and currents in a long closed canal, the
amplitudes and initial phases of the currents produced by tides of
successive amplitudes at a station at a moderate distance from the
head of the canal may be determined by the method that has been
described. The tides and currents set up when another section is
added are derived therefrom. By continuing the process, the primary
tides and currents in a closed canal of any length may be computed.

429. Hxample.—The primary currents produced by tides of succes-
sive amplitudes at the entrance to a canal 60,000 feet in length, and
16 feet in mean depth, determined by the same procedure as that set
forth in figure 76, are:

At entrance (station 60) At head (station 0)
TIDE CURRENT TIDE
a B B A a
Se ees ee eerie. 2: 0 LO Anas ee — 4°50’ 3s — 6°10’
es ens ee Mes AS 0 k(G} Oyen ae wai — 4° 3.2 — 5°20’
DHS a De SSRN ENE aes (0) pA es ey BONY 2.4022 — 4°30’

The computation from this data of the primary currents at the
entrance to a canal 80,000 feet long and of the same mean depth,
when the tide at the entrance has an amplitude of 3 feet, is shown at
the bottom of figure 76. For the initial computation the tide at
station 60 is taken as the same as at station 40 of the 60,000 foot
canal, the amplitude of which is 3.12 feet and the initial phase is
—3°50’. The corresponding amplitude of the current is, by inter-
polation from the tabulated data, 1.73 feet per second, and its initial
phase, for a zero phase of the tide at station 60 is —4°10’. Since
the phase of the tide at station 60 is —3°50’, the phase of the current
at this station is —4°10’—3°50’=-—8°. The coordinate amplitudes
of the velocity at station 60 are then:

1.73 sin (—8°)=—0.240 —«‘1..73 cos (—8°) =a

The current at station 70 is derived by adding the velocity increment,
stations 60 to 70, determined by the tide at station 65, and the resulting
coordinate heads, stations 60 to 80 derived therefrom. The recom-
putation from the corrected tides at station 60, develops heads in
satisfactory agreement with those first found. In the final computa-
tion, the current at station 80 is determined by adding to the corrected
velocity components at station 60 the storage due to the tide at
station 70. The current at the entrance to the 80,000 foot canal is
found to have an amplitude of 2.27 feet per second, and an initial
phase of —10°30’.

The final determination of the amplitude of the tide at station 60
of the 80,000-foot canal is 3.13 feet and its phase is —8°20’. The

2 3

a

Ss

Ss

7

Primary Tides and Currents.
Closed canal, 60,000 fect long, mean depth 16 fect

t) 10 12 13

i i 16

17

138 is 20 2t 22 23

24.

Sta. |Asina|Acosa

21 0-

2)

u
I |IAsinajTAco B inp |® cose aS B Bb cu i) sy

oy] 7

tang H H°
P |p/B | PD | la/g|Bla/g|(ieysing 6+ 64995 sin HTH cos Hl Hsin "| Hoaall’

Computed Adjusted.

9 2

a 0.0) |-

0,0 ey =

70,

iS

15]2.228

=0,)86|-10 2.2

FIGURE 76.

192750—40 (Face p. 222)

pire rina i
pte no = oF}
Ge eeu aes 3

ee Vi btS. aac

“ fneved
ct a !
( ‘ z —
aur
fs 268.2
4

4 Jae 0

;

a oe

223

corresponding amplitude of the tide at the head of the canal is, by
interpolation from the tabulated data, 3.34 feet; and its initial phase
is —5°30’—8°20’=—13°50’. The primary tides and currents at
any other station on an 80,000 foot canal could be determined in a
similar manner by establishing their relation to the amplitude of the
tide at station 60.

Perhaps a better method for computing the tides and currents in a
very long closed canal is to determine those that would be produced
in the successive subsections by tides of several amplitudes at the
head of the canal. If the subsections are 20 stations in length, the
current produced at station 10 by a tide of given amplitude at the
head of the canal is derived from the velocity increments from the
tide at station 5, and the coordinate heads, tides and currents at
station 20 computed therefrom. The coordinate amplitudes of the
tide at station 25 can then be set forward with fair assurance and
the currents at station 30 determined by adding the velocity incre-
ments due to the tide at station 25 to the coordinate currents at
station 20; and so on to the station at the entrance. The amplitudes
and phases of the tide and current at any station on the canal can
then be plotted against the several computed amplitudes of the
tide at the entrance, and those corresponding to an entrance tide of
a given amplitude taken off these diagrams.

430. Characteristics of the tides and currents in a long closed canal of
uniform cross section.—The primary tides and currents in a canal
140,000 feet (about 26.5 miles) in length, of uniform cross section, 16
feet in mean depth, produced by an entrance tide of 3 foot amplitude,
as computed by the step by step process just outlined, are as follows:

Primary tides and currents in closed canal 140,000 feet long, with a mean depth of
16 feet at mean tide ,

[C=100]
Tide Current
Station —
(1,000 feet : :
z Ampili- Ampili-
ee Phase mala 7D Phase

140 3.0 0 3.2 —44°30'
120 256 —18°30’ 2.8 —50°30’
100 2.0 —36°10’ 2.4 — 54°50’
80 2.6 —48°20’ 2.0 —57°50’
60 2.0 —55°40’ 15 —59°20'
40 2.8 — 59°20’ 1.0 —60°20’
20 2. 86 —60°40’ a) —60°40’
0 2.9 —60°50’ Ome Tees eee

The angular lag, ¢, of the current increases from 15° in the entrance
subsection, station 120 to 140, to 90° at the head of the canal. To-
ward the entrance the flow becomes largely frictional, while near the
head it is essentially frictionless.

224

The range of the tide decreases from 6 feet at the entrance to 5.0
feet at a point 40,000 feet (about 8 miles) up the canal, and thence
increases to 5.8 feet at the head of the canal. High water at the
head of the canal is 60°.8/28.98=2.1 mean solar hours later than at
the entrance. The rate at which the tide progresses up the canal is
far from uniform. In the first subsection next the entrance it pro-
eresses at the rate of 8.2 fect per second, while in the upper 40,000
feet it progresses at an average rate of over 200 feet per second. The
rate of advance of a progressive wave in a canal of the given depth
would be /16g=22.7 feet per second.

The instantaneous profiles in the canal at successive lunar hours
are shown in figure 77.

In an even longer canal of the same depth the rate of progress of
the tide is found to decrease slowly from the entrance for some dis-
tance up the channel and thence increase rapidly toward the head. —

431. The primary current at the entrance reaches a maximum of
3.2 feet per second, and the strength of the positive current occurs

3 1°
I
xe HI
IV XI
V Vil pe SSS
35 Vi

Sta [40° 120) 100) SEO mNGOn OND O MEE

FIGURE 77.—Instantaneous profiles in closed cana} 140,000 feet long and 16 feet mean depth.

ad

1.5 mean solar hours before high water, or 1.6 hours after midtide.
The strength of the current decreases nearly uniformly to the head
of the canal. Near the head of the canal the strength of the positive
current is 30 minutes later than at the entrance, and occurs nearly
at midtide at the head of the canal.

Considerable deformations of the primary currents are to be ex-
pected in so long and shallow canal; but the many successive approxi-
mations necessary to bring the deformations of the tides and currents
into concordance would render their computation excessively laborious.

432. Computation for canal of varying cross section.—If the surface
width and the mean depth of a closed canal are not the same through-
out, the values of the coefficients, J, p, and la/g are determined for
each subsection, and the form of the computations is modified in the

225

same manner as those of a connecting canal of varying cross section,
illustrated in the second example (pars. 392-401).

MIDSTREAM CURRENTS

433. The computations developed in this chapter should afford
substantially as reliable an indication of the mean velocity at a given
cross section of a tidal canal as is to be expected of a computation of
the mean velocity set up by steady flow with a constant head. In
both cases, the reliability of the results depends upon the completeness .
of the data on the actual widths and depths in the canal, and on the
selection of the coefficient of roughness. The procurement of the
data for the computations generally entails much more effort than do
the computations themselves.

434. It should be recollected that the currents which will be en-
countered in the navigation of the canal are those in midstream and
that their velocity considerably exceeds the mean velocity in the
cross section. An analysis of the detailed meter measurements made
in the Cape Cod Canal in 1915, when its designed depth was 25 feet
at low water and its bottom width 100 feet, shows that the average
velocity in a vertical section at the middle of the canal was 25 percent
in excess of the average velocity in the entire cross section. While
the ratio of the midstream velocity to the mean velocity must depend
upon the contour of the bed of the channel in the vicinity of the cross
section, available data indicates that in general the strength of the
midstream current in a channel of regular dimensions should be taken.
as 1.3 times the mean strength.

435. The midstream current also turns later and reaches its maxi-
mum velocity after the mean current in the cross section. In a canal
of regular section this difference in timing usually does not exceed a
few minutes. In a wide natural channel differences of half an hour
or more in the time of the turning of the current near the shore and
at midstream are quite common (Manual of Current Observations,
Special Publication 215, U.S. Coast and Geodetic Survey, 1938).

Cuapter IX

TIDES AND CURRENTS IN ESTUARIES AND INLETS

Paragraphs

SPAR MEOMMCS UAT Y 52 ee te Re ae Ate RE eee ee 436
Piemarcuensuic: vides and Currents...- 22-2 J 42-2 ee ee 437-438
HommeEomsnadealtestuarys. 2a-e. ae see ee we Pe oe ee 438-445
iereyalence.omessuaries Of typical formes 8 ee | ae ee eee 446-448
Effet of local contractions and enlaigements________----_---_------- 449-450
Dec oROnanMMelsKoest Weta tORmMe a= sees fee) nee ee ee ee 451-453
HiectmoOlsinesh-water GisChalges= ae aae = 5255S 2 Se Se ot Se ee eee 454-456
Differences in tidal range caused by earth’s rotation____.__________-- 457-459
Overtides and overcurrents_—=—-=52__--_--= == aS <2) 2 See ee 460-462
TG] (OO ie. Se em ae ee ee ere nee eee nr ee Meee eee SEN Pe GE LE 463
Piecwor artificial changes im an estuary 2_2--_ 2. -222 5220.25 oe 464-477
Abram ets eAeSChipllOn= = 22s2 ce 22a eee ee ee ont pee 478-479
avanraulicsromniletiehannm el Seess Sees an te eee ee ee eee 480-481
sem ComnelailOnMGeees setae A bes Se ae ee ree yee ee ere ee 482-483
OvencuEnemespmril tS sae + alas eee ee ea eee eee ee 484

436. Definition of estuary.—The reversing currents produced by the
filling and emptying of the tidal prism of a river that enters a tidal sea,
generally dominate the river flow for a considerable distance up the
stream. This part of the river usually is funnel shaped, flaring to-
wards the entrance. A river mouth of such a shape is called an
estuary. The term may be applied as well to any tidal channel of
similar shape, even if it does receive any considerable inflow from the
uplands.

437. Characteristic tides and currents in an estuary—In a typical
estuary, the currents often have nearly the same strength and tides
nearly the same range at all cross sections; except in the upstream
reaches where the tidal flow merges into steady flow. The rate at
which high water and low water, and the strength and turn of the
current, advance up an estuary is often so close to gD, the rate of
advance of a frictionless progressive wave in a channel of uniform
dimensions (par. 339), that this is commonly regarded as the normal
rate of progress of the tide.

438. The ideal estuary.—In the preceding chapter it was shown that
the currents in a long closed canal of uniform cross section diminish
from the entrance to the head of the canal and the rate at which the
tide advances up the canal increases toward the head. The uniformity
of the currents in an estuary, and of the rate of advance of the tide,
evidently is due to its shape. It is of interest to determine the special

(227)

228

shape that a closed tidal channel of constant depth must have, in
order that a simple harmonic fluctuation of the tide at the entrance
will produce throughout the channel primary tides of constant range
and primary currents of uniform strength. In the lack of a better
term, a channel of this shape may be called an ideal estuary.

439. Derivation of the form of an ideal estuary—Taking the origin
of distances at the entrance from the sea, the positive direction up-
stream, and the origin of time at a high water at the entrance, let:

D, be the constant mean depth of the channel at mean tide.

2, its width at a point distant x from the origin.

r, its constant hydraulic radius at mean tide.

C, the applicable Chezy coefficient; also taken as constant.
A, the constant amplitude of the primary tide.

B, the constant amplitude of the primary current.

a, the speed of the primary tides and currents.

S, the amplitude of the surface slope at a point distant x from

the origin.

H°, the initial phase of the slope at Ahe same point.

¢, the angular lag of the current.

The relations established in paragraph 373 show that if B, 7, and
C have constant values in a given channel, the values of ¢ and S also
are constant throughout the channel.

The equation of the tide at the entrance is:

SAN COSROE:

Since the tide at a station within the entrance occurs at a later
time, its equation is in the form:

y=A cos (at—f) (296)

in which ¢ (zeta) is a positive angle which varies with z.
The surface slope at a point distant x from the origin, and at the
time ft, is then:

S cos (at+ H°) = dy/or=A sin (at—¢) 0¢/dxr
=A cos (at—f—71/2) 0¢/Oz. (297)

Since equation (297) is identically true:

S=Aodt/oz (298)
BP = eae (299)

From equation (298)

Oe Aor.

229

The integration of which gives, since ¢=0 when r=0:

¢=(S/A)z. (300)
It will be convenient to place:
S/A—n. (301)
So that:
—t (302)

The equation of the current at any cross section of the channel may
be written:

v—B sin (at+ B)
in which, from equation (150):
B=H°—¢—n/2
From equations (299) and (302):
B=—nzr—o—T.
The equations of the tide and current in an ideal estuary are then:

y=A cos (at—nz) . (303)
v=B sin (at—nx—¢—7)=—B sin (at—nr—¢). (304)

These equations show that the tides and currents progress up an
ideal estuary at the constant rate of a/n.

440. The differential equation of the primary current has been
derived in equation (142), paragraph 243:

oy/Ox+ (1/9) Ov/Ot-+ (8/37) Bo/C?r=0

Substituting the differential coefficients and the expression for »
obtained from equations (303) and (304):

An sin (at—nx) — (aB/g) cos (at—nr— 9)
— (8/37) (B?/C?r) sin (at—nr— ¢) =0 (305)

By placing at—nx=0, the equation of condition is derived:
— (aB/g) cos 6+ (8/37) (B?/C?r) sin ¢=0 (306)

and by placing at—nz=72/2:

An— (aB/g) sin ¢— (8/37) (B?/C?r) cos ¢=0 (3807)

230

Multiplying equation (306) by cos ¢ and equation (307) by sin ¢ and

adding:
An sin ¢—aB/g=0 (308)

441. The equation of continuity is (equation 183):
O(w2D) /dx-+ zdy/0t=0
When the depth, D, is constant, this equation becomes:
Dvoz/0r+ Dz0v/dxr-+ 2dy/dt=0 (309)
Substituting the differential coefficients and the expression for 2,
from equations (803) and (804):
—DB sin (at—nzx—¢)02z/0x-+nDBz cos (at—nx— ¢)
—aAz sin (at—nx) =0 (310)
The equations of condition, derived by placing at—nxz=0 and
at—nxz=1/2, are:
DB sin $02/0z+nDBz cos ¢=0 (311)
—DB cos ¢02/07+nDBz sin ¢—aAz=0 (312)
Multiplying equation (811) by cos @ and equation (312) by sin 4,
adding and dividing by 2z:
nDB—aA sin ¢=0 (313)
Combining equations (808) and (313) to eliminate sin ¢:

aB/gAn=nDB/aA
whence: has
TOG) n=a/VgD (314)

The rate of advance of the tide and current in an ideal estuary is
then gD, the rate of advance of a frictionless progressive wave.
442. From equation (311):

0z/z=—n cot dx

The integration of which gives:
e— Ke-™ cote (315)

in which K is the constant of integration. When z=0, K=z. K is
then the width of the estuary at the entrance, which conveniently may
be designated z. Then:
en ea (316)

Or:

log z=log z,.—(az cot ¢ log e))-/gD (317)

a
If, then, the depth of an estuary is constant, and the width varies in
accordance with the law expressed by equations (316) or (817), its

231

primary tides and currents will have a constant amplitude, and will
advance up the channel at the rate of /gD.

443. To determine the amplitude of the currents, the value of
P=1.084 07S and P/S may be computed from the value of S
derived from equation (301):

S=An=Aa/VgD (318)

The value of ¢ may then be obtained from table IX, chapter V. The
amplitude of the current is, from equation (308):

B=(Ag/VgD) sin 6=AYgQ/D sin ¢ (319)

As shown in paragraph 338, the amplitude of current of a friction-
less progressive wave, in a channel of uniform dimensions, is:

Ayg/D

The currents in an ideal estuary are therefore less than those set up
by a frictionless progressive wave:

Example—The mean tidal range at the entrance to the estuary
proper of the Delaware River, at Woodland Beach, is 5.63 feet. The
mean depth of the estuary at mean tide between Woodland Beach and
Philadelphia, taken from maps of about 1918, was found to be 21.5
feet. A reasonable value of the Chezy coefficient, C, is 120. Taking
the tides as simple harmonic fluctuations with a speed of the M,
component, the constants for computing the form of the ideal
estuary are:

A=2.815 feet

a=0.0001405 radians per second
D728

C=120

These values give:
S=0.00001504
log P/iS=5.19235
o—o1 2
The scaled width of the Delaware at Woodland Beach is 23,000
feet, the logarithm of which is 4.36173. The logarithm of the width
of an ideal estuary at a point distant x feet upstream, is then, from
equation (317):
log z=4.36173—0.000,0030405z
In figure 78, page 232, the outline of this ideal estuary is shown
in broken lines on a small-scale map of the Delaware. It will be
seen that the general shape of the river conforms quite closely to an
ideal estuary.

Actual shore line

—

ye deal esutialny,
—

Artificial
Island

O 50 OOO 100 OOO
[ic Sere pape he pete lls a ie |

Scale of feet

Woodland
Beach
FIGURE 78.—Delaware River, Philadelphia to Woodland Beach.

444, The actual mean tidal range in the Delaware increases from
5.63 feet at Woodland Beach to 5.85 feet at the contraction at Artificial.
Island, then decreases to 5.09 feet at Philadelphia. The rate of
advance of the tide up the ideal estuary would be 21.5g=26.3 feet
per second. The actual rate of advance of the high water from Wood- ©
land Beach to Philadelphia averages 23 feet per second, and of low
water, 19.5 feet per second.

The amplitude of the primary current, computed from equation
(319) with the given data, is 2.09 feet per second. The actual mean
current velocities at various cross sections of the Delaware, determined
by cubature, have strengths generally of about 2.00 feet per second,
increasing to 2.6 feet at contracted sections.

445. The equation of the tide in an ideal estuary (equation 303)
shows that high water at a point distant x from the entrance occurs
when afp—nz=0, or when f=nz/a. Similarly equation (304) shows
that the current turns from positive to negative, or from flood to ebb,
when at;—nz—@¢=0, or when t;=(nz+¢)/a. The interval between

233

high water and the turn of the current is then ¢/a. If then the Dela-
ware were an ideal estuary, the primary current at any station would
turn from flood to ebb 37°.35/28.98=1.29 hours after high water.
The actual currents turn in this direction from an hour to an hour and
a half after high water at the station.

The general characteristics of the tides and currents in the tidal
portion of the Delaware conform therefore to those deduced for an
ideal estuary.

446. Prevalence of estuaries of typical form.—The depth of a natural
tidal channel is far from constant, and the variation in its width which
would be required to produce currents of constant strength departs
somewhat from that of the ideal estuary deduced in the preceding
paragraphs. However, as a natural channel carrying a constant steady
flow tends toward a general uniformity of width, a tidal estuary tends
toward the funnel-shaped form of an ideal estuary. In a tidal channel
which has not such a form, the currents have different strengths from
section to section and the bed tends to scour where the currents are
the stronger, and to fill where they are the weaker. Tidal channels in
alluvial material therefore mold themselves into the typical estuary
shape. The result of this process is strikingly shown in the natural
channels through the tidal mud flats bordering a sheltered bay or
coastal sound, in which the wave action does not cause enough littoral
drift to contract the outlets. A glance at a chart of such a region, or a
view from the air, shows that the many channels cut through these
flats by drainage from the uplands have molded themselves into the
typical estuary form, generally with a sinuous alignment.

447. Large rivers which enter the sea through an alluvial coastal
plain also usually cut for themselves a typical estuary channel; unless
they carry down silt and sand at a faster rate than can be molded by
the tidal currents, when they maintain a generally uniform cross
section to an ever-growing delta at their junction with the sea. A
delta generally is found at the mouth of a silt-bearing river which,
like the Mississippi, enters a sea having a small tidal range; but the
burden of detritus may be sufficient to form a delta at the mouth of a
river where the tidal range is large. Thus deltas are found at the
mouths of the heavy silt-bearing rivers which enter Puget Sound,
although the diurnal tidal range in the sound is generally 10 feet or
more.

448. Many rivers enter the sea through submerged valleys, which’
ordinarily widen toward the sea and have the general shape of a self-
made estuary. If the valley has become filled with alluvial deposits of
fairly uniform consistency, the tidal flow generally has molded an
estuary of typical form, often subdivided by islands and _ shoals.
The entrance to an estuary from the open sea usually is contracted
by deposits from the littoral drift along the coast line; but this contrac-

234

tion does not affect greatly the tides and currents inside of the entrance.
In short, long natural tidal channels, other than tidal straits, are
usually of estuary form; and if they are not too deep, their tides and
currents ordinarily have the general characteristics of those of an
ideal estuary.

449. Effect of local contractions and enlargements upon the range and
rate of the tide.-—Variations in the consistency of the bed and banks of a
natural estuary result in local contractions and enlargements of the
cross section, so that the strength of the current is no more uniform
than is the current velocity in a natural upland stream whose bed and
banks are of similar material. The consequent variation in the
amplitude of the surface slope produces variations both in the tidal
range and in the rate at which the tide advances up the channel. The
nature of these variations may perhaps be developed most readily from
a diagram.

In figure 79, CP» is the generating radius
of the primary tide at the downstream, or
initial, end of a short section of a tidal
‘ estuary. If the currents were of uniform
strength, and the tides of constant range
throughout the estuary, the generating
radius of the tide at the upstream end of
the section would be CP,, equal in length,
but lagging behind CP, by an angle deter-
mined by the rate of progress of the tide,
A gH. The surface head in the section
would then be CH, equal and parallel to
PoP, (par. 244); and the generating radius
of the primary current would be CB, making an angle of —¢—90°
with CH (par. 248). Upland inflow disregarded, this current is due
wholly to the discharge at the section produced by the filling and
emptying of the tidal prism upstream therefrom. The phase of the
current CB has therefore a fixed relation to the phase of the tide CP,.

If, because of a local contraction at the section, the discharge pro-
duces a current of increased amplitude CB’, the amplitude of the head
is increased to CH’, but the angular lag, ¢, of the current with respect
to the head is decreased, so that the angle H’CH is greater than B’CB.
The generating radius of the tide at the upstream end of the section
becomes CP’,. The angle P,CP’, is nearly or quite equal to BCB’,
and the angle P,P,P’; is equal to HCH’. It is apparent from the
figure that the increase in the current results in a decrease in the tidal
range and an increase in time of the tide at the upstream end, with a
consequent decrease in the rate at which the tide travels through the
section. The decrease in the tidal range at the upstream end tends to re-
duce the discharge at the section, and checks the increase in the current.

FIGURE 79.

235

450. A local increase in the strength of the current in a section of an
estuary therefore decreases the tidal range upstream, and retards the
progress of the tide up the estuary. A local decrease in the strength
of the current tends to increase the tidal range, and speeds up the
progress of the tide. The decrease in the range as the tide passes
through the contracted sections usually results in a less range in the
wider and deeper sections upstream, while because of the tendency
toward an increase in the wide and deep sections, the range of the
tide normally increases as it approaches a contracted section. The
larger tidal ranges are therefore found ordinarily at the contractions,
and the smaller ranges in the wide and deep sections of an estuary.
The advance of the tide up the estuary is more rapid where the tidal
range is increasing than it is when the range is decreasing.

451. Deep channels of estuary form.—Submerged valleys, unfilled by
alluvial deposits, afford some long closed tidal channels, flaring toward
the entrance like an estuary, but so deep that the frictional resistance to
flow is very small. The convergence of the shores of the ideal estual
becomes less as the depth increases and the frictional resistance to
flow decreases. In a channel so deep that the flow is essentially
frictionless, ¢ is so close to 90° that cot ¢ is practically zero. The
width of the ideal estuary then closely approximates, from equation
(316):

2%

The ideal estuary becomes an endless channel of uniform width and
depth. It follows, therefore, that in a closed channel of finite length
the tides maintain a constant range, and the currents a constant
strength, only when the channel is so shallow that the frictional
resistance to flow is material. When its depth is so great that the
frictional resistance is negligible, the tides and currents take the
general characteristics of those produced by frictionless flow in a
closed canal, discussed in chapter VII. The wave lengths of the
principal tidal components become so long in a deep channel (par. 326)
that the length of nearly all natural channels is but a fraction of these
wave lengths, and the range of the tide characteristically increases from
the entrance to the head. If the effective length of the channel approxi-
mates one-quarter of the component wave lengths corresponding to its
depth, the range of the tide at the head of the channel may be very
large. In timing, the tides approach the condition of a stationary
wave, which rises and falls simultaneously.

452. Tides and currents in the Bay of Fundy—The Bay of Fundy,
on the Atlantic coast of Canada, just north of the State of Maine,
affords the outstanding example of the heights to which the tide may
rise at the head of a fairly deep natural channel. An interesting

192750—40 —16

236

description of the tides in the bay is given by Marmer in “The Tide”,
from which the figures herein have been abstracted. The bay ex-
tends 170 miles inland and there subdivides into two comparatively
small and shallow branches. The entrance to the bay is 85 miles wide,
and has a mean depth of 280 feet. The bay gradually narrows to 30
miles at the junction of the branches, where the mean depth becomes
130 feet. The mean tidal range increases from about 13 feet at the
entrance to 40 feet or more at the heads of the two branches, reaching
50 feet at spring tides. This is the greatest tidal range in the world.
The midchannel currents at the entrance to the bay have a strength
of 1% knots. The tide tables indicate that high water progresses 90
miles in 15 minutes in the deep water in the main part of the bay, but
the progress of the tide slackens in the shallower branch channels.
The current turns nearly at high and low water.

The mean depth in the bay may be taken at 240 feet. The corre-
sponding wave length of the principal lunar semidiurnal component,
M,, is 663 miles (par. 326). The length of the bay is therefore nearly
one-quarter of this wave length. As shown in paragraph 346, this is a
critical length of a closed canal of uniform dimensions, at which the
tides are limited only by frictional resistance. While the analogy is
far from accurate, it affords an explanation for the great tidal range
at the head of the Bay of Fundy.

453. Other examples of the increase in the tidal range in deep chan-
nels.—The Gulf of California, inside of the peninsula of Lower California,
is a deep channel extending inland over 700 miles from the Pacific
Ocean to the delta cone at the mouth of the Colorado. The mean tidal
range decreases from 4 feet at the entrance to 3 feet in a zone about
300 miles up the gulf, and then increases to 22 feet at the mouth of the
Colorado. In Cook Inlet, in Southwestern Alaska, a deep, funnel-
shaped channel about 200 miles in length, the mean tidal range in-
creases from about 12 feet at the entrance to 30 feet at the head.
Long Island Sound affords another and often quoted example of the
increase in the tidal range in a fairly deep closed channel whose length
and depth have the relation which should lead to this increase. The
sound has a prevailing depth of 65 feet and a length of 70 miles. This
length is approximately one-quarter of the wave length of the principal
semidiurnal tidal components at the given depth. The entrance from
the sea, at the eastern end, is contracted by a chain of islands, in the
passage between which the currents are strong, but these passages
are so deep and so short that the currents do not appear to produce
any considerable head. Inside the entrance the tidal currents are
weak. As is to be expected under these conditions, the tidal range
increases from 21 feet at the eastern entrance to 74 feet at the western
end of the sound. High water travels through the sound in about
half an hour.

237

In contrast, the Lynn Canal, in southeastern Alaska, is a narrow
fiord, 80 miles long, with a prevailing depth of 1,000 feet or more. Its
length is but one-twentieth of the wave lengths of the principal semi-
diurnal components at the depth. The mean tidal range increases
from 12.6 feet at Barlow Cove, near the entrance, to but 14.6 feet at
Skagway, at the head of the fiord. The tide is so nearly a stationary
wave that high water at the head occurs but 5 minutes after high water
at the entrance.

454, Hffect of fresh water discharge.—The fresh-water discharge of a
river increases the ebb currents in its tidal reaches and decreases the
flood currents. In the wide part of a typical estuary, nearer its
junction with the sea, the tidal discharge from the storage in nearly
the whole of its tidal prism may be so much greater than the fresh
water discharge that the latter has but little effect upon the currents

Trenton gage — feet

O 50000 lIOOO00O cfs
Discharge of Delaware River above Trenton

FIGURE &.—Relation of high and low water to fluvialdischarge at Trenton, N. J.

The discharge from tidal storage steadily diminishes upstream, the
ebb currents increase, and the flood currents decrease until a point
is reached at which the flood current disappears. Above this point,
the current fluctuates in velocity, but does not change direction.
The mean tide elevation in the river slopes upward from the sea at
an increasing rate as the ebb currents become the stronger. If the
river has so ample a cross section that the slope is small, the fluctua-
tions of the tide may extend far up the stream, diminishing as the
backwater from a dam diminishes, until at some point the tides
disappear, and with them the tidal storage, and the last traces of
tidal fluctuation in the river current.

Quite obviously, the range of tides in the upper reaches of a tidal
estuary diminishes when the fresh-water discharge increases, and may
disappear when the river is in flood; as the backwater from a dam
diminishes and eventually disappears with the increasing river dis-
charge. The observed heights of high water and of low water at a
tidal station may be plotted against the upland discharge to afford a

238

diagram showing the effect of the discharge upon the tides. Such a
diagram for the Delaware River at Trenton, near the head of tide,
prepared from selected monthly mean high and low waters and dis-
charges during the period 1922 to 1926, is shown in figure 80.

455. The tidal part of many of the larger streams entering the
Atlantic Ocean in the United States, terminates abruptly in the rapids
at which these rivers drop into the Coastal Plain, or into the sub-
merged valleys in which their tidal courses lie. The upstream tidal
reaches usually have the capacity to carry the ordinary river discharge.
During periods of low discharge the flow in these reaches becomes
almost entirely tidal, and in many cases the tidal range then increases
toward the head of tide, instead of gradually decreasing upstream.

456. Distribution of the currents due to fresh-water discharge.—As
fresh water has a less specific gravity than salt water, the salt water
usually underruns the fresh at the turn of the current, so that the ebb
continues on the surface while the flood current is running in beneath.
Numerous meter measurements made at various depths at the mouth
of the Hudson River show that the strengths of the ebb currents
generally are relatively less than the strengths of the flood in the
deeper part of the channel (Special Publication No. 111, U. S. Coast
and Geodetic Survey).

457. Difference in tidal range on the opposite sides of a wide estuary
because of the earth’s rotation —Unexpected as it may seem, the rota-
tion of the earth produces a measurable difference in the tidal ranges
on the opposite sides of a wide estuary. Consideration will show that
the earth rotates under the moving water in the channel, as it rotates
under a Foucault pendulum. At a place whose latitude is \, the rate
of rotation is 360° sin ) per (siderial) day or 0.000,072,9 sin \ radians
per second. In the northern hemisphere the currents, if unrestrained,
would rotate clockwise at this rate with respect to the earth. Since
the direction of the current in a channel is restrained by the banks,
the rotation sets up a slight transverse slope of the water surface.

Designating the rate of rotation of the earth about its axis, in
radians per second as w (omega) and the velocity of the current in
the direction of the channel by v, the transverse component of the
velocity, due to the earth’s rotation, if unrestrained, would be wv sin X.
Since the steadily exerted force required to restrain a body from mo-
tion at a given velocity is twice that necessary to accelerate it to the
velocity, the pressure acting on each unit of mass of the flowing water,
to restrain it in the direction of the channel is 2wv sin \, and the trans-
verse slope to produce this pressure is 2w” sin \/g. The difference in
level between the two banks of the channel is then 2wvz sin X/g, 2
being the width of the channel.

458. Ordinarily the flood current in an estuary is near its strength
at high water, and the ebb current near its strength at low water.

239

Looking upstream, as is customary in regarding channels which lead
in from the sea, the rotation of the earth therefore tilts the water
surface upward to the right at high water, and upward to the left at
low water; with the consequence that the tidal range on the right
(ascending) bank is greater than that on the left bank by 4wvz sin X/g.
Since w=0.0000729, and the value of g is not far from 32.16, this
increase in range becomes, when Zz is expressed in statute miles and v
in feet per second, 0.05 vz sin X._ If z is expressed in nautical miles of
6,080.2 feet, and v in knots, the difference in range is 2.92vz sin X/g, or
0.09 vz sin X.

459. The observed differences in the tidal ranges on the two banks
of a wide estuary conform fairly well with this formula. Thus at the
entrance to Delaware Bay the distance between the two shores is 10
nautical miles, and the average current at high and low water is
about 1 knot. The entrance is at latitude 38°20, whose sine is 0.62.
The difference in range between the two shores from the formula is
0.56 fect while the observed difference is 0.6 foot. At the head of the
bay the width is 4 nautical miles, the current is 1.3 knots, and the
latitude 39°23, giving a calculated difference of 0.3 foot, while the
actual range on the right, ascending, bank is 0.2 foot greater than on
the left. A similar concordance with the formula is observed in other
tidal waters.

\s 18 21 ray 3 6 ) 12 is 18 al R4
Octal Octala

Standard Time-—hours
FIGURE §1.—Tide curve of Delaware River at Philadelphia October 11-12, 1924.

460. Overtides and overcurrents in an estuary.—As is to be expected,
the tide advances more rapidly up an estuary, or any long closed
tidal channel, at high water, when the depth in the channel is the
ereatest, than it does at low water when the depth is the least. The
further a tidal station is up an estuary, the earlier is the time of high
water with respect to the time of low water. The time interval from
low water to high water, or the “‘duration of the rise,” steadily be-
comes less as the distance of the station from the entrance increases,
and the time interval from high water to low water, or the ‘duration
of the fall” becomes greater. The tide curve takes the typical saw-
tooth form exemplified by the tides on the Delaware at Philadelphia,
shown in figure 81.

240

As pointed out in paragraph 155, these deformations of the tide
curves are reproduced by overtides and compound tides whose periods
are multiples or sums or differences of the periods of the principal
tidal components. The corresponding deformation of the velocity
curve at a tidal station is the accumulated effect of the tidal distortions
at the stations upstream upon the rate of tidal storage and release of
water, and is consequently greater than the deformation of the tide
at the station. A typical shape of the velocity curve at Philadelphia
was shown on figure 50, page 155.

461. It should be noted that the deformation of the tides and cur-
rents as they travel up an estuary is due primarily to the difference
between the depth in the channel at high and at low tide, and depends
therefore on the ratio of the tidal range to the mean depth. Although
the deformation may be increased because of the stronger ebb and
weaker flood currents resulting from fresh-water discharge down the
estuary, the latter is not the essential reason for these deformations.

462. Slope of mean river level—Since in each section of an estuary
the ebb current runs out at the lower tidal stages and the flood current
runs in at the higher stages, the frictional resistance to the flow of the
ebb is greater than that to the flow of the flood current. As a con-
sequence the mean river level in an estuary has an upward slope from
the sea, even though the fresh-water flow is negligible. In a channel
deep enough to be navigable by ocean shipping at low tide, this slope
is very small. In a shallow estuary it may be considerable.

463. The tidal bore-—The successive instantaneous profiles in a
tidal estuary show the water surface advancing up the channel as a
long wave, outwardly resembling, in a general way, the advance of a
wind wave toward the shore. In nearly all estuaries the slope of the
front of the advancing wave is very small. This slope steepens as the
depth of the channel decreases, and as the currents increase with the
rate of rise of the tide. The rate of rise of the tide rarely is sufficient
to create an excessive slope on the front of the wave even when the
estuary is so shallow that much of its bed runs bare at low tide; but
if the range of the tide is so large that its rise is exceptionally rapid,
and if the fast rising tide encounters a strong outflowing current, the
advancing wave may trip and break, like a wind wave breaking on the
shore. The incoming tide then rushes up the shallows in a breaking
wave, generally called the tidal bore, but otherwise known as the
‘“Aegre” or ‘“Hygre” in England, the ‘“Mascaret” in France, and the
“Proroca”’ in South America. <A bore is formed in one of the shallow
tidal branches at the head of the Bay of Fundy; it forms also in the
mouth of the Colorado, at the head of the Gulf of California; and in
the shallow waters at the head of Cook Inlet, Alaska; but does not
appear to form in any other estuaries on the North American Conti-
nent. Because of the large tidal range at many localities on the

241

coasts of England and France, bores occur in a number of the shallower
estuaries of these countries. The most noteworthy tidal bore re-
ported is that in the Tsien-Tang-Kiang River, which enters Hangchau
Bay, some distance south of Shanghai, in China. The tidal range in
the mouth of the river, at ordinary spring tides, is given as 25 feet,
sometimes reaching as much as 34 feet. The bed of the river runs
nearly bare at low tide. The incoming tide advances in a breaking
wave which is described as from 8 to 12 feet in height, rushing with a
loud noise up the estuary at a rate of 14 miles per hour (Wheeler,
Practical Manual of Tides and Waves, pp. 142-144).

EFFECT OF ARTIFICIAL CHANGES IN AN ESTUARY

464. Comprehensive enlargement—A comprehensive enlargement of
the channel in a long tidal estuary, to afford greater depth and width
for navigation, generally increases the tidal range in the upstream
reaches and increases the rate at which high and low water travel up
the channel. The increase in range depresses the plane of mean low
water, and other low water datums. Additional excavation is there-
fore required to afford the projected increase in depth at a designated
low water datum. Thus, after the navigation channel in the upper
part of the estuary of Hudson River was deepened from 14 feet to
27 feet at mean low water, the mean low water datum at the head
of the improvement was lowered by a foot. The increase in the depth
of the navigation channel between Philadelphia and Trenton, from 12
feet to 25 feet, also depressed the low water datum at Trenton by a
foot.

465. Contractions.—A radical local contraction of an estuary by
training works, piers, or land reclamation, decreases the tidal range
upstream. The consequent reduction in the volume of the tidal prism
decreases the currents below the contraction and tends to increase
the tidal range at and below it. The removal of a marked local
contraction at midlength of a long estuary similarly increases the
storage and release of water upstream, increases the currents below
the contraction and may decrease the tidal range at and below the
site. A decrease of about half a foot in the mean tidal range on the
Delaware at and below Philadelphia, shown by a comparison of the
tide gage records prior to 1890 with those after 1900, usually is
ascribed to the contractions at the extensive training works which
were constructed in the lower part of the estuary during the interval.
The decrease in range may have been due partly, as well, to the major
enlargement of the river at Philadelphia during this period. This
enlargement included the removal of several islands which had so
contracted the cross section as to create excessive currents. The
enlargement was followed by a considerable increase in the tidal range
at the head of the estuary.

242

466. Dams.—The construction of a dam across an estuary, to
maintain the upstream reaches at low tide level in the interest of the
reclamation of tidelands, or at high tide level in the interest of naviga-
tion, recreation, and sightliness, obliterates the tidal storage up-
stream. The tidal currents in the downstream reaches are diminished,
and disappear at the dam. The tidal range at the dam is increased
by an amount dependent upon the length and depth of the remaining
part of the estuary. The accumulation of silt in the channel below
the dam ordinarily is to be expected.

467. Character of computations of the effect of enlargements or con-
tractions.—In the preceding chapter a method was developed for
computing the tides and currents in an artificial channel of such
regular dimensions that the Chezy coefficient in the successive sub-
sections could be selected with sufficient assurance from precedent.
A somewhat different problem arises in estimating the changes in
the tidal ranges and currents that may be expected from projected
enlargements or contractions which merely will modify, without
essentially changing, the characteristics of the flow in a long tidal
channel. The latter problem is somewhat analogous to an estimation
of the changes in the slopes and currents of an upland river because
of similar enlargements or contractions. In both cases the imme-
diate effect upon the currents at the locality is easily determined, but
a reliable computation of the consequent effect in other parts of the
channel can be secured only from an elaborate and painstaking
analysis of the existing flow in the successive subsections, based on
adequate survey and records.

468. Fortunately, a computation of the changes in the tidal ranges,
tidal datums, and currents because of projected enlargements or con-
tractions of a tidal channel is called for but rarely. If a closed chan-
nel is relatively short, the datum throughout it can be taken as the
established datum at the entrance, whatever the scope of the proposed
improvement; for while, as shown in paragraph 427, the tidal range
at the head of such a channel may be greater than at the entrance, the
increase in range and the lowering of the low water datum at the head
of the channel generally is too small to be of real consequence.

Projected local enlargements or contractions of a long tidal estuary
rarely are so extensive that any material change in the low water
datums need be apprehended. Because of the daily variation in low
water, a change of even a foot in the low water datum, resulting from
major channel enlargements, is not immediately apparent. An early
redredging of the channel often is required in any event to remove
material which has slid in from underwater slopes or has been deposited
from other sources. If a projected improvement has been cut so close
that a shortage of a foot or so in the depth at low water is of any real
consequence to shipping, its further enlargement is to be foreseen.

49

Except, perhaps, for providing initially a reasonable margin of in-
creased swept depth over areas that must be drilled and blasted, the
sensible procedure in nearly every case is to lay out the work from
existing low water datums, and to determine any required changes in
these datums by direct observation after the improvement has been
made. Only in most exceptional cases will doubt or controversy over
the consequences of the effect of enlargements or contractions justify
a prior computation. In the following paragraphs, an outline is
suggested of computations which should afford results in which some
confidence may be placed when the flow is essentially tidal. If the
fresh-water flow dominates the currents and tides, recourse to a
hydraulic laboratory might be necessary.

469. Computation of changes in mean low water datum.—If the tides
are of the semidiurnal type and if the ordinary fresh-water flow is
small in comparison with the tidal flow, or if the adopted mean low
water datum is established from the tides during periods in which the
fresh-water flow is inconsiderable, the changes in mean tidal range at
stations along an estuary, resulting from proposed enlargements or
contractions of the channel, should be substantially proportional to
the changes in the primary tides corresponding to the mean tidal
fluctuations. The mean primary tides before improvement may be
determined from the tide records, and those after improvement
computed by the formulas developed in chapter VIII, paragraph 422,
with coefficients derived from the corresponding primary currents
before improvement.

470. Primary tides and heads before wmprovement—To afford the
requisite data for the computations, tide gages must be established at
suitable stations from the head of tide to a point at which the cross
section of the estuary is so large that the effect of the improvement
upon the currents will become too small to be considered. These
stations should be placed at the more marked changes in the cross
section of the estuary, and at such distances from each other that the
water surface between them will not depart materially from a plane
surface. They establish the ends of the subsections into which the
channel is to be divided. From the tide records during a period of 15,
or preferably 29, consecutive days of low upland flow, average tide
curves are prepared for each station as described in paragraph 304,
and the corresponding primary tides computed from the heights at
successive lunar hours as explained in paragraph 360. The coordi-
nate amplitudes of the primary tides are then determined. Their
differences between the successive stations give the coordinate
amplitudes of the primary heads between the stations, from which
the amplitudes, H, and initial phases, H°, of the heads in the sub-
sections are determined.

244

471. Primary currents before improvement.—The velocity stations
are midway between the established tide stations, and the storage
stations, generally, should be midway between the velocity stations.
The coordinate amplitudes, A sin a and A cos a, of the primary tide
at the storage stations are obtained by linear interpolation between
those at the established tide stations. Taking any representative cross-
section area of the estuary as a base, Mj, the values of J=aU/M,
between the velocity stations are computed from the surface areas U
at mean tide, a being 0.0001405 radians per second. The summation
from the head of tide of the values of JA sin a and JA cos a then gives
the values of (B/m) sin B and (B/m) cos 6B at the velocity stations,
from which the values of 6 and B/m may be determined. The average
or effective cross-section area, (/, in each subsection may be deter-
mined from a consideration of a sufficient number of plotted actual
cross sections. The multiplication of B/m by m=M,/M then gives
the amplitude B of the primary current in the subsection before
improvement.

472. Subsection coefficients before improvement.—Since the values of
H* and 6 in each subsection have been found, the angular lag, ¢, of
the current is determined from the relation expressed in equation
(290), paragraph 373:

¢=H°— B—90°.
This value should also satisfy the relation, from equation (289):
sin ¢=Bla/gH.

While the values of ¢ computed from these two equations should not
be widely apart, a complete agreement cannot be expected. The
vaiue of ¢ should therefore be computed from equation (290), the
length, /, in equation (289) taken as the virtual length of the subsec-
tion, and the value of the coefficient Ja/g computed from the relation:

la/g=(A/B) sin ¢.

The value of the coefficient p in each subsection is determined from
the relation, from equation (288):

p=B tan ¢.

473. Subsection coefficients after improvement.—If the area, U, of the
water surface between any of the velocity stations is changed by the
proposed improvement, the coefficient J=aU/M, must be recomputed
for the value of MM) originally chosen. The effective cross section
M,’ in each subsection after improvement, determined by a procedure
paralleling that used in the selection of the value, \/, before improve-

245

ment, gives the value of m’=M,/M’. The coefficient la/g remains un-
changed. Since p= (32/8) (a/g) Cr its value after improvement is

Pi=p(C?r/Cyr,)

in which CO and C,, r and 7; are the values of the Chezy coefficient and
the hydraulic radius before and after improvement. Often C, may
be taken as the same as C, so that

Pi=pr/ry.

The value of 7, should be computed by a procedure paralleling
that used in the computation of 7.

474. Completion of computation —The primary currents that would
be produced in the improved channel, if the tides were unchanged,
are first computed. If the values of J are not changed, the phases, 8,
are those already determined, and the amplitudes, B, are derived by
multiplying the values of B/m, previously found, by the new ratio m’;
otherwise the values of JA sin a and JA cos a are recomputed from
the primary tides at the storage stations and the values of (B/m’)
sin 6 and (B/m’) cos 6 found by their summation from the head of
tide. The coordinate amplitudes of the heads in the subsections cor-
responding to these currents are then computed from the values
ascertained for Ja/g and p,; the corrected coordinate amplitudes of the
tides derived therefrom; and the process repeated until the tides and
currents are in satisfactory concordance. The elevation of mean low
water after improvement is then found by multiplying the mean semi-
range of the tide at the station, as established by comparison or other-
wise, by the ratio of the computed amplitudes of the primary tides
after and before improvement, and subtracting the result from es-
tablished half-tide level.

475. Computation of changes in mean lower low water datum.—lIf the
tides are of the mixed type, and the adopted datum is mean lower low
water, the changes resulting from an extensive channel improvement
might be computed on the assumption that the ratio of the mean
range to the diurnal range will remain the same after and before the
improvement. The primary tides after and before improvement could
then be computed as outlined in the preceding paragraphs. The ratio
of their amplitudes at a station would then give the ratio of the ele--
vations of mean lower low water below established half-tide level after
and before the improvement.

476. Application to the approach to a sea-level canal.—The changes
that may be expected in the tides and currents in a confined approach
to a sea-level canal because of the flow in and out of the canal entrance,
could best be computed by determining the subsection coefficients in
the approach channel from tidal observations made before the canal

246

was opened. The computations of the tides and currents in the canal
would then be extended through the approach channel. The co-
ordinate amplitudes of the tides at the storage stations in the approach
channel used in the initial computation would be determined from the
primary tides at these stations derived from the observations.

477. Computation of the effect of dams or other works decreasing the
tidal prism.—The effect of a projected dam, or of other works which
would decrease materially the area of the tidal prism, upon the cur-
rents in an estuary or other channel, is definitely ascertained by mak-
ing a cubature of the channel with the prism unimpaired, and a
cubature from the same tides with the reduced prism. While some
increase in the tidal ranges below the dam is to be anticipated, the
counterbalancing effect of the increase ordinarily is not sufficient to
warrant consideration. Similarly any question that might arise on
the effect of the excavation of a considerable tidal basin in the upper
reaches of an estuary may be settled by comparative cubatures.

TIDAL INLETS

478. Prevalence of inlets—The littoral drift of sand and shingle
along the seacoast tends to build up beaches across the entrances to
the identations of the shore line. This process has formed the coastal
sounds and lagoons which are the prevailing feature of the coast line
of the United States from Maine to the Rio Grande, and which are
found occasionally on the Pacific and even the Alaskan coasts as well.
Inlets into most of these sounds are preserved by the currents set up
in these channels by the filling and emptying of the tidal prism. The
entrances to nearly all tidal estuaries are similarly contracted by
littoral drift, sometimes sufficiently to produce typical inlet channels.

479. Typical shape of inlet channels—The material carried by
littoral drift into an inlet channel is removed by the currents through
the inlet. At an inlet into a coastal sound, it is deposited in fan-
shaped bars in the approaches both from the sea and from the sound.
A typical natural inlet channel has a deep, narrow gorge through the
barrier beach, from which it spreads in both directions with diminish-
ing depth. The sea approach to the gorge often is through ill-defined
and shifting channels between sand bars. In the sheltered waters of
the sound, the bars may even build up into islands. If the basin is
small and shallow the approaches may become so prolonged and con-
stricted that the currents are no longer sufficient to cope with the
encroaching littoral drift, and the inlet closes. The entrance to a
large estuary, in which the ebb currents predominate, is often en-
circled by a crescent-shaped bar, well out to sea.

480. Hydraulics of inlet channels —The improvement of tidal inlets,
to afford stable and adequate channels for navigation across their
ocean bars, or for other purposes, has an important place in harbor

247

engineering. It often is accomplished by constructing jetties to im-
pound the littoral drift and to concentrate and direct the currents
over the sea bar. In the design of these works consideration must be
given to the cross section of the channel that can be maintained by
the tidal flow, and to the effect of a contraction or enlargement of the
channel on the tidal ranges in the basin and the consequent currents
through the inlet. A mathematical analysis of the relation between
the capacity of an inlet channel, the tides in a basin of given surface
area, and the currents in the inlet, is necessarily based on the assump-
tion that the inlet channel is of determinable length and regular cross
section, and the basin so deep and of such limited area that its tides
have the same timing and the same amplitude at all points. The
approach channels of a natural inlet depart so far from these ideal
conditions that the computation of the tidal currents in them is as
uncertain as is the computation of the currents in an irregular shoal
reach of an upland stream. Even a channel between parallel jetties
is apt to have an unpredictably irregular cross section. Furthermore,
the tides at stations on a wide and comparatively shallow basin do not
rise and fall simultaneously, but become progressively later the more
distant the station from the entrance. Space will not therefore be
taken for a mathematical analysis of inlet tides. Their outstanding
characteristics may be inferred from elementary hydraulic relations.

481. It is fairly evident that the frictional resistance in the con-
stricted channels through an inlet must reduce the amplitude of the
tides in the basin, and delay the rise and fall of these tides, so that high
and low water in the basin are later than in the sea off the inlet. If
the constricted channels of the inlet are relatively short, the currents
must become excessive before the friction head can be sufficient to
have any material effect upon the tides in the basin. Stable short
inlets through erodible material therefore are usually so large that the
tidal range inside the inlet is practically the same as that outside. If
the improvement of such an inlet is so designed that the discharge,
determined from a cubature of the recorded tides in the basin, will
not produce excessive currents, no apprehension need be felt that the
improvement will have any material effect upon the tides in the basin,
or reduce the tidal discharge. Again, the straightening and deepen-
ing of inlet approach channels which have become so filled and pro-
longed as to throttle the tidal range in the basin may be expected to
increase the currents in these channels, and increase the tides in the
basin, until the channels have been viven a sufficient capacity to nearly
equalize the tidal range in the basin and in the sea. In either case
the maximum cross section of a self-maintaining channel is determined
by the volume of the unimpaired tidal prism jn the basin.

248

482. Observed relations between the volume of the tidal prism and the
capacity of inlets on the Pacific coast of the United States —A compilation,
made by Prof. M. P. O’Brien of the University of California, printed
in Civil Engineering, May 1931, shows that the area, at mean tide, of
the cross section at the throat of the entrances to the estuaries and
bays on the Pacific coast of the United States, conforms quite closely

to the relation:
Mele QOMV2 =>

in which M is the area of the entrance in square feet, and V is the
volume of the tidal prism of the basin between MLLW and MHHW,
in square mile-feet.

It should be observed that the tides on this coast are of the mixed
type, whose sequence is such that lower low follows higher high water.
The diurnal range, from MLLW to MHHW,, therefore affords a measure
of the stronger ebb currents.

483. A study made in office of the Pacific Division, United States
Engineer Department, by Mr. Grimm, principal engineer, shows that
the area of the cross section over the ocean bars of the larger estuaries
of the Pacific coast of the United States, at MLLW, is from 1.04 to
1.26 square feet per acre-foot of tidal prism in the basin between
MLLW and MHHW. The corresponding average strength of the
ebb currents is about 2 feet a second.

484. Overcurrents in inlets —The currents in some inlets are much
distorted by the overcurrents produced by the variation in the area
of the water surface in the basin, and in the area of the cross section
of the inlet, with the rise and fall of the tide. The curve of the flood
velocities in such an inlet may rise rapidly to a maximum, fall off, and
again rise to a second maximum, before turning to the ebb. The ebb
currents may go through a similar variation.

CHAPTER X

OFFSHORE TIDAL CURRENTS, REDUCTION OF CURRENT
OBSERVATIONS, AND CURRENT PREDICTION

Paragraphs
SEIN? UIC 2) CUI RE MSs Lo =o rr ae a Se er Te 485-486
ola CUIRLe Mb saClAPT AM Smee ea os 2 Bee LG eke ee 487-497
FLCC CiMONCO RC UEC (Smeeeeemen eens 2 ed ee SN ee a eee Efe 492-496
EipmuenicaanalysisiOl bidaleecurrents. = 22) 309-2 2 tee 496-491
Nonharmonic reduction of current observations_____________._______ 500-505
Viv iO! CUGHRGIN = Be ee a Ree be oe ee ee eee ee 509

OFFSHORE TIDAL CURRENTS

485. Rotary tidal currents —The tidal flow heretofore considered has
been that in a confined channel, in which the currents periodically
reverse their direction and pass through zero at each reversal. A
consideration of the tide producing forces, developed in chapter I,
shows that their direction is rotary rather than reciprocating. As is
perhaps to be expected, the action of these forces on the whole mass of
water in the oceans tends to produce rotary movements of the current
at offshore tidal stations. At such stations, the currents usually veer
around the compass during the tidal cycle, and have no periods of
slack water. These are called rotary currents. At most offshore
stations in the Northern Hemisphere the direction of the current
turns clockwise, and in the Southern Hemisphere, counterclockwise.
The velocity usually varies during the semidiurnal tidal cycle between
two maxima, in approximately opposite directions, and two minima
whose directions are nearly at right angles to the directions of the
maximum velocities.

486. Nontidal currents—The periodic tidal currents at offshore
stations are generally weak and may be much modified by permanent
currents of fairly constant strength and direction produced by the
circulation of ocean waters, and by temporary currents due to winds
and other meteorological causes. The Gulf Stream and the Japan
Current are well known permanent currents.

487. Polar current diagrams.—Offshore currents are conveniently
represented by laying off the current strengths at say hourly intervals
on radiating lines (radii vectores) drawn from a common center (pole)
in the direction of the current. The curve through the ends of these
vectors is the polar curve of the current. The time is marked on the

(249)

250

vectors. Since the directions and velocities of the current are
repeated, with some variation, at intervals of the periods of the tidal
cycles, and since high and low water at any tidal station in the same
region are repeated at nearly the same intervals, the times marked on
the diagram generally are re-
ferred to the times of high and
low waters, or of the principal
current phases, at a well-estab-
lished tidal station.

488. Shapes of polar current
curves.—In regions where the
tides are of the semidiurnal type
the currents are nearly identi-
cally repeated during each suc-
cessive semidiurnal tidal cycle,
and the current curve usually
has an elliptical shape, exempli-

fe) |Knots fied by the mean current curve
; at Nantucket Shoals Lightship,

FIGURE 82.—Mean current curve for Nantucket Shoals
Lightship, referred to tides at Boston. figure 82, taken from. the Man-
ual of Current Observations,
United States Coast and Geodetic Survey (Special Publication No.
215). The times marked on the diagram are referred to the times of
high and low water at Boston. Thus “H—2” marks the current 2
mean solar hours before high water at Boston, and ‘‘L+3” the current

3 hours after low water at Boston.

489. In regions where the diurnal inequality of the tides is con-

L+3

LH+3
Oo 0.5 1.O Knots
ea ee) Se ee

FIGURE 83.—Tidal Current Curve, Swiftsure Bank Lightship. Refer-red to predicted time of tide at
Astoria, Oreg.

siderable, the currents during the two semidiurnal cycles have a
corresponding inequality, and the daily tide curve describes a double
loop, exemplified by the mean current curve at Swiftsure Bank Light-

2o1

LL+3
N
LL+2
is LH+I
LL+] Ds L HH-2 LH+2
eae ||
IKWNO
IX S APLIHL- 2
| Y HL-|
Ae
S Hs HL

L =
(@) E 2.5 Knots
Lacs. eee Eee ES eee |

FicuRE 84.—Tidal Current Curve, San Francisco Lizhtship. Referred to predicted time of tide at
San Francisco (Golden Gate), Calif.

ship, off the entrance to the Strait of Juan de Fuca, figure 83, taken
from the same source.

The times of the currents on this diagram are referred to higher high
water (HH), lower low water (LL), lower high water (LH) and
higher low water (HL) at Astoria, Oreg.

490. At some tidal stations the second current loop may become
very small, as shown in the mean current curve at the San Francisco
Lightship, 10 miles off the entrance to San Francisco Bay (fig. 84).

The current swings through a nearly complete circle, and then
swings backward and forward through a limited are before it resumes
its swing around the compass. The behavior of the currents at this
and other similar stations varies greatly with the declination of the
moon. “At the time of equatorial tides the curve has two nearly
equal loops and the current swings around the compass twice during
the day. At the time of tropic tides, the secondary loop becomes
very small or vanishes altogether, and the current makes but one
daily swing entirely around the compass.

491. Combination of constant and rotary currents.—In figure 85,
PP,P, is the current curve, and QO, at its geometrical center, is the
pole, of a rotary tidal current. The vector OP then represents the
direction and velocity of the tidal current at a given time.

192750—40 17

202

If a constant (nontidal) current at the station has the direction
and strength O’O, the resultant of the tidal and constant currents at
the given time is O’P. Since P may be any point on the tidal curve,
the current curve of the resultant is the same as the curve of the rotary
tidal current; but the pole is
shifted to O’. This shift is in
the direction opposite to that
ot the constant current and
through the distance repre-
senting its velocity. If the
velocity of the constant cur-
rent exceeds the tidal, when
the latter has an opposing di-
FIGURE 85.—Combination of constant and rotary rection, the pole O shitts to a

currents. 3 ‘
point O’’, outside of the curve.
The direction of the resultant current then swings to and fro in
the limited arc between the tangent vectors O’’P, and O’’ Py.

The position of the pole of the diagram in figure 84 shows a pro-
nounced constant set of the current toward the north west at the
station.

HARMONIC ANALYSIS AND PREDICTION OF TIDAL CURRENTS

492. Current tables—Advance information of the time at which
the currents in tidal waterways will change direction, and will reach
their strength at the flood and ebb; and of the maximum velocities of
the surface currents in the navigation channel at each flood and ebb,
is of such value to navigators that yearly current tables giving this
information for the tidal waterways in and adjacent to the United
States are prepared and printed by the United States Coast and
Geodetic Survey. The tables give the predicted times of slack water
as the current turns from ebb to flood, or ‘‘slack before flood,” and
from flood to ebb, or ‘‘slack before ebb,’’ and the times and velocities
of the maximum flood and ebb currents, on each day of the year,
at a considerable number of reference stations. They also give the
corrections to be added to or subtracted from these times to obtain
the predicted times at a large number of secondary stations, and the
factors for reducing the predicted current strengths at a reference
station to those at the secondary stations. Most of the stations
listed are in confined channels at which the currents are of the revers-
ing type. Needless to say, the velocities in the tables are not the
mean velocities in the cross section of the waterway, which have
heretofore been dealt with, but are the surface velocities at definitely
located points or stations, so selected as to represent the currents
which will be encountered in the navigation of the fairway.

493. Preparation of current tables ——The fluctuations of the tidal
currents, like the tides, are caused by the tide producing forces of the

253

moonandsun. The currents at any station may therefore be resolved
into harmonic components of constant amplitudes, whose speeds
are the same as the speeds of the tidal components. The mean
amplitudes and the epochs of the several current components at the
selected reference stations are determined from an harmonic analysis
of the actual current velocities and directions at the station, measured
by float or current meter, at hourly or half-hourly intervals for a.
sufficient number of days. The dials of a tide-predicting machine are:
set at the component current amplitudes reduced to the current year;
and at the component phases at the beginning of the year; and the
current predictions at the reference stations are run off like the pre-
dictions of the tides. At stations where the tide is of the rotary
type, the harmonic constants of the east-west and north-south com-
ponents of the tide may be similarly computed, their resultants in
the prevailing direction of the maximum and minimum currents
ascertained, and the predicted times and strengths of the currents in
these directions run-off from the machine.

494. The corrections to be applied to the predicted times and
strengths of the current at a designated reference station to obtain
those at a secondary station are derived from the average intervals
between a lunar transit and the times of slacks and strengths at the
two stations, and the average tidal current velocities at the strengths
of the current. The compilation of this data is termed the non-
harmonic reduction of the observations, as distinguished from the
harmonic reduction by which the harmonic constants at the reference
stations are obtained.

495. Accuracy of tidal current predictions—The actual times of
slack or strength of the current at a station occasionally differ by as
much as half an hour from the predicted times, and in rare instances
by as much as an hour. Comparisons of the predicted and observed
times show that more than 90 percent of the slack waters have been
within half an hour of the predicted. Both the times and the strengths
of the currents in tidal estuaries may be greatly altered by unpredict-
able variations in the fresh-water discharge, and in inlets and straits
by the storm tides and lesser variations due to winds and other meteor-
ological disturbances.

496. Methods employed for current observation and reduction.—The
procedure adopted by the Coast and Geodetic Survey in taking, re-
cording, and reducing current observations is set forth in detail in the
Manual of Current Observations (Special Publication No. 215, U.S.
Coast and Geodetic Survey). The harmonic reduction and predic-
tion of tides has been explained in chapter IJ. The harmonic con-
stants of the tides, besides providing the means for tidal predictions,
afford an understanding of the variations in the tide, and of the tidal
datum planes to which works in tidal -waters are referred. Because

254

of the variation in the currents at different points in a cross section of
a tidal waterway, the harmonic constants at individual current sta-
tions are not of such general interest, but a summary description of
some of the processes employed in their computation may not be out
of place.

497. Harmonic analysis of reversing currents.—Current measure-
ments at a station in tidal waters usually must be made by a party of
some size, from a suitable boat, anchored accurately in position.
Consequently current measurements generally are fewer than the tidal
observations used for harmonic analysis. If hourly current obser-
vations for a 29-day period are available, the harmonic constants for
the groups of the principal lunar and solar components, M and S,
including their overtides, and the N», K, and O, components, usually
are determined directly from the observations by precisely the process ©
used in determining the tidal harmonic constants. These compo-
nents are sufficient for current prediction. The overcurrents, such
as M, and M,, generally are proportionally larger than the correspond-_
ing overtides. If a longer series of observations is available, other
components may be included. If hourly observations for a 29-day
period are not available, harmonic analyses are made both of the
currents at the station during the limited period of the observations,
and of the concurrent tides at a standard tide station. The mean
value of the amplitude of each current component and of its over-
currents is then found by multiplying the amplitude computed from
the observations by the ratio of the established mean amplitude of the
corresponding component of the tide at the base station to the tidal
amplitude computed from the concurrent observations. The epoch
of each current component is found by applying the differences be-
tween the initial phases of the current at the current station and the
tide at the base station, computed from the concurrent short-term
observations, to the established epoch of the tide at the base station,
corrected for the difference in the longitudes of the two stations.
The predicted hourly heights of the tide at the base station, instead
of the recorded heights, usually are preferred for this comparison,
since accidental meteorological disturbances of the tide may not
produce corresponding changes in the current at another station.

498. Prediction of currents in tidal straits —It was shown in para-
eraph 256 that if a channel is so short that its currents are but little
modified by the storage and release of water in its tidal prism, and if
the fluctuating surface head between the entrances produces such
strong currents that the flow is essentially frictional, or “hydraulic,”
the current lags behind the head by but a small angle, and the square
of the successive strengths of the current is closely proportional to the
nearly concurrent maximum surface heads. The amplitude and phase
of each component of the surface head in the strait may be determined,

by the procedure set forth in paragraph 239, from the amplitudes and
phases, referred to a common origin of time, of the corresponding
components of the tides at stations at the two entrances. The tabu-
lated epoch of each of the tidal components is the difference between
the phases of the equilibrium and actual tidal components at the
station. To transform these epochs to a common origin of time, they
may be converted into Greenwich epochs, by adding the longitude
of the station multiplied by the subscript of the component (paragraph
120). After the harmonic constants of the head have been deter-
mined, the predicted heights and times of the two daily maximum
heads in the strait may be run off on a tide-predicting machine. The
relation between the square of the strength of the current at a selected
station in the strait and the corresponding head, and the lag of the
current with respect to the head, are both determined from the
averages of an adequate number of current measurements at the
station. From these relations, the predicted times and heights of the
heads are readily converted into predicted times and strengths of
the current. By applying suitable scales to the tide-predicting ma-
chine, the times and strengths of the currents may be read off directly.

499. Harmonic analysis of rotary currents—Data on the rotary off-
shore currents are provided principally by hourly measurements of
the current directions and velocities at the lightships operated by the
Lighthouse Service of the United States. The south-north and west-
east components of the observed currents are analyzed, and their
harmonic constants in each direction determined. It is not difficult
to show that the current curve of the resultant of each of the harmonic
components, produced by combining its coordinate components in
the two directions, is an ellipse. The resultant currents of the com-
ponent are a maximum and a minimum in the direction of the major
and minor axes of the ellipse. The azimuths of these maximum and
minimum currents are determined from the coordinate amplitudes
and epochs of the component, by a formula whose derivation and
application need not be repeated here, and the harmonic constants
of the component in these directions determined. The direction of
the maximum and minimum velocities of the resultant of all of the
components nearly coincides with the axes of the principal lunar
semidiurnal component, M>. By transforming all of the components
to these axes, the strengths and times of the current in these directions
may be predicted.

500. Computation of average times of reversing currents.—The succes-
sive times of slack water and of the strengths of the current at a station
ordinarily are taken off a plot of the hourly or half-hourly current
measurements. The respective intervals after the preceding pre-
dicted high or low water at an established reference tidal station in
the vicinity, or the intervals after the times of slack and strength at

256

an established reference current station, are ascertained and averaged.
If the tides and currents have much diurnal inequality the intervals
of the greater flood and ebb strengths preferably are referred to the
times of higher high and lower low water. By adding the established
intervals between a lunar transit at Greenwich and the times of the
tide or current phases at the reference station, the Greenwich intervals
at the given station are then determined. These Greenwich intervals
are preferred to lunitidal intervals reckoned from the time of a lunar
transit at the station, because the difference between the Greenwich
intervals at any two stations gives the difference between the respective
times of their currents directly, without any correction for the differ-
ence in the longitudes of the stations. If the current had a simple
harmonic fluctuation with the speed of the M, component, the dura-
tion of each increase in velocity from slack to strength, and of each ©
decrease from strength to slack, would be one-quarter of the semilunar
day of 12.42 mean solar hours. To establish a single time interval
for all four slacks and strengths at a station, the ‘“‘mean current hour”
at the station is computed by averaging the Greenwich intervals of
the strength of the flood, the slack before flood increased by 3.10
hours, slack after flood decreased by 3.10 hours, and strength of flood
increased or decreased by 6.21 hours, after bringing all of these sums
into the same semilunar day by adding or subtracting 12.42 hours
as ay be necessary.

501. In estuaries and tidal rivers the fresh-water flow may be so
great that the current remains in one direction and the velocity varies
from a maximum to a minimum without passing through slack.
Again, the overcurrents at some stations are so large that the current
reaches two maximum velocities during each flood, or ebb, or both.
The direction of the current may even reverse between these maxima.
The measures taken in these special cases need not be elaborated here.

502. Reduction of average current strengths.—Since the tidal currents
in estuaries and other closed channels, and in inlets to a closed basin,
are due to the filling and emptying of the tidal prism of the channel
or basin, the successive strengths of the tidal flood and ebb at a current
station in the channel are nearly proportional to the concurrent
ranges of the tide at a representative tide station on the waterway.
The average tidal flood and ebb strengths, determined from a short
series of observations, therefore may be converted into long-term
averages by multiplying them by the ratio of the established mean
tidal range at the tidal station to the average observed range during
the period of the current observations. Obviously, this correction
is not to be applied to any constant component of the current which
may be produced by fresh-water outflow, or other cause, during the
period of the observations.

207

503. Because the strengths of the flood and of the ebb occur at
different heights of the tide, the areas of the cross sections of the
channel are not the same at both and their velocities would differ
somewhat even if the flow were wholly tidal. For the purpose of
applying the correction, the tidal parts of flood and ebb strengths are
considered to be equal. The tidal current strength at the station
during the period of the observations is then taken as one-half of the
arithmetic sum of the mean observed flood and ebb strengths, and the
nontidal current as one-half of their algebraic sum, with the flood cur-
rent positive and the ebb negative. These tidal current strengths are
corrected to their long-term values by applying the factor derived
from the comparative tidal ranges at the reference station. The cor-
rected average flood strength is then derived by adding, algebraically,
the nontidal current to the corrected tidal current strength; and the
corrected ebb strength by the algebraical subtraction of the nontidal
current.

504. At stations in tidal straits, in which the flow is largely frictional
and determined almost entirely by the surface head between the
entrances, the average tidal current strength derived from a short
series of observations is multiplied by the square root of the ratio of
the established mean range at a suitable tidal station in the water-
way to the average observed range during the period of the current
observations.

505. Average polar curves of rotary currents —The rotary currents at
offshore stations usually are weak and irregular. To prepare anaverage
current curve at a station where the tides and currents are of the
semidiurnal type, such as that shown in figure 82, the directions and
velocities of all currents observed within half an hour before or after
a predicted time of high water at the reference station are summed
and averaged to give the average direction and velocity at the time
of high water at the reference station; those observed between half
an hour and an hour and ahalf after high water, to give the average
direction and velocity 1 hour after high water at the reference station;
andsoon. The reference times usually extend from 2 hours before to
3 hours after both high and low water at the reference station. Cur-
rents of the mixed type, such as those shown in figures 83 and 84, are
similarly grouped at the nearest hours at, before and after, higher
high, higher low, lower high and lower low water at the reference
station.

506. Any average constant current at the station may be determined
by resolving either the original observations or their hourly compila-
tions into south-north and west-east components. The algebraic
average value of these components in each direction quite evidently
is the component of the constant current in that direction. The
summation of the component velocities to derive these averages and

258

the subsequent subtraction of the constant component current, is
facilitated by adding to each component velocity an arbitrary con-
stant sufficiently large to make all of the quantities positive. The
direction and velocity of the resultant constant current may be
obtained from its components, after the subtraction of any arbitrary
constant that may have been added for the convenience of computa-
tion. The algebraic subtraction of the constant component of the
velocity from the hourly current components in either direction, gives
the hourly components of the tidal velocity in that direction. The
curve of the average tidal velocities proper may then be constructed
by finding the resultant hourly tidal currents. If the period of
observation is less than a month, the tidal velocities may be reduced
to better mean values by multiplying them by the ratio of the estab-
lished mean tidal range at the reference station to the average range
during the period in which the current observations were made.

507. Wind currents.—Analyses of the current observations at light-
ships have afforded useful information on the strength and directions
of the currents produced by winds in open waters. The results
indicate that as a general rule, along the Atlantic coast, the velocity,
in knots, of the current, produced by a wind of some duration, is
about 114 percent of the wind velocity in miles per hour; and along the
Pacific coast, about 2 percent. Because of the rotation of the earth,
the direction of the current tends to lie to the right of the direction of
the wind in the Northern Hemisphere, and to the left in the southern.
A Swedish mathematician, V. W. Ekman, has shown that if the depth
of the ocean was unlimited, the surface wind currents would have a
direction 45° to the right of the wind in the Northern Hemisphere,
and 45° to the left in the southern. (Arkiv for Mathematik, Astro-
nomic, 1905). A comparison between the recorded deviation of
vessels from their courses and the direction and strength of the winds
causing the currents to which the deviations may be attributed, is
said to confirm these relative directions of wind and current (Marmer,
The Tide, p. 165). Near the coasts, the direction of the current with
respect to the wind is modified by the configuration of the coast line.
Thus the current observations at the light vessels from San Francisco
to Cape Flattery show that the winds from the northeast, southeast,
and northwest quadrants produce currents which set 20° to the right
of the wind direction, winds from the southwest quadrant produce
currents 20° to the left, and winds from the south and west produce
currents which set with the wind.

It need not be remarked that these offshore currents are of more
concern to the navigator than to the engineer.

APPENDIX [|

EQUIVALENTS AND CONSTANTS
EQUIVALENT VELOCITIES

1 knot=1.69 feet per second=1.15 miles per hour.
1 foot per second=0.592 knot=0.682 mile per hour.
1 mile per hour=0.868 knot=1.467 feet per second.

LUNAR TIMES

Mean interval between lunar transits=12.42 mean solar hours.
1 mean lunar hour=1.035 mean solar hours.
1 mean solar hour=0.966 mean lunar hours.

MEAN SPEED, Myo, OF SEMIDIURNAL LUNAR TIDE

In degrees per hour, 28.9841 log 1.46216
In degrees per second, 0.008051 log 7.90586 —10
In radians per second, 0.00014052 log 6.14774—10

TABLE XI.—mgt in degrees and minutes, for integral values af t from 0 to 69

t 0 1 2 3 4 5 6 7 8 £
|
|
OBS ee 0 28°59’ 57°58’ 86°57’ | 115°56’ | 144955’ | 173°54’ | 202°53’ | 231°52/ | 260°51”
eS eee 289°50’ | 318°50’ | 347°49’ 16°48’ 45°47’ 74°46’ | 103°45’ | 132°44” | 161°42” | 190°42’
Vie See 219°41’ | 248°40’ | 277939’ | 306°38’ | 335°37’ 4°36’ 33°35/ 62°34’ | 91°33’ | 120°327
Oat ae 149°31’ | 178°30’ | 207°29’ | 236929’ | 265°28’ | 294927’ | 323°26’ | 3529257 21°24’ 50°23”
Ae so, 79°22’ | 108°21’ | 137°20’ | 166°19’ | 195°18’ | 224°17’ | 253°16’ | 282915’ | 311°14’ | 340°13/
Gan a aes wii 38°11’ 67°10 S6n08 125°87 154°8’ 183°7/ 212°6’ 241°5’ 270°4’
(See 299937 328°2/ 357°! 26°0/ 54°59’ 83°58’ | 112°57’ | 141°56’ | 170°55’ | 199°54’

Acceleration of gravity, g, at sea level, varies from 32.089 feet per.

second at earth’s equator, to 32.234 at poles.

Taking g=32.16 log 1.50732
m,/g=0.000,00437 log 4.64042 —10
g/M,= 228,890 log 5.35958

CIRCULAR CONSTANTS

ASS log 0.49715

In /360=0.01745 log 8.24188
8/3r=0.8488 loz 9.92882
3n/8=1.1781 log 0.07118
4/37/8=1.0854 loz 0.03559

(259)

AppENpDIx II

REDUCTION FACTORS F(Mn) AND 1.02 F, AND CORRECTION
FACTOR i/B.

DERIVATION OF F’ (Mn)

1. As pointed out in paragraph 171, the mean range, Mn, at a tidal
station varies slightly with the inclination, /, of the moon’s orbit.
The factor F(Mn) is applied to reduce a mean range derived from
observations extending over a month or year to the true mean value
during the 19-year period in which the orbit tilts to and fro as the
moon’s node makes the circuit of the ecliptic (par. 35). The values
of F(Mn) are derived from the relation between the mean range and
the harmonic components of the tide at a station.

2. Relation of high water to the amplitude of the Ms, component.—At
most tidal stations the M, component is so much larger than the
others that high water occurs near the time at which the ordinate of
this component is a maximum.

In figure 1, CP is the generating radius of the M, component at an
instant when its ordinate is a maximum, CR is the radius vector of the
resultant of all of the compo-
nents at that instant, deter-
mined by drawing successive
lines parallel and equal to the
generating radii of the other
components (par. 51), and TH
Bini is the ordinate of the resultant

on the tide curve. At the high
water immediately ensuing (or preceding), the radius vector of the
resultant is CR’, nearly, if not quite, coinciding with the Y axis,
its ordinate on the tide curve is T’H’, and CP’ is the position of the
generating radius of the dominant component. Let At be the time
interval in which R moves to R’ and P to P’; v, the corresponding
angle between CP and OP’; and let T’H’—TH—ay.

3. The height of mean high water above sea level is the mean of the
successive values of TH plus the mean value of Ay. Since the succes-
sive values of TH occur at intervals equal to the period of the prin-
cipal component, and at the instants at which this component is a
maximum, their mean is the amplitude, Mo, of this component,

(260)

261

increased or decreased by the constant values of any of its overtides
at these instants (par. 78). As the overtides are relatively small, the
mean value of 77 may be taken, for purposes of computing a cor-
rection, as M»). Ay is always positive, whether high water occurs
before or after the high water of the principal component. In the
long run, for every value of Ay occurring when high water is in the
lead, an equal value will occur when high water lags behind. Neg-
lecting the effect of overtides, the height of mean high water above
mean sea level is therefore the amplitude, M2, of the principal com-
ponent plus one-half of the numerical mean value of Ay.

4. Representing, for generality, the ordinate of the dominant com-
ponent as A cos (at+-a), and the ordinates of the other components
as B, cos (0,t+ 8,), By cos (bst+- Bo), ete., the equation of the tide takes
the form:

y=A cos (at+a)+B, cos (b)t+ B:)+By cos (bf + B.)+ ° + - (1A)

Since Ay is the change in y due to a relatively small increase, At, in f,
its value is approximated by differentiating the right-hand member
of equation (1), and is:

Ay=
—[Aa sin (at) a) +B,b; sin (bif>-+ 81) +-Byb2 sin (bof)+ 62+) - * > JAt 2A)

in which % is a time at which the ordinates of the dominant com-
ponent isa maximum. Such times occur when at+a=0, 27, 47, 67,
ete. The value of t is given by the equation:

atyta=2nr
whence:
thy=2nr/a—a/a (3A)
where 7 is any integer.
Substituting this value in equation (2A):

Ay=—[Aa sin 2n7+B,b, sin (2n7b,/a—ab,/a+ B,)
+ Bobo sin (2n7b2/a— ab:/a+ Bo) + oe eS JAt. (4A)

Since the generating radius CP of the dominant component moves
through the angle v with the speed a in the time Af:

At=—av
Placing for convenience

2n7b,/a—ab;/a+ pia 2nrb,/a— ab,/a+ Bu=o, etc. (5A)
Then, since sin 2n7=0, equation (4A) reduces to:

Ay=—[B,b, sin t+B2b2 sin % * * * Jav. (6A)

262

An expression for v remains to be found.

5. The maximum values of y, equation (1A), occur when dy/dt=0,
or when

— Aa sin (at,+ a) — B,b, sin (6,t+ B,)—Byby sin (Ost Go) =— + ae)

At these maxima, the radius vector, CR’, is so close to the Y axis
that the angle R’CP’ may be taken as equal to PCP’=v. Since the
generating radius of the dominant component is at CP’ at the maxi-
mum values of y,

at; ta=2nr-+v.
whence
t,=2n7/a—a/a+ov/a. (SA)

Substituting this value in equation (7A):

Aa sin (2nx+v)+B,6, sin (2n76,/a—ab;/a+vb,/a+ B;)
+B5b. sin (2n7b2/a—ab,/a+vb,/a+ B.)+ °° > =0. (9A)

The first term in equation (9A) reduces to Aa sin ». Simplifying
the remaining terms by substituting z,, 72, etc., for the equivalent
expressions given in equation (5A), the equation reduces to:

Aa sin v+B,b; sin (7,-+6,v/a) +.B2b, sin (a%+630/a)-+ - - -=0
Expanding the sine functions:

Aa sin v+8,6, sin 2, cos vb,/a+B,b; cos a sin vb,/a
+ Bb, sin x cos vb./a+Byb2 cos z2 sin vb,/a+ -*- -=0 (10A)

The fractions 6,/a, 63/a, etc., are the ratios of the speeds of the various
components to that of the dominant component. For semidiurnal
components these ratios are close to unity, and for diurnal compo-
nents close to one-half. The angle vis not large at any time unless the
tide approaches the diurnal type. The values of sin vb,;/a, sin vb./a,
etc., are therefore approximately equal to b,v/a, b v/a, etc., respectively,
and the values of cos vb,/a, cos vb2/a, etc., are nearly unity. Sub-
stituting these values, equation (10A) becomes:

Aav+B,b, sin 7, +.B,b;70/a cos 2; +.B.b2 sin 2.+.Bb,"v/a cos %2+ * > *=0

whence:

ee By,b, sin #,+.B,b. sin %-+- >> * (1A)
iat Aa+B,bi7/a COS XY + Bb27/a Cos Lo-|- see

263
Substituting this value in equation (6A):

Aq= —Pibi sin Bobs sin tyr + + +)?
y A@+B,b2 cos 2,-+Brbs? COS 2:

esate sin? Ty + Bb? sin? Yo+ $ +2B,B5b,b, sin Ly sin Lo+ :
Ad+B,b,? cos 7, + Bb? cos + +: (12A)

6. Mean value of Ay.—The symbols 2, x, ete., in equation (12A)
represent angles in the form (equation 5A):

r=2n7b/a—ba/a+ B

where 7 is an integer.
As successive integral values are assigned to n, x increases by:

27b/a=27(b—a)/a+2r.

At each increase in n, the value of x increases, therefore by
27(b—a)/a. As the speed, 6, of any semidiurnal component does not
differ greatly from a, the speed of the dominant component, the
fraction 27 (6—a)/a is comparatively small for such components.
The successive values of x steadily increase (or decrease) with each
increase in n by an angle which describes a small fraction of the cir-
cumference. The speeds of the diurnal components (except My,)
differ by a relatively small amount from one-half of that of the domi-
nant component M,. For these components the value of x steadily
increases by a little more or less than 180° with each increase in n.
In either case the values of x fall uniformly, in the long run, over the
entire range of angles from 0 to 27, and the mean values of the trigono-
metric functions of x in equation (12A) become their true mean
values as x varies from 0 to 27. The mean value of sin 2x between
these limits is one-half, while that of cos z,and of the products of the sines
of the differently varying angles 7, x2, etc., is zero. Aside then from
the effects of the M; component and the lunar overtides, the mean
value of Ay, becomes:

Ayo= }s(B,b/?+.B,7b2’+ °° *)/Aa? (13.A)

7. Mean high water in terms of the harmonic components.—Since the
height of mean high water above mean sea level is the amplitude of
the dominant component increased by one-half of the mean value of
Ay, it is given by the expression:

MHW=4A-+ 14(B,7b+By7b?+ * + -)/Aa?
= Af +4(Bb2/A?a?+ B,b.?/A’a?-+ + + *) (14.A)

264

in which A= M2,a=my); B,, Bo, etc., are the amplitudes of the other
harmonic components (except M, and the lunar overtides) and 6,,
bs, etc., are the respective speeds of these components.

Since the speeds of the lunar overtides are two, three, and four times
the speed of M:, the successive increments of z in equation (12A)
for these components, as 7 increases by successive integers, are 47, 67,
and 8a respectively. The successive values of the trigonometric
functions of « in that equation are therefore all identical. Similarly
the successive increments of x for the M, component are each equal z,
and for the M; component 3/2.¢ For all of these components the
mean value of sin? zis not %. The effect of these components on the
elevation of mean high water does not therefore follow the law
expressed by equation (144A). These components are however gen-
erally too small to affect the elevation of mean high water appreciably,
and the terms to be added to account for them need not be developed
here.

8. Mean tidal range —The elevation of mean low water below mean
sea level may be derived in the same manner as the elevation of mean
high water above: sea level, and with the identical result. The
expression for the mean tidal range is therefore:

Mn=2A(1-+ ¥,(By2b,2/A2a?+By?bs?/A2a2+ + + -)) (5A)

The factors 6,2/a?, 6.2/a?, etc., are close to unity for thesemidiurnal
components, and close to }4 for the diurnal. The ratios B,?/A?, B,?/A?,
are very small for those components whose amplitude is less than one-
twentieth of that of the M, component. Omitting the components
that rarely if ever exceed this ratio, equation (15A) becomes:

Mn= 2M.[1 + V (S75825/ Mecmas? + N.?no?/M.?m,.? + Ko2k,2/M,2m,?
+ Ky?k,?/M.?m,’-+0,70,?/M2?m3?+ P?p,?/M2’m2? + Q,q:?/M2"m2’)|
(164)

9. The numerical value of the mean tidal range derived from
equation (16A) is always substantially less than that derived from
direct observation. Aside from the effect of overtides and the approx-
imations introduced in the derivation of the formula, this deficiency
may be attributed to the fact that any accidental variation in the
water elevation occurring near the time of computed high water
increases the observed high water by substantially the maximum
amount of the variation if positive, but decreases the observed high
water by but substantially the minimum amount of the variation if
negative. In the long run, therefore, these variations effect a cumu-

265

lative increase in the observed high water, and, similarly a cumulative
depression of the observed low water. Equation (16A) establishes,
however, a logical basis for determining the corrections to be made
for the changing inclination of the moon’s orbit to the Equator.

10. Numerical value of F(Mn)—The amplitudes of the various
lunar tidal components during any particular year (or month) are
determined by applying the appropriate factor f=1/F to the recorded
mean values of these amplitudes (par. 125). For solar components,
the value of fis unity. The expression for the mean tidal range during
any particular year is then:

Mn’>= 2fM,{1 + il (Soso/fMo.my»)?-+ (fNon2/fM.m,)?+ (fKoke/fMom»)?
+ (fKaki/fMome)*-+ (f0,0:/fMsm»)?+ (Pipi/fM.m_2)?
cm (fQiqi/fM2m:)?}} (17A)

The factor to be applied to reduce the mean tidal range, as deter-
mined from observations during a particular year, to its true mean

value is therefore:
F(Mn)=Mn/Mn’ (18 A)

in which the value of Mn is given by equation (16A) and the value of
Mn’ is given by equation (17A).

11. The computation of the value of # (Mn) for the true ratios of
the amplitudes of the actual components of the tide at a tidal station,
and for the successive values of the reduction factors f corresponding
to the inclination J of the moon’s orbit to the equator, would be a
very laborious process, not justified by the accuracy of the results
secured. A sufficient approximation is afforded by taking for the
ratios of the semidiurnal components the ratios of the mean values of
the coefficients of the corresponding equilibrium components, set
forth in table IV, paragraph 129. The ratios of the amplitudes of
the diurnal components to M, vary widely at different tidal stations,
but these amplitudes have a fairly consistent ratio between themselves.
The index for the amplitude of the diurnal components is therefore
taken as the ratio of K,+O, to M,2 at the tidal station, the ratio of
the diurnal components to K,+O, being taken as that of the mean val-
ues of the coefficients of the corresponding equilibrium components,
as given in the same table.

12. Equation (16A) may be written:

Mn=2M,{1 + (S_82/2.Mom.)?-++ (None 2M.m,)?-+ (K3k»/2M.m,)?
+[(K,+0,)?/M.?] [(K,k,/2 (K,+0O,)m,)?-++ (O,0;/2 (K,+0;)m,)?
+ (Pip:/2 K+ O,)mz)?+ (Qiqi/2 (K,+0,)m:)’}}.

266

Applying the numerical values of the speeds of the various com-
ponents and the mean values of the coefficients of the corresponding
equilibrium components, this reduces to:

Mn=2M,{1.0717 +0.03585 (K,+0,)?/M,7}. (19A)

Designating the reduction factors of the several components as
f(M.), f(Ne), f(Ke), ete., and their squares as {?(Mo), /?(Ka),. etc.,
and noting that f(M2)=f(N2) and f(Q,)=f(O;) and that 1/f(M2)=
F(M,); equation (17A) similarly reduces, after applying the same
numerical values to the amplitudes and speeds of the components, to:

Mn’ =2M.f(M,){1.009 + F2(M.) [0.0583 +.0.0043/?(K,)]
+ F?(Mz) ((K,+0:)/Mz)?[0.0025-+ 0.0230f2(Ky)
+0.0103/2(0,)]}. (20A)

Designating for brevity the expressions within the brackets in
equations (19A) and (20A) as R& and R#’ respectively:

F(Mn)=Mn/Mn’=2M,2/2M,f(M,)R’ =F (M2) R/R’.. (217A)

The M, component may be considered the dominant one when it
is not less than K,+0O,. The values of F(Mn) for a given value of
the inclination of the moon’s orbit, J, and of the ratio (K,+0,)/Ms,
when the latter does not exceed unity, may then be found from
equation (21A) by substituting in this equation and in the expression
for R’ the values of F(M,), #(K2), ete., corresponding to the value
of J, as given in the tables contained in manuals on the harmonic
analysis of the tides.. The determination of the values of /(Mn) for
values of (K,+0,)/M, exceeding unity becomes more complicated
and need not be here described. The values of F(Mn) are shown
in table VI, paragraph 173.

DERIVATION OF 1.02F,

13. The factor 1.02F; is applied to the low- and high-water in-
equalities, DLQ and DHQ, derived from observations during a month
or more, to reduce these inequalities, and the consequent elevations
of mean lower low and higher high waters, to their astronomical
long-term means (par. 189). The diurnal inequalities are due to
the diurnal components of the tide at the station. Since the equi-
librium components have the same relation to their long-term means
as the actual components, the expression for the reduction factor
may be derived from the diurnal equilibrium components. For this
purpose only the K,, O,, and P; components need be considered;

267

since, as shown in table IV, paragraph 129, the amplitudes of the
other diurnal equilibrium components are relatively small.

14. The resultant of the diurnal components may be termed the
diurnal wave, and its varying amplitude designated D,. The diurnal
wave increases one of the two daily high waters of tides of the semi-
diurnal and mixed types, and decreases the other. Since the diurnal
wave keeps in general step with the semidiurnal tidal fluctuations,
the consequent diurnal inequalities during any period is taken as
proportional to the mean value of D, during that period.

15. Long-term mean value of D,—As K, is the largest diurnal
equilibrium component, the approximate long-term mean value of
D, is, from equation (14A):

Dm=Ky|1+- (Oy01/2 Kyky)?-+ (Pipi /2 Kak;)’].

The corresponding long-term mean value of the resultant of the
K, and O, components only is:

Rm=K,[1 + (O,0;/2Kik;) ale
Whence:
Dm/Rmn= 1 -— (Pyp,/2K,k,)?/f1 + (O,0;/2K,k;)7].

By substituting the speeds and the mean values of the coefficients
of the components, the long-term mean value of D, is found to be
approximately 1.02 times the mean value of the resultant of the K,
and O, components only.

The amplitude of the resultant of the K, and O, components
fluctuates between K,+O,, and K,—O, during the period of one-half
a tropical month. Its mean value may be written:

In which C is a constant which need not here be determined.
The long-term mean value of D, is then:

Dm=1.02C(K,+0,). (22 A)

16. Monthly mean value of D,—During a month in which the moon’s
declination is J, the amplitudes of the K, and O, equilibrium com-
ponents are K,f(K,) and O,f(O;) respectively, f(K,) and /(O;) being
the reduction factors for this value of 7. Since P; is a solar com-
ponent, its amplitude remains constant. It combines with the K,
component into a resultant whose amplitude fluctuates between a
maximum and a minimum in a period of the half tropical year. The
angle between this resultant and the K, component changes but

192750—40——18

268
little in a half tropical month. Designating the length of this re-

sultant at the middle of the month as K’, the mean value of D,
during the month is, very nearly:

Dm’ =C(K’ +0,f(O)) (23A)

in which C has the same numerical value as in equation (22A).

The correction to be added to the value of K, for the month, to
give the value of K’, is to be derived.

17. Correction for P,;.—As shown in paragraph 122 the equation of
the K, equilibrium component is:

4,=K, cos (T+h—90°—»’)
and, from equation (69) that of the P; component is
ee COs (T—h+90°).

The angle between them is:
6’ =2h—180°—»’.

in which h is the mean longitude of the sun (par. 105). Its value on
any given day of the year is substantially the same from year to
year. It increases at the rate of 0.041° per solar hour, or about 1°
per day. v’ is a small angle, which varies with N, the longitude of
the moon’s node. Its values corresponding to values of N are tabu-
lated in manuals on the harmonic analysis of tides. The value of 6’
on any date may be corrected for »’ by taking the value of h on half
as many days before the given date as there are degrees in »’ when
v’ is positive and after the given date when »’ is negative. When
so corrected the value of 8’ is

p’ =2h—180°

The length, K’, of the resultant of the K,; and P; components is
easily shown to be

K’=VK2+ P?—2K,P, cos 8’
— 1 K2+P2—2K,P, cos 2h

Placing K’=ck,

c= K’/K,=+V1+ P?/K2—2(P,/K;) cos 2h (24A)

Taking the value of P,/K, as the ratio of the mean values of the
coeflicients of the corresponding equilibrium components, or as
0.0880/0.2655=0.3315, equation (22A) becomes:

c=71.11—0.663 cos 2h (25A)

269

The value of ¢ on any day of the year may be computed from equa-
tion (25A) by substituting the value of h on that day. At the vernal
and autumnal equinoxes, March 22 and September 21, A=0 and 180°
respectively, and ¢ has a minimum value of 0.668. At the summer
and winter solstices, June 22 and December 22, h=90° and 270°,
and c has a maximum value of 1.331.

The correction to be added to the value of K, for the month, to
give the value of A’, is then

K’—K,=cK,—K,= (ec—1)K, (26A)

18. Expression for 1.02 F,.—Substituting in equation (23A) the ex-
pression for A’ given in equation (26A)

Dm’ =C{K,f (Ky) + (e—1) K, + 0,f(0;)]
and the reduction factor is:

Dm/Dm'=1.02C(K,-+- 01) /C[((e—1+F(Ki)) Ki +0,f(0))]
=1.02(1-+K,/0,)/[((e—1+f(K,))K,/0,+fO)]

The ratio K/O, of the mean values of the equilibrium components is
taken as 1.4066.
The reduction factor is written:

1:02F,
in which:

F,=2.4066/(1.4066 (e—1+/(K,)) +f(O,)]

By substituting the values of c, f(K,) and f(O;) at the middle of
each month, the values of 1.02 7, may be found as shown in table
VIII, paragraph 189.

APPROXIMATE VALUE OF (K,+0O,)/M,

19. The statement was made in paragraph 175, that in the lack of
better information the ratio (K,+0O,)/M, for entering table VI is
taken as 2 (DHQ+DLQ)/Mn. It is not difficult to see that the
daily high water inequality, DHQ, closely approximates D; cos a,
where D, is the length of the resultant of the diurnal components and
a is the angle between the position of its radius vector at high water
and the Y axis. At the next low water the radius vector of the re-
sultant of the semidiurnal components has moved through approxi-
mately 180°, and that of the diurnal components through approxi-
mately 90°. The daily low water inequality is therefore about equal
to D, sin a, and the sum of the two daily inequalities to

D, (cos a+sin a)

270

As cos a and sin a are both essentially positive, the factor
(cos a+sin a)
has values lying between the comparatively restricted range of a
minimum of unity, when a is 0 or 90°, and a maximum of 1.414
when a—45°. The value of DHQ+DLQ is the mean value of
D, (cos a+sin a).

The value of C, in equation (22A) of this appendix may be shown to
be approximately 0.66. The mean value of DHQ+ DLQ is then close
to but generally less than K,+O,. The value of Mn is similarly close
to, but a little more than 2M). It follows therefore, that very roughly:

(DHQ+ DLQ)/Mn= (Ki+-01)/2M,
2(DHQ+ DLQ)/Mn= (Ki-+-01)/M2

The ratio 2 (DHQ+DLQ)/Mn generally is somewhat less than that
of (K,+0,)/Mgz, but the values of /(Mn) in table VI change so slowly
with this ratio that no large error is introduced by using this approxi-
mation in entering the table.

CoRRECTION FACTOR 1/B

20. As stated in paragraph 261, chapter V, the correction to the
primary current therein designated as 7 is such that the corrected
velocity:

B sin (at+ 6) +71=Blsin (at+ B) +7/B]

satisfies the general equation of motion (equation 112) when the sur-
face slope has the simple harmonic fluctuation, S cos (at+H°), and
the velocity head term is dropped. Placing, for convenience, 1/B=z,
equation (112) therefore becomes:

S cos (at+ H°) + (1/9) 0B[sin (at+ B)-+2]/0t+ B*[sin (at-+ 8) +2]?/C?r=0

or:
S cos (at+ H°) + (aB/g) cos (at-+ 8) + (B/g) 0z/ot
+ B{sin (at+ B) + 2]?/C?r=0. (27A)

‘he values of B and 6 are such that equation (145), paragraph 243,
S cos (at+ H°)+ (aB/g) cos (at+ B) + (8/37) (B?/C?r) sin (at+ B)=0

is identically true for all values of ¢. Equation (27A) therefore may
be written:

(B/g) 02/0t— (8/37) (B?/C’r) sin (at+ B) + (B?/C?r)[sin (at+ B)+2]??=0.

271
Dividing by B?/C’r;
(C*r/Bg) 0z/0t— (8/37) sin (at+ 8) +[sin (at+ 8) +2)?=0.
From equation (153), paragraph 244:
C’r/Bg= (8/37) tan ¢/a.
Giving:
(8/37) tan d2/dat— (8/37) sin (at-+8)+[sin (at+8)+<2)?=0. (28A)

Expanding the last term of equation (28A), the differential equation
for z becomes:

(8/37) tan ¢0z2/dat— (8/37) sin (at+ 8) +sin? (at+ 6)
+22 sin (at+ 8) +2?=0. (29A)

The correction factor, z=71/B, is therefore a function of the angular
lag, ¢, and the phase, at-+8, of the primary current.

21. Since z is relatively small, a first approximation to its value
for given values of ¢ and ai+ 6 may be derived by dropping its square
from equation (29A) and neglecting its effect upon the sign of the
velocity.

Rearranging, equation (29A) then becomes:

(8/37) tan ¢0z2/0at+2z2[+sin (at+ B)]
— (8/3) sin (at-+ 6) +sin? (at+ B)=0 (830A)

in which the positive sign is to be applied when sin (at+-8) is positive
and the negative sign when it is negative. Angles are in radians.

Equation (30A) does not appear integrable, but the values of z for
a given value of ¢, and itor successive values oi at+ 8 increasing by
sufficiently small increments may be derived by a somewhat laborious
arithmetical solution. The increment selected, in degrees, will be
designated Aat®. Its value in radians is then mAat°/180. Since
differential equations remain approximately true when small finite
increments are substituted for the differentials, the first term may be
written: .

(8/3) tan @Az/(rAat°?/180)

in whichAz is the increase in z due to an increment oi Aat° degrees in
the phase of the primary current. If Aaf° is sufficiently small, the
value of Az does not differ materially from the increase in 2 during
the preceding increment in the phase. Designating the preceding
value of z as 2, the first term of equation (830A) then becomes:

(480/n?Aat®) tan 6(2—2) =b(2—2)

272

in which the coefficient b= (480/7’Aat°) tan @¢ may be computed from
the given value of ¢ and the selected increment Aat°.

In the second term of equation (30A) the negative sign is prefixed
to sin (ai-+ 8) when this function is negative. The factor+sin (at+ 8)
is then positive for all values of at, and will be so distmgu‘shed by
writing it as sin (af+8). The algebraic sum of the last two terms, for
values of at at the selected intervals, may be designated as —R.
Equation (380A) then becomes:

b(z—2) +22 sin (at-+ 6) —R=0
whence

2=(29—R/b)/[1+2 sin (at+ B)/B]. (31A)

22. The values of z for successive values of at-+ 8 may be computed
from equation (31A) after an initial determination of 2) has been made.
By taking A at° as an integral factor of 180°, these values are repeated
after at+ 8 has passed through 360°. Taking then 2 as zero at any
value of af+ 8, such as zero, the resulting values of z may be succes-
sively computed through 360°, a corrected initial value of 2) derived,
and the procedure repeated. Since the divisor of the second term of
equation (31A) is greater than unity, the new values of z successively
approach and finally coincide with those previously found. The
process is in fact abbreviated, since the values of z repeat themselves,
with the sign reversed, after passing 180°.

23. Second correction.—When the flow is largely frictional, and ¢
consequently is a relatively small angle, the values of 2 derived from
the foregoing procedure are so large that their squares are not negli-
gible, and are sufficient, also, to reverse the sign of the velocity when
the primary current is small. A further correction, 6, is therefore
required. Designating the first determination of the correction factor
as 2, the corrected current becomes B [sin (at+6)+2;+6].

Equation (112) then takes the form:

S cos (at+H°) + (aB/g) cos (at+ 8) + (B/g) (02/0t-+- 06/02)
+B{sin (at-+ B)+2,+6]?/C7r=0

which, by a procedure paralleling that in paragraph 20, may be
transformed into:

(8/37) tan 6(06/a0t+ 02,/a0t) — (8/37) sin (at+ 8)
+[sin (at+ 8) +2)?+26[sin (at+ 8) +2,]+°=0

273

The approximate values of 6 may be derived by dropping its square
and neglecting its effect on the sign of the velocity. The errors intro-
duced by these approximations are, it may be observed, much less
than those resulting from the same approximations in deriving the
initial values of z. The resulting equation may be written:

(8/37) tan $06/adt-+ 26{ + [sin (at+ B)+2,]}
+ (8/87) tan ¢02,/ad0t— (8/37) sin (at-+ 8)
+ [sin (at-+ 8) +2,)’=0 (32A)

in which the positive sign is to be used when the primary current,
corrected by 2, is positive, and negative when it is negative.

By using the same increment, Aat®, as in the first determination of 2,
equation (32A) becomes:

6(6—6)) +26[sin (at+ B)-+2,]—R=0 (383A)

in which 6 has the value previously determined, sin (at+ )+2,
is the numerical value of the velocity as first corrected and:

—R= (8/37) tan ¢02,/O0at— (8/37) sin (at+ 8) +[sin (at+ 8) +2)? (384A)
The first term in this expression for R may be evaluated by placing
(8/37) tan ¢02,/dat= (8/37) tan bAz,/(rAat?/180)=bAz,

in which Az, is the average of the increments of 2, for the preceding and
ensuing increments of af+ 8.

It may be observed that F is the residual by which the first member
of equation (28A) differs from zero when the first approximation to
a value of z is substituted therein. .

From equation (383A):

d= (6—F/b)/[1+-2[sin (at+ 6) +-2:]/6] (35A)

The values of 6 for successive values of af+8 may be computed
from equation (35A) by the same process as that employed in comput-
ing z from equation (31A).

A second correction may be applied, by the same procedure, if the
corrected values of 22,6 give residuals of more than negligible
magnitude when substituted in equation (34A).

24. The increments Aat° used in computing the correction factors
shown in table X, paragraph 261, ranged from 23°, for small values of
¢, up to 10° for the small values of 7/B when ¢=80°.

274
When ¢=0, equation (28A) reduces to:
— (8/37) sin (at+ 8) +[sin (at+ 8) +2?=0
Whence
2=4/ + (8/37) sin (at+ 8)—sin (at+ 8) (36A)

As the positive sign is applied when (at-+8) is positive, and the
negative sign when it is negative, all of these values are real.

The limiting values of 1/B=z, for ¢=0, shown in table X, are
derived from equation (386A).

Paragraph

Acceleration head___--__- Ree Rs cere se eee 231, 233, 26101

SFSU BOC Ol: See eee eh ee eee et eo eer ee 2 2 259
Accuracy of:

iid Te CHC MOINS or AP i ee he OS ws aye en Pld ee 138

‘Cramer Tomer iG oN c |S eee oe, Cee epee be ee ee a 495
Pe eeOr nailer See Nr ee ee ee 153

Sei c) xe ee A 2 lo le eet ot ee ees a 146

reser cena ete hers Dt hse ts 2S eo ore NL 144
Amplitude of component:

19) Chine Gl eae enna Be 8 ee Pe a ee eee 49

SU MUM UUONMOL es oe ee eo i ee ee ale ls ren i ee 91-101

UT an ed LL et oy an ieee = Breen fen Ny a oe Se ene tae pe 102, 1238-129
Anomalistie month:

IIS Fitna wa er ee ey Se ee 8 i es ee 43

Ae 1y1 0 Gl empeeeer patees tere eget oh OS SRN eC ee a = SE eee 62

WORP a. ss SOS at ae oe eR rene ee pee Be meg ee ere oni 62
ANoln@li@rm, Chevatiaveye | es eR Se eae ee ee a res ore oS Slee 60
STU CDED, CLEC aes ee ag eS ap ee ee 43
ASE DE SED UCAS Sk RS De ee ce ee wn ne ees RS 145-148

Migayn low, canGl loo yes Cla oe eee ee EEE ee 186
ASNGS IGS ys oe] Se ee ee a en ee 50
ASE ETSI? TRO UOIRS Se, Sages eS ed een ne ov Sa 97-99
ANTINWOLTDDURG UAG a (PSS SS aa sy A See a Re ee 212-215
Average tide curves—preparation of:

HOTERE VERS Oe CUETEMLS Sere ey sean es eee lees es A ee SE 304-306

JENOER. THOU OUNP ETON Sas, ee ees es ee aE le eee 505, 506
BOI, (HG ee Dee ts ps ee OO Nay 5 463
Canal:

"DUC IDL to KS SY Uh te ee SS geod Se gts eo 295

HERI EETOMESSeh Owe llntrata epreen Re hes ee en eee es eee en ee ee re 316-349

TNACUIOM! GOmMsie Rela oo | 8 ee ee ee es eee ee 351-432
Cape Cod Canal:

Campmtamonsomprimary COrrent.—-- 2-22 4225-2222. 25-- 240, 249-250

OF Ghiaieriionit eS 2 2s es ie Beer Se see ee ee 263, 272-274

IN@asCRReG| GUTS es Se Ee eee Ree See Se ee 279
Chesapeake and Delaware Canal:

Computation of primary tides and currents___..-.-------------- 392-401

Pertti we, A eee San ek SOs 407-414

@omputedsandsmeasured currents. —-=------=2----—==-=--<--e— 414
Sremcocticiont tor tidal fow...-----.-«2----------=--2-4-=s2-se== 223
Closed channels:

ihwied!_ i205. e ee ee ee eee es ee 295

IMIG HIOMlESS HCAS Gael GuiaKRIMS Wal Soe ane 335-337

‘Cathigall I@meniigi2 8 52 es ae eee eee eee nee 346

Compson frietion considered. =... -_ ---2=-=-=-=<2--2=-2-——— 420-432

Paragraph
Clyde River, effect of improvement=--- ----..- --= 25" 2 =" 3555) en 168
Comparison, determination of datum planes by_---.--------------___- 196-207
Components of the tide:
Defined... 2.2). 32 SA a ee ee ee 49
Combination-ofja2 223522 esse ee ee eee 51-59
Speeds v. .- ss ee ase kes Base eee ok 64-76
Computation of hourly heights-=-_.2.- =. 2-22) eee 78-90
Amplitude and: phase... 22252552225. 52-642 oe eee 91-101
Mean values,and:epochs:-. 2° _~2 5-22 2 5l. Se ee 102-131
Component day and hour defined a 83-84
Component currents. = 2-=2 -2 28222242 ee ee eee 291-294
‘AmalysisiOl +35. 22. eee Se oe Ae ee ee ee 493-506
Compoundicurrenticomipone mises se = een 292
Compound tidal components:
Defined 22.22. . 2 eee eee te ee ee 73
Speedsiet Mae Saws Se ee re ge ee eee 75
Hquilibrium rar sume nits 52 = =e ae 121
Reduction factors.2-2 52.22 3a 2 es ee eee 128
Connecting canals:
Deéfinedw: 3.252252 b235 Sees eee ee ee 295
Frictionless flow mz. 222422232 L282 2a. 2 3 ee ee 316-334
Critical lengths... 22 25 of Bonk ee ee Fe ee 347
Hrictionaletlows como uteaGlom se see ee 351-419
Continuity of tidal flow, equations of2 —552_ 22 __ = eee 296-298
Contraction head... 22202224222 322 ee ee ee 236
Cook Inlet, Alaska:
WideSis: vise jSece S228 = ee eee ee ee 453
Bores. ous see foe ee oe ee ee ae re 463
Coordinate components of primary tides and currents____________--_-_ 364
Corrections:
Jaone Moravern nukes Le teaorounlss ilo = ee ee ee ae 171-179
Of diurnal inequalities.-_25.22. 2-32 2 eee 189-191
Of primary current, for friction term_-22=5- 2 = 260-263
Hor other distortions =2 28.2552 en. ene 264-276
Cubature-of tidal channel--) 2-22-2222. e222 300-315
Currents in tidal channel:
Detimitiens-42 25622... 0 et Se ee ee ee 5
Generaliequationiof MoOtOne 22225224" 2 se eee 216-221
IDG WAKO Ot COMMUNI = Soe so ee oe oe ee nee eee eee 296-299
Relationitorsurtace headi=” = 52 =" =oe an = eee 237-259
Primary current, defined: 222595. sa0 2) 2s se ee 242
DiStorRbvon's of, prim aitaya CUNT eT eee ee 260-277
Hrictionless. How: si2. b- 42 3a See 257, 287-290
Determination by: culo at une ses ee ee ee 300-315
Im lone (canal sy frictionless itl oye ea 316-349
Friction considereds 3 2 2 Se 351-432
Midstréam. currents. 6 ee eee 433-435
In an estuary 260-35 ees he ee 8 ee 437
In an inlet ooo eee eee ee ee 480-484
Offshore currents. Se: Bie eae ee ere ee 485-490
Harmonic analysis of currentses: oo) ee) 22 ee 492-499
Nonharmonic/analysis--- 20 S52 eee eee 500-506
Wind:currents. 2. |) 2 ae ee Dee 507

Paragraph
oo AIPPEOI TREN OINDG) oe ey ee ee ae ee ae 492
Currents:
Ere ee a ee ey foyer a Doe eae nS SB ee 2 ee 6, 485-490
ENGIN Sis LOler OUAG VACULUCMES= 22 se.2- 6 9. he te a kote eee e 499
SUN SePm OMUTE SSS ya geo eh a a 505
Dams in estuary:
JEIRSCE Om WIGS Coch Cubans oe se ee ee ee 466
© SMU AAOME OMEMeC Cha pests he ote ete es tw ee Se 477
Datums:
MiG ale escriptiome 2s. 5 6 eee ee a De a ee ee eee 3, 157, 193-194
ING erie ct) ewe Ret mte yt Bie ee a ee a eA ae st oe ee Be IE 159-163
lene tices evel eee Bp. ae ees She aE eye pan a) 164
LOW? Bin Maen WHO CERN oe es eee ee ee ee 165-168
Mean low and mean high water_______- i SAYIN eat Bs 4S Fae 169-180
Ofsspnimoatid Cseereat me see ce wey ee he eee A on ey eee ee oe 181-185
Mean lower low and higher high water_.......2-.._..-.......+- 187-192
Weatcrmmation Dy Comparison... Vo -2 22. 22s bo 196-207
TEASE EOIM OUR Se a ea aes ae eA eg eae Ry a is pe te a ees gh = 208-211
Computation of effect of channel improvement_________________- 467-475
PEVaMaoneore in and mM OOM. 22.2262 2 oe es ee 35
Declinational:
OL SNLIOR (ANIC AS Se See ae a al Oe ee el ene eee Pe 187
VRGTOEOS asa Fics Sa a ne De ee Pe eee ee 152
Delaware River:
Curnenizcunvessiromicubatunei== =. =e ee. lee 2 eae ee ee 311-313
Comparison wath ideallestiany_ 2 2025.22 22k lee eae 443-445
iectrotatireshmwaterG@ischanrcemsen meee Sarma es Le eee 1. 2 ae 454
Be CUMO le IMM ORO VEIN Ets meee eters en em ye Ne ee ee 464-465
Discharge:
iroushEd alvchanneleneqmatlons == eee een ee Te 296-298
ID eiginmimejn@m lo? CUSANAUIRE— 5 ee ee ee ee 307-314
Relitionsto storace. in lone icanalye: = 5 ae Sees ee 388-391
repomderance, m connecting Canale = = 2222222. = ee 416-419
Distortions of primary current:
DrewoMonmiOLimiciionetenmens stu ep = tae, Cee ee Be Sa 260-263
Othenmcishortionssiny short secuonas =... 2. = 52. ee 264-276
Hiomlonomesni alae mien Bel ae see Oo ee ee ee 404-415
D farsa, see, SRA A eee eee a ee ee eS 153
LD Revsecazll' Thayer GRO ICS eee aS ne ee ne a See re 152, 189, 190
ecnenalll tpcngee 0 Se MN ee OSes ee et el ee ed ere ee ey 4,152
rinanrmingormoiuides=! 10220. 22S. J a a eee See 139-140
Mimecnmimomnicerang alls. A. eS Nee ond lel See ee 460
i apmcnnnemm—mOenmeOe eh te. (24 2a Al) ae oe ee Se ee ee 5
BecomuniCunmon MOONS OTbib . 22. Lo) ep ss eS ee ee 43
icin inemacinicdeeass tie ee ee LAS. See oe Sees 30
Miimamation on residuals) of components____._---.-._------=--=------ 100
Timihieniae: [neal = ae eee _ SE aie te ea a ci pe een eres ec 234-235
Epoch:
Dated). . 2th Se a eae | ee A RE Dalene setae ee ee eat 103
SerAprNGeiTOtipOt een M ry ce oe ek Lhd oe ee ee = 117-119
FERC angie hee ek eee eS ee ee te 120

ranienicnict mui cee eee se J, Be) ee ee eee 149

Paragraph
Ei quiliorimm yar own emit nee ae eee ates Wai
Biquilibriumitide ss: 622 535u a5 See ee 2. 2 ae ree ee 26-32, 104-111
Variation wiathedeclimatiommo tenn @ ons = ee Bi
Vairlations wathidistamGexoi mano @ nies en 43
Componentsiofse = 2S ee eee ee ee 115-116
Mean valuevof cocthcients2 22 ==... 3 eee 129
Establishment of the Portis sees 2 eee 13
Estuary:
Definitions. 22 22232 228..6 ee eee ek ee eee eee 436
Tides and Currents itis =e So. ea ee 437-445
Effect of local contractions and enlargements__________________- 449-450
Of fresh-water discharge 2). 2.<3--22ec2e 2. 454-456
Of rotation of earth upon tidal ranges. ===. _-_-- 5 eee 457-459
Slope of mean tiverslevel- = 2-222. 2-2) 32.2) 8s 462
JBN Olt Be nTeTeN! ClaainGrSs mm — eee 464-477
Hloodscurrent)detined © 22 a2. ee ee 5
Fresh water flow:
Inieubature*e. 308 ein el ee Se ee 307-314
Hilectvon currents mestwamya a= = =a a ee 454-456
Mricttonmhead S02. 2 2 ee eee a oe eee ee 232, 269
Friction term:
Derivationit= 2 = 22 Meee se ee ae ee ee 219
Hor harmon calllyaazatylineael@ ciitnyese ss 224-226
Brietional’ tidal flow 2.020 8 A es ee ee 256
Mrictionless«tidaliilows..222 22s. 22. = he ee 257, 258
IH qwatons nes horissectlomyonceh ances ee 287-290
Tin alone: Camas 2 See le eee ee 316-324
Ima ‘connecting Canale oh. i oi ee 325-334
Ima losed: canal.) = 2.02 hs Lee soe eee 335-337
Kindy tidesan Bay Of. eo 1 ee ee ee 452
Gaces chides sea soso l sas lee meg eg ce eae 212-215
Generating sracius:- oficomponent ya efile cles 49
Oftiined ds ee Ce Be ae SO ce a ae 241
Greenwich epochs sso. 2s Se ee ee A 120
Lunitidalintervalse oi. 0252. ea. ee 12
ATRIA ST Stal FR lee ee eek he ee 9
Flali tide gle yee ce 1k es Sa Shere Th ese pe eee ey ee 164
Establishment, by comparisons 95 sos se ne 197
Harmonic analysis of:
FEV GG Se Ne fs cee SS 2 eee: eee eS aol ee UN et ge 77-134
@urrents ss = Se a fe Spe ee I a 496-499
Harmonic components:
Described 3 Sos ee Oe 49
Combination of: = 42.56 See ee ee 51-59
Speeds of tidal 2.0 2 Sas aan ae ee Oe 64-74
istOfo 6.32 & se to Se ee ee, Bee = oe 75-76
Atcertalm stations. 2 seas see oes eee ee tS a eee 134

Harmonic tide planes «2225 Sie eae ee ee) Se 193

Head: Paragraph
Av@RSISTDUNO TS oe ern ee eS 231, 233, 267
OCR UDC OMe 5S Mays SS ce ape ee ar ea ce 236
BML AMCeraM Gene COMER === a Ec Rear eos Bee ae ee OL. 234, 235
IPIGUROM ae ee ee ee ee BG
SHUT ace en gp a ee Se en 229
Relaiomeopuidalgheteht= 042) eee 237-241, 364
Mompnimanyercimrnent= 22. Shee 2 we ee 249-254, 373
* SLOG 2 kw — a Reh sa er ere Cg a ei 1 a, 230, 270
High water:
UE ede! _«. . 225 eee Re SO an ie rr eee a 2
Literal 2 Se UE oe bck Be ee eg aves et 8
LUGE et ee pe ak ee a oa A erent 169-180, 199
Higher high water:
Presinic lee ee em eer So eee ee oe Sey das 2) FEO, 2, 150
AGN pe els UE i Ueno eT RE Fe” EE 187-192, 203
SGI Rem Tle Owes ames eye PER 8 Ee, TT ee See 256
Mmevence Or MaArmlonic COnstaniis.=-2-- == 39 - oan "So eas See 130, 131
mene Oe Cenne nya =) oe on wn St we eee. oe ee 228
BiMinan ce eG etined aie Hens Je we ee oe ee eee Sil 7/
ese mC tine tee: Mares Se Nae es ee es Oe ee oy ee 49
Tnlet:
Ben IN CUM nee fe eS Oe 2 = SUC Ge eee eee A te ee 295
HAGGMTCTCORO Keer Lei, AS Ee ns sl ee ee eee 478, 479
Jai GLFNUINGS Git Sess de Bee Oe eye eee ee meee see | Ae _. 480-484
Escurimeamcouspramie, detined= =. 22.2.2. fat ee ee 326
Shape in connecting canal, frictionless flow_____.____________ 326, 328, 332
lin Ghosee! Caingil, ineveinomlass mloyy = = 9 ee 336
PRORLESSIVCRI AN Chime omnes ae 2 Bn 22 ade ee ey 339, 340
Connecting canal, frictional flow... ..-...=...----..--__- 415
Sioseducanalimemonal MOw. 2952 28-026. 2 = See 430
HET Se COTE ROIN@ QING Pee PEt eee Po ek ee ob eye ee 35
Pomeitude amd roht ascemeon Of... .2..22-.---5 00 ss 105, 114
Lag of primary:
(CHMPRETINE Sn oh ot rl Se ees ae eee eee Gee 254, 258
ela ionstOnarapliinderom current. 982 9 2. ee ee 373
Roam eeislancnsound. wides ius. 2 Ff 453
oneitude, correction of lunitidal intervals for..__.._.......2-__._-_- 10
Baneindcommoons node, defined... =~... -=-2- 522 244- sen an ne 36
C@ornrechonsotmmea»nerangedtOre 225022 2s s cet ek ee 171-179
Low water:
Deiter en eee a ee LS isle See ee eee 2
Ztieaeveell __ — BSI a saree le a as, en eles Ie 8
Sint) ees Le pS eT oe a ee eae oe eS ee 165-168
MAGRID ot ee ee ee ee pe 169-180, 199
Lower low water:
Reh eer eee Sap. SELL he Sh oS ee 2
ISIS es 5 a eo a 187-192, 203
WammrbCIAGme nino ties a. 2 2 os fo eL ae ee 2, 143
MIGBIN. = 352423208 at See eee eae ae ee oe ee 181-185, 202
ign righ TEN OUI ee Se ee Se Sk SL eC ee ee 60

CEES 11 a enn pe 8S 2k se SO ae

Paragraph
Tanitidal intervals... 2. Se = Se eee ee eats
Relation to epoch of M; components. —-- 2222225255 .5. 555 142
Ofcurtents abe s Sees Be ee Sa ose eee 500
Mean: current: hour... 2 55 22 eee ee 500
Mean depth, relation to hydraulic radius_—--__2_—2 >" eee 299
Mean high and low waters22-- 25 22 S222 22) 2. eee 169-180
By comparison. S25 52 2 eee ot 199-201
Computation of effect of changes in channel_____________-__-==_ 467-474
Mean higher high and lower low waters_=—=——) =~ 9-232 eee 187-192
By) COMPArisSONe = eee ee ee 203-206
Computation of effect of changes in chanmel_________________=== 475
Mean high and low waters of spring tides________-_---------_-_- 181-185, 202
Mean range__--__-----+-------------------------------+---------- 4, 158
Correction for longitude of moons node222_ === ===: == === ae ihr malhe3
Rela tionetonticalnco nap OMe nts eee em eee ee Appendix IT, 2-9
Mean séaclevel 2 2250 oe 222s Sie oo ae eee 159-163
IBY y COMM OAS = 222s fsbo es 2 See ss Se eases 222 =25----- 198
Mean values of amplitudes of tidal components_--—__-___-____________ 123-128
Of equilibrium cociicients sess eae aes eee 129
Meeting .obttides: =. = 222 3 8as 52 eee 334, 391
Meteorological! tides, defined: — 922222 == =" = = eee 47
Components] 23. 2 =" eae S28 Se ee 70, 71
Midstream currents..o. 202 2a ie Ea ee eee 433-435
Mine tides. . 2-5.) Sh - 26 2 Si ee ee eet Se Bee 170, 188
Mixed hidese= = en's Sa = Se = ae ee ee 139-140, 149-152
Moons declination and inclination oforoiyss === === a = 35
Ibommmhiucls OMnOCle_ .- = 2 se ee so ae See 22 Se a= s55----- 36
Hiccentriciby. Of Orbit- ie \oae= see ee 43
Neap tides:
Defined 2. 2k Se Se ee ae a eee eee 2
Related-to moons.phases= 224552 5. 222 soe ye ee ee 20
ANG. Wilh Guael Sip oman oromeISS = 2 Bo ee == 143
Mrect ot N, componentsupoue 2) 28) === == 147
Mean low and high water of22522. =) 5-25 aa 186
Nodes: moons, defined: . 222223 922 2 35, 36
Nodes: incclosedscanal $202 22 a2 Yo es se ee ee 348
Of seichesie. 2 SL Se ee ek ee ee ee 350
Offshorencurremts222 - oa = es ee ee ee 485-490
Average Curves Of-2 )_ 2222 2s Sau ea ae ee eee 505-506
Orbit, moons:
Inclination and: nodes=2- Hee ete ee 35
SENG Ge Tate it yee a 43
Owwericurentise as ae 291, 349
Ihnen SWUCNA eS = See ee Boa a He oe Sas Sess ses ehes=se55----- 460, 461
Tneanvinl@ts. 2-82 2es ee ee a ee ee ee ee 484
Over tides: ;
Wehne Gases eS Se ee ee 2,
Speedsae 22 222. 22 Ses ee ee eee 2 75
Equilibrium arguments_--_--_-___--------=--=---=---_-----=-__— 121
Reduction factors: 2:52 = 228 Sk eee eee ee 128

Mffectrom fOr Of tC el CUT ye meee eee ea 155

Panama Canal Zone: Paragraph
SAINI OP GUUS See ee ier 185
Difference in mean sea level at entrances_-_________.____________ 159, 162

seat KemGLe TING Clemens weir cigs oes ee De 15

Pe reM URC OMUN Ve eRe See ee Be ee a eta 43

21 3048 258. ecoe poe See oe ee 146

PELE ES, LLLDSE,, (CLIT SUG |e we 43

2 BLESSED WG GSo2 5 Ea 145
ELL EG p D2) SDI ers eee ee ee pe ey Be en eo 147, 148
Dia mmowrncemoh wateriofe 22 .. J 82 Se oN la 186

Pe TLL EL Lin, GiGi TINS |S 2k eS sirens a cap pally Vane baa he ee 60

Period of:

Bmuapanente wens | ee mae Sait fe ee, bee 49
S WO CIC 2 I es ee eS ee RS Sm, oe 0 aeRO ULE ee ADE 54
Ma HOMS On IMOONT ANG SUM sole ehh ee 62
SIG OS EOE Se oe Se ee ee ee gee eee Ee IP ee ey Rie ges 350

PURSE DEG. oe Sa NE a ae ees see a oe eS I, 0 etd Sah 144

LP LSRES: Cif TTD) Oia ee eee ae ee ead Pe VEE eae Pa 20

Phase of component:
ce sail cee a” Se pate EN Oe, a Ee ee Se ah eee 49
PmGMUAOMeOtMibialey -o22 Abs 5S se > 22 ss Ee PAN Ae Ne ieee 91-96

Pe MmeMeREC MeCN CS rs. foe oe a ode he ee ee 487-491

Peete abies prOuGIne 2252 6. 4. 2k ao S)) ek = See ae 22-25

Prediction of:

GE ee ts FSS ER a ho EE Se Ao Rok LS 135-138
DUDES ce Se ere ee eae eas eo PR, See Se 492-498

PPE MMC MINeMU WeMMed: 02 52 5o 5. owes yo ba sess RS 242
mAnpluiude, trom surface head and slopes_--_-=2.2...21. 5.22902 245-247
BOW NISy. (UUDG SE SoS Sc te ne ee Ek eS ay 248
PME icine eee meet meas ae Pe Ns, ots 6 Ee EE ee 254, 373
issicnoieTiniicsam mean gees 22 2 8 a, ol ee 255-259
wo USNOIPIO ISS Ss Se) Se eee 260-277
Ar oneacOMMeCuINesCANale sa eae Sos le a Se -- 359-385

Se tOn eu menite eam = se ens Senee mT oe lo ee 420-482

Primary tides:

DST S = pete NE Se eS ee ee ee 359
Wetermimation, of entrance tides=—. 2 22... 2 se 360
LPB, (UCR SS ee ee ee 295

rE INC RWVELC ae te epee ee Ne Sees 2 Se oe eS oo 338-344

Ranges, tidal:

resi pe ENS ey We Sy a Sag re 4
SyMMOONS 2 See ee ee eg ee RE ee a 158
Mean, correction for longitude of moons node_____________-_____ 171-179
Eelation to harmonic: components... + --__=-__-2-=-_2_ Appendix II

Divert. . 2.22552 ae ee ne eee OE, ee eRe ee ey 4 152

OO GTC. 22. obs ee oe eee) Soe See es 152
LBB D SI LOSE C2 on A a eo 234, 235
Se bcePilon igictions, / DAD ee i ee eee ee epee Ses, e © 124-128
RermiantiaOmeomponents. Mi: 9... as - Le ee te 51-59
JESUP DRESS VS. NO Spe ee aa 342, 343
emer sucamonourrents, Gelned_..-- .- _- -4---s-s--4 2 se-ee eee ence 5
PERONEAL CI SNO lee me = 2 ek Lie a el See 497

MOMMaMoOniceama lysis Ole p a. ao. oe ee ee Se 500-504

282

; Paragraph
INOW CUMREMIS 2 222-555-5225 igifee £0. Sea Ae ee ae 5, 485-491
Average Curves: 2) 2. Ue Ue ee ee eS ee ee 505, 506
Harmonie analysis of. 35-2 2s 22. Ss oe ee 499
Rotation of earth:
Effect ontides:.<$2 2828.42. bs 22S... 4 33
Onitidalirangevonloppositeishoresea= = == === === eee 457-459
On windicumrents:.. =. eee se 2 elle ee 507
Seiches- 22 eh eee ow ae eee 350
Semidiurnal part of equilibrium) tides] 92= 222" = = eee 38-40
Semidiurmalltypevor tides 22: S== 52 22 Se eee ee 139, 140
Sequence of highvand low waters2= === 2-25 2255223552522 = eee 151
Shatlow, water componentss=422= 4-24 === == ee 72, 73, 349
Slack water:
Detinedet. aus. 2-8. = See ee ee oe ee 5
aimevof 7. 2 ea. Se = ee ee sees 492, 500
Slope; of mean river level in esiuary= ==> =-—-=——— = 462
Slope:
Surfaces s22 22 2th ce Se te eee ee ee 218-221
Relation! totprimaryscurrent ess 55 a= Soe = ee 237-248
Speed of:
Component, ‘defined2. =. 2222-32256" 22-4. =. eee 49
IMIOVEnTIEMNTS Ot Gwin AIAG! WOON — 3 ee 62
‘Fidal:components. 2-484 _ aoes= ee ee ee 64-76, 116
Spring tides:
Defined =. ..2 - 22h usec eee ee eee eee eee ee 2, 143
Ordinary Se. = 206 Sse ek Se ee ee ee 157, 181
Relation to moons phases: =--- 522-2225). 252) 3 eee 20
“0io) Mla, Ghavel Shy, COMM NONVEM SS so eH eee 143
iIPhaserage tis. 6 pls 2 oe ee eae 144
Miceya lone aincl lovaln Wweliens OF j5-.s55--2 2252525 5-5225252-- 181-185, 202
Stati ease co 2. iil es hes Se) oe ee eee 212-214
Stationaryewa Veceer ee 7 + aes oe eee spe pe eee 345
Stencils, for computing components==2=2 9222 = ee eee 87-89
Storace station, defined! == 2.524. (2220 2 = 5 ee eee 369
Strait:
Hydraullite flow Ins 2-222. bes ee ee eee 256
Prediction of current ims - 235-252-062 42255525=! 2 ee 498
Strength of current:
Definedas 222.22. subi ecoese es See se eee eee 5
Prediction of. 222203 225500 2 ee eee ee ee eee 492-504
. Relation to. surface:slope! = 22 2-25. eo eee 255
Surface head:
Defined. 2 25 a0 pee ee eS ee eee 229
Computation, for harmonic iluctuations= = === === =5 == === 238-241
Computation, in Sections of lone canal== ===> — = = = 373
SUIS CLULE | GON SEIC 1a ape 350
Synodie period:
Defined :.2 32 22a ee ee en 54

Month ow... ooo Be an ee ee ee eee 62

Tide curves: Paragraph
SUPE. sed Se ae rrr ree aeSy ee AEs ee 48
JU EDLE. cece ote rete Sane ee 304
DIDDOSiG 2 oe ee a Neen 305-306
“lL? ELEM. .io see lees Se ee ire ies ees 2 212-215
lt Pn LuGie ICRC Bee Bee ee rr re 14-21
iropical month and year____.-._..=__~-------- ae 35
PiarOes Ringl SoetCCh= se = ee 2 ee ee a 62
Merapeemetpes. Oelmned 9. 8 2 a -------- i =o 152
Tropic tides:
we BEET 2 a Se ee ee 149
Milernmlat obec GelOwn Water Ola so> = 8) =e Sse eee = 186
coc URS If UGGS 22 2 eee eee ee ee 139-140
OC ELDGEN [NEG 01. 2 2c OR = Se Ee 230, 270
Memrimesiiiior pOenmMeCG 4) S228 So ae ee Les 367
Wave:
UGIGUN aot ee ee ee 326, 327
1 RD TRE NSHSI NS es = ee a a a 338-341
EACSTHTROYERAEVSISTL PSs se ge re 342
SUPE AE ETI oo ec ee a ee 345
USE PSI Ls a a a 507
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