H. O. Pub. No. 605

GRAPHICAL CONSTRUCTION OF
WAVE REFRACTION DIAGRAMS

By

J, W. Johnson, M, P, O’Brien and J. D, Isaacs

vv

: JANUARY 1948
>

UNITED STATES NAVY DEPARTMENT
HYDROGRAPHIC OFFICE

H. O. Pub. No. 605

U.S. HYDROGRAPHIC OFFICE

TECHNICAL REPORT NUMBER 2

GRAPHICAL CONSTRUCTION OF
WAVE REFRACTION DIAGRAMS

b

a

M

Wh

By

Umwersity of California
Department of Engineering

Berkeley, Calif.

JANUARY 1948

ISSUED UNDER AUTHORITY OF
THE SECRETARY OF THE NAVY

WASHINGTON, D.C.

For sale by the Hydrographic Office, Washington, D. C.

and the Superintendent of Documents

J. W. Johnson, M. P. O’Brien and J. D. Isaacs

Price 50 cents

TABLE OF CONTENTS

Page

Graphical construction of wave refraction diagrams by the wave front method ___- 1
Introductions Ss see Se eee ee ee ES ap cn 1
IRETACHOM Hii & Sumiains Snore In@.._. 22-222 s2252ss-5e-2-25-5eenseeeeeesee 4
Consunic tommotene frac ul omg Cl tel 2. 1 sae 4
Refraction diagram, Monterey Bay, California___.-_________________________ 8
Construction of refraction diagrams from aerial photographs_________________ 15
Determination of refraction coefficients from aerial photographs_______________ 17
Graphical construction of refraction diagrams directly from orthogonals____________ 18
Dnytrodu chime a= Se as es pe a ney a a eer 18
Developmentiotithesmethodse 3 5- = aa a eae oe a eee eee ee 18
Applicationiofithenmethod a2 o 2 see ee ae ee ey ee ee ee 20
Discussionvohitlesmne thos 22s es Le ae i a et ap 21
Conclusions see 2225.55 Sie 5 eS oe Se ea re eT 22
References 2 sseece: oak Nas Ses ed cre a ep ee | Ser pm 22
MxamplesiofuRetraction 2. 2= = so heee= a2 setae ene «ere = ye mye ee eae 32
Appendix, Theory and plotting data for refraction scales_________________________ 44

CONTRIBUTION FROM THE DEPARTMENT OF ENGINEERING
UNIVERSITY OF CALIFORNIA
BERKELEY

II

GRAPHICAL CONSTRUCTION OF WAVE REFRACTION DIAGRAMS

BY

THE WAVE FRONT METHOD

INTRODUCTION

The height, period, and direction of waves in
deep water at an offshore point may be estimated,
either by direct measurement with suitable instru-
ments or directly from synoptic weather maps by
the forecasting method of Sverdrup and Munk.’
When waves move shoreward from deep water
and approach the shore line at an angle, the wave
crests are bent because the inshore portion of the
wave travels at a lower velocity than the portion
in deep water; consequently, the crests tend to
conform to the bottom contours. Figure 1 shows
the bending, or what is called “refraction,” of
waves near ashore line. The results of refraction
are a change in wave height and in direction of
travel. The amount of these changes is best
estimated by use of a “refraction diagram.”
Such a diagram might be prepared entirely from
aerial photographs, as was done in figure 2, but
generally they are constructed graphically. A
refraction diagram may be considered to be a map
showing the wave crests at a given time, or the suc-
cessive positions of a particular wave crest as it
moves shoreward. Only crests several wave lengths
apart are required to show the bending of the
waves and thereby permit the construction of a
set of lines which are everywhere perpendicular
to the wave crests (fig. 2). These lines are known
as “orthogonals”, and the wave energy between
any two orthogonals is considered to remain con-
stant in estimating variations in wave height.
The power transmitted by- a train of sinusoidal
waves is,

P=C,-5 6H?

Here, C, is the velocity of transmission of the
energy, w is the weight of water per unit volume,
b is the length of crest (perpendicular to the local

1 Wind Waves and Swell, Principles in Forecasting,
Hydrographic Office, Misc. Pub. 11275.

direction of travel) and H is the height from
trough to crest. Ocean waves are not exactly
sinusoidal, and their departure from this form
increases as they approach the condition of break-
ing, but this formula is sufficiently precise for
estimating wave heights and can be corrected by
empirical results in the vicinity of the line of
breakers. If no energy flows laterally along the
wave crest, then, in a steady state of wave motion
the same power should flow past all positions
between two orthogonals. Indicating the condi-
tions in deep water by the subscript zero,

The quantity ol e is termed the refraction co-
efficient. It will be designated pe K,. The

4 C,
quantity oe represents the effect of a change

&
in depth on the wave height. It will be desig-
nated as D. The wave height in any depth of
water then may be written as,

H=H,-.D.Ky

It is the purpose of this report to present
methods of determining K,, the refraction co-
efficient.

It should be noted that the values of both D
and K, depend upon the depth and that they are
usually opposite in effect. Refraction commonly
tends to increase the length of the wave crest and
thus to reduce the height while D, representing
the effect of shoaling, tends to increase the height,
except in a relatively unimportant range of
depths where the waves first ‘feel the bottom.”
For reference, values of D are presented in table 1.
(For additional details, see plate I, Breakers and
Surf, H. O. No. 234, U. S. Navy, where it is

fs
ve
tJ
=)
<x
>)
wn

Figure 1.—Aerial photograph of swell, breakers, and surf north of Oceanside, Calif., 17 August 1945 (Utility Squadron 12).

2

SUNOLNOD WOLLOS

SY¥SNVSNE
SLS3Y9 SAVM

XXXXX XK

S TVNOSOHLYO

14- 31V9S
| en a |
ooo! 00s fe)

—Wave pattern fram aerial photograph shown in figure 1.

Figure 2

designated as H/H’,. or Tables of the Functions
of d/Z and d/L,, HW-116—265.)

TABLE 1

Coefficient of shoaling, D

CH OS seer 0.002 0.005 0.007 0.01 0.02 0.04
Dose 2 Sees 7, WH OD IL Be 1.45 1.23 1.06
Ohne == 0.056 0.08 0.1 0.15 0.2 03 0.4

Dewees 1.0 OH 5 CB oO 68H 5GB 4 Oe

The graphical or analytical determination of
wave refraction coefficients assumes (1) that the
velocity of the wave crest depends only upon the
still water depth under the crest at each point,
(2) that the wave crest advances perpendicular to
itself, and (8) that the wave energy is confined
between orthogonals.

REFRACTION AT A STRAIGHT SHORE LINE

When the shore line and offshore contours are
straight and parallel, refraction may be treated
analytically by utilizing what is known as Snell’s

Law,
Sina C
Sina, C,

Here, a is the angle between the wave crest and
the shore line in a depth such that the wave
velocity is C (fig. 3). The change in angle deter-
mines the increase in crest length, and thus the
value of K, is fixed by the depth, which deter-
mines C for a particular C,, and a, the angle in
deep water. Referring to figure 3, the value of
K, may be computed from the relationship,

bo igi b
Cosa, ° Cosa
or
/b. Cos a,
i Cos a
where

one! (& Sin as)

For example, if a, =45° and the depth and period
at the point for which K, is to be computed, are
such that C/C,=0.5,

a=Sin“! (0.50.71) =20.8 deg.
Cos a=0.935 and Cos a,=0.707
K,=0.87

WAVE FRONTS

Se —_

CONTOUR

SHORELINE

Figure 3.—Wave refraction assuming a gradual change in wave
velocity.

For convenience the relationship between a,
@, depth, period, and K, have been summarized
in graphical form in plate II, Breakers and Surf.
This graph is included herein as figure 4.

A thorough understanding of the nature and
magnitude of refraction effects at straight coast
lines is helpful in constructing refraction diagrams
for complex hydrography. The beginner should
study figure 4 in order to develop judgment
regarding the hydrographic conditions necessitat-
ing graphical analysis and as a basis for checking
approximately the numerical values of K, result-
ing from a graphical determination.

It is noteworthy that, on a straight shore line,
the reduction in wave height by refraction is less
than 10 percent when the initial angle in deep

- water is less than 36 degrees.

CONSTRUCTION OF REFRACTION DIAGRAMS

Ideal waves in deep water move forward with
their crests parallel, but over a shoaling bottom
the reduction in wave velocity causes the crest
to swing around in the direction which will
decrease the angle between the crest and the bot-
tom contour. The preceding statement obvi-
ously requires a quantitative definition of what is
meant by “deep water” and by “shallow water.”
The usual definition is that in

deep water, Te
L,=£T?=5.12T°
ie 2a
shallow water, d<=y

10. 200° SOO

£00 200

Figure 4.—Change in wave direction and height due to refraction on beaches with straight, parallel depth contours.

However, the meaning of these terms may be
examined in the light of refractive effects in shallow
water by considering the velocities and angles
involved.

The velocity and length of a wave decreases in
shallow water as shown in the following tables:

d/L. 0.4 0.3 0,2 Oi
(OKOr 0. 98 0. 96 0. 89 0. 71

The angle through which a wave crest will
turn between deep water and any value of d/L,
may be obtained from figure 4. By selecting a few
values to represent the effect, table 2 shows that
the limits of ‘shallow water’ depend upon the

angle «, and the accuracy to which the diagram —

is to be constructed. If a,—70 degrees and if the
construction is at an accuracy of +1 degree, the
diagram should start in depths even greater than
d=0.5 L,, but if a, is only 10 degrees, a negligible
error is introduced if the diagram is started from
d=0.8 tby.

As a working rule, to be modified if circum-
stances so indicate, refraction diagrams should
start from straight wave crests in a depth equal to
half the deep water wave length or at d=0.5 X 5.12
ee

TABLE 2

Values of « as a function of d/L, and a,

d/Lo 5 ;

Aw 0.5 0.4] 0.3 0.2 0.1
iN
NZ ae

70 69 68 64 57 41

Mico 49.5 | 48 46 42 32

30 30 29 28 26 21

10 10 10 9. 95 8.5 7

The velocity of a wave in deep water is 5.12 T.
As the wave moves into shallow water, its velocity
decreases and, if the crest makes an angle with
the bottom contours, the wave velocity will vary
from point to point along the crest. Graphical
construction of a refraction diagram consists
simply in moving each point of the crest in a direc-
tion perpendicular to the crest by a distance equal
to the wave velocity times the time interval
selected. The initial form of the wave is a straight
line in the deep water area, as previously defined.
Figure 5 shows scales constructed in such manner
as to give the advance of the wave crest at any
value of d/Z, on a chart of any scale S. (These
scales have been printed on thin paper and are
available for distribution.) The two scales of
figure 5 differ only in that scale A gives the wave

advance during an interval which is twice that of
scale B.

To construct a refraction diagram, the chart
first 1s contoured with an interval which will
represent accurately the details of the bottom
topography. Each contour on the map is con-
verted to mean sea level-—or any other desired
stage of the tide—adding the proper constant to
the chart soundings. For the wave period selected
the deep water wave length is computed from the
relationship, 2,=5.12 T?. The contour values, in
depth in feet below the tide stage selected, are
then divided by Z, (in feet) to give contours in
terms of d/Z,. Thus, in figure 6, for example, the
contours in fathoms have been re-labeled in terms
of values of d/Z,. Additional contours of d/Z, may
be added if considered desirable. In figure 6, for
example, contours of d/Z, of 0.5 and 0.4 have
been added.

Generally, it is sufficient to draw every nth
crest, where the value of the crest interval, n,
depends upon the scale of the chart and the
complexity of the bottom topography. The crest
interval is determined by the scales used and may
be expressed as 7, a multiple value of wave length,
or as a time interval, ¢. The crest interval does not
have to be an even value, nor does it have to be
the same for the entire chart, simce more crests
often should be drawn where the bottom topo-
graphy is particularly complex. The two trans-
parent scales (fig. 5) for plotting the wave advance
are provided so that the crest interval in one scale
(scale A) is just twice that for the second scale
(scale B). These scales are applicable to charts
of any scale and of any wave period. The only
variable between refraction diagrams prepared by
the use of the scales is the crest interval, this
interval being a function of the scale of the
hydrographic chart. Formulas are provided for
computation of the crest interval, n, or time
interval, ¢t, for any particular refraction diagram
(fig. 5).

It is often advantageous, as well as sometimes
necessary, to draw a refraction diagram for a
particular locality in several steps; First, the
over-all pattern for a long stretch of coast line is
drawn on a relatively small scale chart, following
the waves from deep water to within a few
thousand feet from shore; finally, the results are
transferred to larger scale charts, and a detailed
pattern is constructed of the waves close to shore in

. bays, harbors, and other areas of particular

importance. Where the tidal range is large, it

AMOLVHOSY1 SOINVHOSN GINS
VINYOSINVD JO ALISNSAINN

SWVYSVIG NOLOWH4SY SAVM JO NOILVEYVdSYud YOS S3A1IVOS

=
S
<
m
= SGNOO3S #18 = (000°0S)(E9100) = }
S) ONY
Bp 2(01)
ro) SHLONIT 3AVM #I°8 = (000'0S)(€910 0) =U
m
- SI SNOILISOd 1S349 3AISS399NS
: 000‘os
N33M136 JONVISIC 3H1‘@ 3IVIS ONISN N3HM- 93S O1 1 ONY ——, — JO 31V9S dYN V HO4
21d NXE
(@ 31V9S)
S .
<
m
b
()
<
E SGNO93S ‘ GOIN3d JAVA = 1
?)
a GIONVAGY SHISN31 3AVM JO Y3EWNN = U
N33M138 WWAMaLINI 3NIL = ¢ Sf, WHOS 3H NI 3179S dUN = S 3U3HH

1 2 i i zi
3e900=4 Gv =$ 9100-4 ‘g31IvVOS HOS s9ZE00=+ any |9Z800=u fv 31V9S Od

FIGURE 5

i

2

776699 O—48

may be necessary to construct several diagrams for
different stages of the tide. On the California
coast, where the range of tide is approximately
5 feet, diagrams prepared for an average stage
of tide usually will suffice.

REFRACTION DIAGRAM, MONTEREY BAY,
CALIF.

It is desired to prepare a refraction diagram for
Monterey Bay and obtain values of K, for points
along the coast from Monterey to Santa Cruz.
The diagram is to be prepared for a mean tide
condition of 2 feet above M. L. L. W., direction of
advance in deep water from W. N. W., and T=14
sees. Thus, £,=5.12 T?=5.12 (14)?=1,000 feet
and the depth which is the dividing contour be-
tween deep and shallow water is L,/2—500 feet.

U.S. C. and G. S. charts that are available to
show bottom topography are No. 5402, scale
1:214,000 and No. 5403, scale 1: 50,000.

The contours appearing on these charts are in
fathoms. The equivalent values in terms of
d/L, are as follows:

100 fathoms; Gs (6) (100) +2 9 602

1B 1,000
. (6) (50) 42.

50 fathoms; Tes 1,000 =(.302
. d _(6)(40) +2 _

40 fathoms; OO =(.249

d_ (6)(30)-+2

30 Halon) 1,000 0.182
20 fathoms; p= OER =0.12
15 fathoms; Eo abo 9.022
10 fathoms; p= OF? =0.062
5 fathoms; i Oe 0.032

For accuracy in preparing the refraction dia-
gram, it is necessary to use the chart with the
largest scale (chart 5403); however, this chart
does not extend to deep water (that is, beyond a
depth of 500 feet) ; hence chart No. 5402 must be
used, necessitating carrying the waves part way

on this smaller scale chart and then transferring
the front to the larger scale chart (No. 5403) near
to the shore line. (Usually it is desirable to
draw refraction diagrams on tracing paper placed
over the hydrographic chart; however, for illus-
trative purposes in this report, the diagrams are
drawn directly on the charts.)

Figure 6 shows a portion of USC and GS Chart
5402 with 14 wave crests marked 1, 2,.... 14.
Crest 1 lies in deep water and is drawn as a
straight line. The southerly portion of crests
1-14 remain in deep water (that is, where the
contours of d/Z, have values greater than 0.5),
and the distances between them are equal to a
constant multiple of Z,. The northerly portions
of the crests advance into shallow water with the
result that the distances between the crests
decrease. The position of each crest is deter-
mined from that of the crest behind it by locating
a few points on the new crest and drawing a
smooth curve through them. These points are
shown as small circles in figure 6. The points are
located by means of scale B in figure 5 which has
been cut out and placed on the refraction diagram
as illustrated in figure 6. For example, to locate
point ¢ on crest 12:

1. Lay the scale on the chart so that the dashed
center line and the line of d/Z,=0.302 on the scale
intersect with the contour d/Z,=0.302 on the
chart (point a).

2. Move the scale so that condition (1) remains
satisfied and the lower side of the scale is tangent
to crest 11 on the diagram at the end of the d/L,=
0.302 line on the scale (point 6).

3. Mark point c where the d/Z,=0.302 line on
the scale reaches the upper side of the scale.

4. Thus, other points as d, e, and f are found by
making use of the d/Z, contours of 0.242, 0.182,
and 0.122, respectively. The process is repeated
until a sufficient number of points are found to
determine the position of crest 12.

In a similar manner to that outlined in items
1-4, above, other crests are located until the
diagram is carried into a locality which is within
the limits of a larger scale chart. On figure 6 it is
noted that crest 14 is within the limits of the area
shown on chart 5403. This crest is then trans-
ferred from chart 5402 by taking offsets from a
convenient longitude (122°), correcting for scale
ratio, and replotting on chart 5403 (fig. 7). Thus,
the offsets in inches at each minute of latitude on
chart 5402 are multiplied by the ratio 214,000/-
50,000=4.28 and are shown plotted on chart 5403

WAVE PERIOD=I4 secs. ¢.
_ WAVE. DIRECTION = WNW ~

UNITED STALES= WEST COAST
CALIFORNIA

POINT SUR: TO SAN PRANCISCO

SOUNDINGS EN FA PHTOMS
AT MEAN LiSeKH LOW WATE

Figure 6.—Chart 5402.

with a smooth curve drawn to give the position of
crest 14. (Note that charts 5402 and 5403 have
been reduced by photostat for inclusion in this
report.)

Starting with wave crest 14 on chart 5403,
crests 15 to 23 are plotted by the method illus-
trated in steps 1 to 4, above. Because the bottom
slope in the area covered by this chart, in general,
is relatively uniform, crests can be plotted using
a larger interval between crests; thus, crests 15
to 23 were plotted using scale A of figure 5.

On chart 5402 the spacing between crests was
determined by means of scale B and the corre-
sponding values of n and ¢ are:

.n=0.0163 ZT 7.8 wave lengths
(14)
t=0.0163 i ane seconds

Similarly, on chart 5403 scale A was used, and
the values of n and ¢ are:

n=0.0326 (DUO) 5, wave lengths
(14)
t=0.0326 a =117 seconds

At a few localities where the bottom configu-
ration is irregular, intermediate crests on chart
5403 (fig. 7) have been added by the use of scale
B. These localities are between crests 16 and 17
near a side canyon of the Monterey Canyon and
shoreward from crests 21, 22, and 23. Even this
spacing is inadequate to describe the refraction
which probably occurs at such locations as Santa
Cruz Harbor, Monterey Harbor, and at the head
of the Monterey Canyon at Moss Landing. For
greater detail in these areas, charts with a larger
scale should be used. Photostat enlargements,
of the areas from chart No. 5403 could be made,
but for more accuracy, the original USC and GS
work sheets should be used. Thus, figure 8 shows
contours from Hydrographic Chart 5415, which
covers Monterey Harbor and vicinity with a scale
of 1:5000. Inspection of chart No. 5415 shows
that the bottom contours extend seaward only to
the 17-fathom contour, which is about the location
of crest 23 on chart 5403. Crest 23, consequently,
has been transferred to chart 5415 by the offset
method described above. Crests 24 to 33 have
been constructed by the use of scale A. The
crest interval on this diagram is as follows:

10

(5,000) _
Cm

(5,000)
14

n=0.0326

0.83 wave lengths -

t=0.0326

=11.7 seconds

Further detail near the Monterey breakwater
and within the harbor could be obtained by en-
larging that area from chart 5415.

Upon completion of a plot showing the wave
crests, the orthogonals are drawn on the diagrams.
The orthogonals are started at the shore, or at
some specified depth contour, and carried seaward
as perpendiculars to the wave crests until deep
water is reached. In shallow water they are
curved lines but become straight lines in deep
water. A small triangle and a short straight edge
are convenient in constructing the orthogonals.
The triangle is adjusted to be tangent to a wave
crest, such as at a point a (fig. 9) where the orthog-
onal is to be started (crest 1). The straight-edge
is held against the triangle and the triangle then
slid along the straight-edge to permit a perpen-
dicular to be drawn through point @ and to point ~
b half way to crest 2. The process is repeated
for crest 2 with the perpendicular being drawn
shoreward to point 6 and then extended seaward
to a point ¢c midway between crests 2 and 3. The
procedure is repeated until the wave crest in deep
water isreached. If desired, a smooth curve then
could be drawn through the points where the per-
pendiculars cross the crests. The irregularities
in the wave crests indicate how closely the ortho-
gonals should be located. In figure 7 orthogonals
have been drawn every two nautical miles starting
at the Monterey Municipal Pier. With the ex-
ception of the vicinity of Moss Landing and north-
ward this spacing appears suitable for most of
the coast between the orthogonals A-—F (fig. 7).
As mentioned above, if the refraction coefficients
are desired for the Moss Landing shore line, a re-
fraction diagram should be prepared on a larger
scale map, and the orthogonals should then be
drawn at a smaller spacing. Additional orthog-
onals should be constructed northward from Moss
Landing to better define the refraction coefficients
in this area.

It is to be noted that the orthogonals shown on
chart 5403 (fig. 7) ended at crest 14 and then had
to be transferred to chart 5402 (fig. 6). In all
probability, not all the orthogonals can be car-
ried to deep water on the smaller scale chart.
Only a few of the orthogonals actually are re-
quired to give a measure of refraction between

iU3

24

fAURS — WES!
CALIPOBNIA

MONTEREY BAY

WAVE PERIOD = I4 SECS.
WAVE DIRECTION = WNW

Figure 7.—Chart 5403.

1334 40 31V9S
odse 0002 0051 O00l 00S oO 00s

WOH1V4 |= TVAYSLNI YNOLNOD
SQNO93S 1 = GOINad JAVM

Ava AaYaLNOW
WVYSVIG NOILOVYessy

200,25/2/|

OS 27

\

a

SaNooag

SHLONST SAV

|

4 3

= TWAYSLNI SAIL
€8'0 = SONVAGV S/AVM

~

€0bS LYVHO WOYS
Q34uYN3S4SNVYUL E2 1S3YND

OF 25 «727

00/65 127,

OOPS 127

Figure 8.

12

f

FINAL
POSITION

deep water and crest 14, thus, the points on crest
14 which are midway between orthogonals A-—L
(fig. 7) have been transferred to figure 6. It has
not been necessary to carry orthogonal J, seaward
from crest 14 as orthogonals H and J give a
measure of the refraction for this portion of the
wave.

As an example of the computation of the re-
fraction coefficient, Kz, a point on the 5-fathom
contour midway between orthogonals K and L is
considered. ‘The procedure is as follows:

On chart 5403 (fig. 7) the distance between
orthogonals K and L at the 5-fathom line is
2.9 inches and at crest 14 the distance is 1.72
inches.

On figure 6 (chart 5402) the distance be-
tween orthogonals K’ and L’ is 0.49 inch at
crest 14 and 0.12 inch at crest 1. Therefore,
the refraction coefficient for the point mid-
way between orthogonals K and LZ at the
5-fathom line is:

N02 ON
ea ae X9.49 = 0-38

TO DEEP WATER \

INITIAL POSITION

WAVE CRESTS

Figure 9.

Computations for refraction. coefficients at
other points along the 5-fathom contour are
summarized in table 3. To obtain a refrac-
tion coefficient for a point nearer to the
Municipal Pier than obtained by orthog-
onals A and B, additional orthogonals have
been drawn on charts 5403 and 5415 (fig..8).
Point M on crest 23 has been selected as a
starting point. On chart 5403 two orthog-
onals, 0.3 inch on either side of point MM,
have been carried seaward to crest 17 in
deep water, where the distance between
orthogonals is 0.13 inch. On chart 5415 the
two orthogonals, 1% inches on either side of
point M/, have been carried to the 5-fathom
contour where the distance between orthog-
onals is 4.72 inches. The value of K, at
this location on the 5-fathom contour, which
is 0.8 nautical mile from the Municipal Pier,

1S
Le OBR a
E=4/ 0 £797 0:37

For the benefit of the forecaster, refraction dia-
grams should be prepared for various periods and

ay

N

Bs

REFRACTION COEFFICIENT, K

10

[TA
LiL
eee as ee

12

DISTANCE FROM MONTEREY MUNICIPAL PIER—NAUTICAL MILES

Figure 10.—Refraction coefficient at 5-fathom line—Monterey Bay.

several deep water wave directions. The coeffi-
cients should be summarized in convenient table
or graph form. If forecasts are being made for
only one point on shore, a plot of K, as a function
of wave direction and period is sufficient. If fore-
casts are being made for several points along a
coast, a convenient means of summarizing the data
is a graph which shows K; factors plotted against
distance along the coast for various wave periods
(fig. 10). A separate graph for each wave direc-
tion would be necessary. Another possible method
of summarizing data would be to show a map of
an area with contours of equal AK, values indicated.
A map would have to be prepared for each wave
direction and period. Figure 11 shows such a
map for Monterey Bay as prepared from the
refraction diagrams shown in figures 6 and 7.

As previously stated, it may be necessary,
where the tidal range is large, to construct sepa-

- rate diagrams for different stages of the tide.

14

It is important to note that the assumption of
constant wave energy between orthogonals does
not apply after a wave breaks. If waves pass
over a submerged reef, it may be necessary to
examine this area critically to determine whether
the waves break at some, or all, stages of the tide.
Should breaking occur, wave heights beyond the
reef would be lower than that determined by use
of K, factors from a refraction diagram (for
examples, see figs. 25 and 33). As a wave passes
over a reef, whether or not breaking occurs, the
crest may break into several crests. Thus, the
further refraction of the wave may not be simple.

The refraction coefficient, Kz, is a function of

TABLE 3

Computation of refraction coefficients for Monterey Bay

[Ka values apply to the 5-fathom contour for waves of 14-second period from W. N. W.]

Data from chart 5403 Data from chart 5402 Ka Distance
a) at mid-
Ss t at 5-fathom point be-
WEAOHE Es CAE) Segment at crest 14 Segment at crest 14 | Segment in deep water ____s«|_-s tween or-
line Col. (4) Col. (9) VGo1 (0) Conan
Col. (2) Col. (7) X Col. (5) | miles from
N Length No Length papa No Length No Length ? municipal
a (inches) : (inches) NO: (inches) : (inches) pier)
@) (2) @) @) 6) 6) a) @) @) 10) a1) a2)
A-B 2.9 A’-B’ 1.87 0. 65 A’-B’ 0. 44 A’’-B’’ 0. 44 1.0 0. 81 3
B-C 2.9 B’-C’ 2.61 9 B’/-C’ - 61 B’’-C”’ 61 1.0 -95 5
C-D 2.9 C’-D’ 2.78 - 96 C’-D’ . 65, C’’-p”" . 64 99 98 7
D-E 2.9 D/-E” 1.73 . 60 D/-E’ . 41 D’’-E” . 40 98 ol 9
E-F 2.9 E’-F’ 1. 46 - 50 E/-F’ . 34 E’/-F”’ . 24 71 . 59 11
F-G 14.42 F’-G’ - 90 . 63 F’-Q’ . 20 F’’-Q”’ -1l . 55 2.59 13
G-H 11.88 G’-H’ 1.15 . 61 G’-H’ - 26 G’’-H”’ 17 . 65 2.63 15
H-I 2.9 H’-I’ 1.02 ~35 H’-J’ . 34 H’’-J"’ .18 - 53 3.43 17
I-J 2.9 I’-J’ - 42 -15 H’-J’ . 34 H’’-J”’ -18 53 3.28 19
J-K 2.9 J’-K’ 2.0 - 69 J’-K’ . 43 J/’-K"” .14 33 48 21
K-L 2.9 K’-L’ 1.72 . 59 K’-L’ -49 K’’-L”’ 5 1) 25 38 23

1 Distance between orthogonals measured at crest 20. Refraction shoreward of this crest should be determined from larger scale chart.

2 Coefficients apply to crest 20.

3 Refraction seaward from crest 14 is based on distance between orthogonals H and J.

depth, of period, and of the initial angle of the
wave crest. In the preceding examples, repre-
sentation of the results was simplified by report-
ing the coefficient of refraction at a constant
depth of 5 fathoms, thus eliminating one variable.
The question now arises as to the magnitude of
the difference in this coefficient if the wave breaks
in a lesser depth, say, 10 feet. For this purpose,
the 30-foot contour may be assumed as parallel
to the shore line and the effect worked out approxi-
mately from figure 4. Using the example of
Monterey Bay, with a period. of 14 seconds and
waves from W. N. W., the refraction coefficient
between stations # and F is around 0.60 at
d/L,=0.03. On a straight shore line, from figure
4 this coefficient and depth would correspond to
@=71° and a=24°. At d/L,=0.01, a=13 de-
grees so that the further change in a between
d/L,=0.03 and d/L,=0.01 would be about 11
degrees. The change in coefficient would be
determined from

Bb, Sasi
Ky= Bao 0 WV b,,
Kw bso Cos 30 _g 6
Kx bio Cos aio | ‘

With the change in angle from 24 degrees at
d=30 feet to 13 degrees at d=10 feet, the value
of Kz at the 10-foot contour is only 97 percent of
the value at the 30-foot contour. Evidently, the
exact depth to which the refraction diagram is
carried does not greatly affect the value. As a

776699 O—48——3

15

working rule, to be modified if special circum-
stances so indicate, carry refraction diagrams
shoreward to at least d/L,=0.03.

If the refraction diagram is to be drawn for the
purpose of determining the local angle between the
shore line and the the breaking crest, then it is
obviously necessary to continue the diagram to
the depth in which the wave breaks. This refine-
ment is unnecessary in determining wave heights
but is necessary in estimating the strength of the
littoral current which depends upon the angle
between the breaking crests and the shore line.

CONSTRUCTION OF REFRACTION DIAGRAMS
FROM AERIAL PHOTOGRAPHS

The graphical method of preparing refraction
diagrams may be replaced by a purely photo-
graphic method, using accurately timed aerial
photographs. Various components of the method
have been described in other reports and the
procedure will be summarized only briefly here.
Steps in the analysis are:

1. Obtain aerial photographs of the shore line
and offshore area, preferably verticals, taken at
an accurately timed interval of approximately 3
seconds and with about 85 percent overlap.

2. Check photographs for altitude and tilt and
determine ground scale. Enlargements corrected
for tilt are desirable.

3. Trace crest of major wave train from selected
photographs in the set and transfer to overlay of
hydrographic chart. The set of photographs will
show different crest angles at the same position
and averaged curves should be drawn.

4. Draw the orthogonals and from the spacing
of the orthogonals determine K, at a selected
depth contour.

5. Determine the average wave period of the
major wave train by (a) measuring from the
photographs the time interval between breakers
or between the instants at which crests pass
identifiable points such as rocks or small patches
of foam or (6) measuring the wave length at points
where the depth is known and compute the period
(or obtain it from available graphs).

Except for 50, this procedure may be followed
even when the hydrography is unknown.

This aerial method of preparing a refraction
diagram has the practical advantage that it deals
with real waves, which vary in period and direc-
tion, and it truly represents the effect of local
irregularities in the bottom. It has a number of
disadvantages among which may be mentioned:

1. Obtaining photographs representing a range
of periods and directions, and possibly low and
high tide, will require considerable flying time and,
almost invariably, a long period of waiting for
the desired wave conditions to occur in good
photographic weather.

2. The swells, which dominate the breaker zone,
are frequently obscured at a relatively short dis-
tance from shore by small steep waves. It is
usually the long period waves which are of interest.

If the refractive effect occurs in a limited area,
as in a small bay or near a harbor entrance, direct
use of aerial photographs as outlined here is
feasible. If the refractive effect takes place gradu-
ally over large areas, the aerial photographs
become too time consuming because of the problem
of ground control. A judicious combination of the
graphical method with measurements of aerial
photographs will yield the most reliable results at
reasonable cost.

DETERMINATION OF REFRACTION COEFFI-
CIENTS FROM AERIAL PHOTOGRAPHS

In the preceding section, the method outlined
utilized the aerial photographs only for the pur-
pose of determining crest positions. The refrac-
tion coefficients, Kz, were then obtained by draw-
ing orthogonals and measuring the spacing between
them just as in the completely graphical method.
Accurately timed aerial photographs, preferably
corrected verticals, may be used to go a step
further and permit measurement of the refraction
coefficients directly. The steps in the procedure
are as follows:

17

1. Secure accurately timed aerial photographs
of the shore line and offshore area.

2. Determine offshore direction directly from
photographs if photography shows waves in deep
water or indirectly from the period, depth, and
angle at the offshore edge of the photographs.

3. Determine the depth by the wave velocity
method (“Underwater Depth Determination,”
U. S. Navy Photographic Interpretation Center
Report 46) if the available charts do not show
hydrography in and near the line of breakers.

4. Measure the photographs to determine the
depth in which the waves break and compute
breaker height from H,=1.3 d;.. Repeat for as
many waves at each shore-line point as photog-
raphy permits.

5. Determine period (a) from interval between
breakers or (6) from water depth and wave
length outside breaker line.

6. Compute Z,, from JT (L,=5.12T?), and
d,/L, and obtain H,/H,’ from plate I, Breakers
and Surf, H. O: 234.

7. From the value of H, obtained in step 5 and
H,/H,’ compute H,’, the wave height which
would have been required to produce the observed
breaker had the original crest been parallel to the
shore and the shoreline straight.

8. If it is assumed that the breakers ob-
served were generated by a train of waves of
uniform period and height, the relative refraction
coefficient is obtained by taking the ratio of H,’
at each point along the shore to its value at a
single reference point, say, the point where H,’ is
maximum.

9. To obtain an absolute value of K, for each
point, it is necessary to find at least one reliable
value by (a) computing the refraction coefficient
for a point along a straight stretch to which
figure 4 applies or (6) obtaining an independent
measure of H, from an offshore recorder, from
photographic measurements of breakers on an
adjacent straight beach, or other means. Divide
H,’ from step 8 by H, from step 9 to obtain the
absolute value of Ky.

10. The experimental values of K, in step 9
apply to the varying depths in which the waves
broke. Usually, no further analysis is necessary,
but it should be remembered that K, should be
corrected back to a common depth, if the difference
between the standard depth and the depth of
breaking changes Ky.

In addition to the disadvantages, previously

mentioned, of the aerial method of determining the
refraction pattern, measurement of the co-
efficient from aerial photographs suffers because
of the fact that waves vary in height along their
crests and there is not a single value of H, appli-
cable to all breakers shown in the photographs.
At a scale of 1:12,000 a 7-inch print, and a speed
of 15,000 feet per minute, for example, each point

on the shore remains in the field of the camera for
about 30 seconds, or long enough for 2 or 3 waves
to break. By measuring all the waves visible at
many points, random variations may, in part, be
eliminated. The accuracy is incréased by making
several photographic sorties in quick succession
while the same wave train prevails.

GRAPHICAL CONSTRUCTION OF REFRACTION
DIAGRAMS DIRECTLY BY ORTHOGONALS

I. INTRODUCTION

A system has been devised whereby the orthog-
onals to refracted wave fronts may be constructed
directly without first drawing the wave fronts.
This method has the advantage of eliminating an
entire graphical step and its attendant inac-
curacies. In trained hands a refraction diagram
can be constructed by this method in about a
quarter of the time required for its construction by
the wave front method. The method requires a
higher degree of training, however, and the opera-
tion does not become so nearly automatic as does
the wave front method. The method is thus
suitable for use by specialist draftsmen or engi-
neers.

Physically the method is carried out by a special
protractor which incorporates the requisite scales.
The protractor is manipulated in steps from con-
tour to contour and at each step indicates the
direction of the orthogonal. One orthogonal is
thus drawn from deep water to shore in each series
of operations. The device has been made in the
form of a protractor so that it would be entirely
adequate for the construction of refraction dia-
grams. A drafting machine can be employed to
an advantage, however, and the protractor used
merely as a graph and tables for obtaining the
required values for the manipulation of the
drafting machine.

A detailed description of the development of the
method and its application follows:

18

ll. DEVELOPMENT OF THE METHOD
A. Assumptions

1. That contours can be drawn at every abrupt
change in slope of the chart.

(a) The depth at a point between contours is a
linear function of its distance from the contours.

2. That, between contours, wave length and
velocity may be considered to vary linearly (the
usual assumption).

(a) The wave length and velocity may be
considered a linear function of its distance from
the contours.

(0) The radius of curvature of the orthogonal
between contours may be considered constant (a
circular arc).

(c) The angle of the arc is equal to the change of
angle of the orthogonal. ;

3. That the undulations of small Betis
extant in most contours are more likely to be a
measure of observational inaccuracies magnified
by rigorous drafting than an indication of the
direction of level bottom. Also, that bottom
features whose dimensions are small compared
with the wave length do not influence the motion
of the wave to any appreciable extent.

(a) If the contours are smoothed out and only
those features preserved that are obviously
characteristics of the hydrography, the result is
more nearly accurate.

4. That the angle of convergence or divergence

of orthogonals at differential intervals is small
compared to the angle of refraction. (Please note
that convergence or divergence is not considered
non existent).

5. That a line drawn through any point midway
between two contours and making equal angles
with the adjacent contours is closely the direction
of a level line at that point (provided the require-
ment implied in assumption No. 1 is accomplished).

B. Derivation

considering Aw<13° for two-place
accuracy, or Av<6° for three-place
accuracy, tan Aw=sin Aa=Aa=
R.—R,

BB’

AA’=00’=BB’=d (a
Assumption No. 4.

and: hae

differential distance).

let c’ be the velocity from A to B (effective)

let c’’ be the velocity from A’ to B’ (effective)

let ¢ be the time required for wave front to move
from AA’ to BB’

yr),
R,=c'’t; Ri=c’t; then Aa=o tet
oe d sin a

then Ae=—5"7 sin a; but =“
and of te

19

> pj .dtin (Ga=@) 0 18 Ne
7. Aa= eects: sim a; or Aa=F eas sin a
2
RAL .

or Aa=F he sin a (1)

R
but for the general case J see @
therefore, Aa=~ tan a

ave

(2)

C. Utilization

These two equations are thus seen to be inde-
pendent of the scale of the chart, and they have
been made independent of the wave period by the
reduction of this factor to the dimensionless ratio
AL
ee

Equation (1) is applicable to all cases where
Aq is less than some predetermined limit depend-
ing upon the accuracy desired. In general, good
results are obtained when Aa is less than 13°.
In practice Aa rarely approaches this limit.

Equation (2) is more readily applied under
ordinary conditions than is equation (1). The
limitation of equation (2) stems from the fact that
as a approaches 90°, tan a becomes infinite.
Physically this situation may occur but is instan-
taneously altered by refraction so that a becomes
less than 90°. However, the application of equa-
tion (2) normally necessitates crossing an entire
contour interval at each step. The value of
Aa changes very rapidly in the region when a
approaches 90°. Thus the instantaneous refrac-
tion over a small distance results in a great change
in the rate of refraction, and the interval must be
crossed in a series of shorter steps. It is there-
fore desirable to employ equation (1) whenever
a exceeds about 80°. Equation (1) readily lends
itself to crossing a contour interval in partial steps
by the judicious selection of R (the distance of
wave advance).

The protractor has thus been constructed with
graphs for equation (2) and a special table for use
when «@ exceeds 80°, which adapts equation (1) to
the graph of equation (2).

The use of the graph suffices for the great
majority of diagrams and, for all ordinary cases,
there is never any necessity to refer to the table.
The table therefore has been made very simple.
The graph requires the measurement of one factor
only (a) and thus is much faster to use than the
table, which requires the measurement of three
factors (a, R and J).

The uses of these two component operations are
summarized in the following:

TaBLe I
. Measured

Equation quantities Use

AL
Graph Ao=p— taniazs! 225.2 25228 (eet aeee eae For values of @ less

ae than 80°.

RA F

Table Aa=> Tees SIN aaa eee a, Rand J___-| For values of @ greater

than 80°.
or
AL
a tan ae

where

R*
@e=tan TF

*Where the angle made by the orthogonal and the contours is greater than
80°, the graph is still used by crossing the contour interval in a series of steps
and progressing a distance R which has some definite relation to J such as
1, 0.5, ete. Equivalent a for such cases comprise the t-ble. Here a. (equiva-
J
may be given the value of 1 and the equation becomes

lent «)=tan— — sin a, but, as a is between 80° and 90° for these cases, sin a

@.=tan— ins (3)

T

Figure 13 is a reproduction of the type I pro-
tractor showing the component scales and their
functions. The protractor is 14 inches in diame-
ter.

Figure 22 shows the type II protractor. This
protractor involves the use of a moveable arm.
When this arm is aligned along the direction of
level bottom, Aq is read directly along the poimter

a and Aa are

AL
Dose
entered on the graphs in their actual dimensions,
and it is thus unnecessary to determine their

at the appropriate values of

The factors le are entered on

J
a circular scale. In this way @, is indicated
directly on the graph for a value of R/J. The
type II protractor facilitates the operation but is
more difficult to construct.

numerical values.

Ill. APPLICATION OF THE METHOD

A. General Preparation

Contours are drawn upon the chart at such in-
tervals that adequately will represent the details
of the bottom topography and which are consistent
with assumption 1, above. Normally (as a rule
of thumb) there will be about as many contours
required as the period in seconds of the longest
period wave to be studied. These contours must
extend to a condition of deep water for the longest
period wave. That is, to d=2.567", where d is
the depth of the deepest contour required and T

20

is the period of the longest period wave to be

studied. A table is then prepared for each wave

period to be studied as shown in table II following:
TaBLe IT

Computation for use of protractor in example
Period: 10 seconds Do=512 feet

1 2 3 4 5 6 7
d d d it AL AL
Fathom | Feet Tie To To Tg | ae
1 6 0. 0117 0/26) |_2-2s2..2)|-222 ee eee
0.11 0. 31 0.35
2 12 . 0235 37
20 AT 43
5 30 0587 87
-18 . 66 a7
10 60 .117 .75
.18 84 21
20 120 . 235 .93
05 95 05
30 180 . 802 - 98
02 .99 02
50 300 587 1.00

EXPLANATION OF TABLE

The first column, d, fathom, is a list of the contours of
the chart. Should the chart be in feet, this column natu-
rally is eliminated. In practical problems it would be
advisable to have contours at 3, 4, and 7 fathom as the
AL
Lave
in the following figures, however, fewer contour intervals
have been used.

Column 2, d, feet, is the result of multiplying column 1
by the factor 6. If a stage of tide is used other than that
of the datum of the chart, a certain constant must be
added or subtracted from this column.

values of should not exceed about 0.20. For brevity

Column 3, fs is the ratio of the depth to the deep water

wave length for the wave period. In this case (i. e., 10-
second period) the wave length is 512 feet and is computed
from the equation L.=5.12T?, where T is the period in
seconds, and L, is the deep water wave length in feet.
Column 4, a is taken from a graph of this function
such as that in H. O. Report No. 234, “Breakers and Surf,
Principles in Forecasting,” plate 1, or HE-116—265, “Tables
of the Functions of d/Z and d/Z..”’, or they can be obtained
by interpolation in Table 4 in the appendix of this publi-

cation.
AL

Lo
between the lines of the depths between which the change
occurred.

Column 5, is the change in the = ratio and is written

Column 6, - ave, is the average of the > ratio between
any two depths.

This is also written between the lines. This figure can

be obtained simply by adding one-half of the A z figure
to the previous value for E

ah , is obtained by dividing column 5 by
ave

Column 7,
column 6.

B. General Procedure '
The general method of drawing a refraction dia-

EQUIV. a
(x)

63.5)
45.0"
42.0°

SENS
SSIS Fiah

gram by use of a protractor is carried out below
for the case where contours are simple and a is
always less than 80°. The steps are fully ex-
plained under each figure. The protractor shown
is the type I.

C. Special Procedure

The special method is shown for the two special
cases. First, when the angle of approach is greater
than 80° to the contours and, second, where com-
plex hydrography is encountered. The steps are
again explained under each figure.

IV. DISCUSSION OF THE METHOD

The method has given results in close agreement
with those obtained by the wave front method, and
apparently more nearly correct. See figures 19
and 20.

Its particular advantages are:

21

EQUIV. a
(a)

EXAMPLE 2.
EXAMPLE 1.
[--@ = 16°
AL =
62 5 a0 A = 0.07
ave
OL S Qa zh\ Bw = Loet
Lave (ReadA&< for AL of 0.7
Lave
f and divide by 10).
AX = 9.0°
e &
Figure 13.

1. Speed—orthogonals can be constructed in
about a fourth of the time required by the wave
front method.

2. Independence of the instrument from the
chart scale and wave period.

3. The elimination of an entire step in the con-
struction of refraction diagrams.

The disadvantages are:

1. A higher degree of training is required for
the use of the protractor than for the wave front
method.

2. It is not possible to draw an orthogonal from
shore seaward as is the case of the final step in the
wave front method. This is not an actual com-
parative disadvantage, however, for in the wave
front method the wave fronts first must be
drawn to shore before the orthogonals can be
drawn seaward.

V. CONCLUSIONS

In the hands of individuals who thoroughly
understand the method and its application the
method gives speedy and accurate results.

22

VI. REFERENCES

H. O. Report No. 234, Breakers and Surf, Prin-
ciples in Forecasting.

HE-116-265, Tables of Functions of d/Z and
d/Lo. Navy: Wave Project, University of Califor-
nia. January 1948.

DEPTHS IN FATHOM

30 20 10521.

for

Step 1—The chart has been checked carefully
changes in slope and the requisite contours drawn in
heavily in ink smoothing them off as is consistent with
assumption No. 38.

This chart is then used for all periods and directions
without further work.

Step 2.—A tracing paper overlay is placed on the con-

tour map and the AL is written between each of the

Lave
contours as computed in Table II. This overlay is then
good for all directions of approach for a particular period,
and can be prepared in a very few minutes. All further
work is performed on the overlay.

Figure 14.

23

DEEP WATER WAVE “FRONT

Step 3.—A deep water wave front (W W’) is drawn in for
the direction to be studied. A suitable interval for orthog-
onals is stepped off and the directions of the deep water
orthogonals are drawn in. These are all straight lines, of
course. The direction selected was from the NW. Deep
water refers to depths greater than Z,/2 as described in the
text, in this case 50 fathoms.

Case 1.—Regular, simple hydrography, no angles in excess
of 80°.

Step 4.—The diagram is started on any orthogonal. If
the K value is required for a partiular point on the beach,
orthogonals can be selected which will reach the point.

Where the == values are small, two contour intervals

can be crossed simultaneously. Thus, the first point select-
ed is P;, where the orthogonal intersects the second con-
tour. ais measured to an estimated line P;P;’ which is the
direction of level bottom at the point P,.

In this case it
is a line tangent to dzd,’.

Figure 15.

24

o

2 \)

Step 4.—Measuring a with protractor pinned at Py,
Aa is then turned from the line P,O;’ by use of the degree
scale. When a drafting machine is used, a; is measured
with an ordinary protractor.

Step 5.— Aa, at P, is determined from the graph on the

protractor.

AL
the ae
turned at P; to the right or so that a, is decreased. The
orthogonal is carried into P2, midway between d3d3’ and
d,d;’. Two contour intervals cannot be crossed simulta-
neously from P» shoreward as the refraction is too great. If
it were not so great to shoreward, Po, could be established
at the intersection of P,O,’, and the contour dyd,’.

Since two contour intervals are being crossed,

values for the two are added (i. e., 0.07). Aa is

Figure 16.

25

Step 5.—Measuring Aa: with protractor pinned at Pi,
the line P,02’ is the new direction of the orthogonal.
When a drafting machine is used, Aq is turned on the

protractor head. Aa is always turned in a direction so
that @ is decreased except in the rare case when the wave
is progressing from shallow to deeper water. In this case
Aa is turned so that @ is increased.

Step 6.—At P2, a, is measured in respect to a line P,P,’
which is drawn in by eye. P2P,’ has a direction midway
between the directions of the contours d3d3’ and d4d,’ at the
points where they intersect the line P,O;. Aa is turned
off at P,; and P3 is established in the same manner as P».
This is continued to the beach. In the example Aa at
P, was greater than 138°. Thus P2’ and P,’’ were estab-
lished using the line P,P.’ as an intermediate contour with

= e values of 10.5 on each side. This step is rarely neces-
sary in practice. The true orthogonal corresponds with
the line P; . P; only at the intersections with con-
tours. For all ordinary purposes, however, the line
Py. P; is an orthogonal.

Figure 17.

26

Case II. a greater than 80°.

Step 4.—(Steps 1 to 3 are the same as in Case I). P, is
established along the deep water orthogonal at a distance
from C; equal to half of one of the proportions of J shown
In this case R was selected equal to J.

PH=0.02 is 1.15°. Thus P,P3,

etc., can be established by turning 1.15° at each point and
Rs, ete., equal to

on the protractor.

The Aa for R/J=1 and

progressing a distance f,, Ro .
Ji, Jo. J; until a becomes less than 80°. Any other
proportion of R/J can be selected at any time. It is to
be remembered that the angle turned at any P carries the
orthogonal a distance of 3:2 beyond the point.

Case IITI.—Contours are not simple.

Steno 4.—(Steps 1 to 3 are the same as for Case I.) As
a, and a, are zero no refraction occurs between dd,’ and

d3d3’. P3.,; can be established on the line 0,0,’ as the limit
of no refraction. a3. and a34 are approximately 90° so
P35 and P34 are drawn in using R/J=0.5. At P35, con-
ditions change and P3, is established as though a contour

existed at P35 with a AL between it and dd,’ of %X

Lave
0.21=0.13.

That is, AL is reduced by a simple proportion to estab-

avec
lish an intermediate contour interval.
progresses as usual to Py, Ps, etc.

The diagram then

Figure 18.

oN

W-Sl 24b-O€-2
“03S O02 —— dOolead
1S3M “S ——NOIL93HI0

Q “vy G—alWih
QV3H V1393H
WVYSVIO NOILIVYEs3yY

VL393H

STIVNOSOHLYO
SNOSNONYS ATLHOITS
NI Gains3y SVH
GOHL3SW LNOYS SAVM
3HL NOI93SY SIHL NI :SLON

SNI7 WOHIV4 O1 IV 148 2

SINOYS JAVM

SNIHOVW ONILIVUG
GNV HdVY9 AS SIVNODSOHLYO—o—
LINOYS SAVM AB SIVNODOHLYO——

SGOHLSW JO NOSIYVdWOD

SIVNOSOHLYO

Figure 19.

28

W-OfV Lv -L2-9
94S 02 dolu3ad
1S3M S NOILDSuIG

92v”S dVW
YyNS td
WVYSVIC NOILIVYs3Sy

YOLOVYLOYNd AB SIVNODOHLYO —o—
INOYS SAVM AB STIVNODOHLYO ——

SGOHL3W JO NOSIYVdWOO

\¥?

S1iNOU3

Figure 20.

£8

8

SNS)
SK SANE

2
/

=
en GE De ee Ee SE
aa Le Tee ES)

SS eA Been
ape ee eee (eee Ds BQN
ie ell Sl a een a ad ee SS CSN
ane) De fe en Ee a ESN
a ne ee ed BS SSS |
a Ea ae Geen SEEN a SE SS ES)
ee hea nel Gees ert) eee] ae) (SS
SS ES eS eS SSSSaaaSaSSSSqSaq=
i SS = es S_ a2ZzZZ
7 a eS SS a Ss Ss 2 2)
eS as a ts = = OZ Aah)
=e 2 Ee as en) Bz BZ 44s
Cy Ss Se ae) Bess es Bz See So 2S
= LS ee

My

os.

$

‘Sb

30

Figure 21.

‘j
8
a

Protractor Arm

0

=

%

Protractor REFRACTION
PROTRACTOR
Type Il

UNIVERSITY OF CALIFORNIA
FLUID MECHANICS LABORATORY
BERKELEY

DATE 11-25-47 DRAWN BYJDI-KK

FIGURE 22

31

EXAMPLES OF REFRACTION

Various aerial photographs and refraction dia-
grams are of interest in providing typical ex-
amples. Figures 23 and 24 show refraction
diagrams for Little Placentia Harbor, New
Foundland. Figure 23 shows waves of 10 second
period from NW carried from deep water, on
Hydrographic Chart 2376, to the mouth of the
harbor. Wave fronts then were transferred to a
larger scale chart (chart 5621) and carried into
the harbor as shown in figure 24.

Aerial photographs which show examples of
wave refraction often are of value in the prepara-
tion of refraction diagrams. With the exception of
figure 27 such examples are shown in figures 25-33.

Figure 25 shows a mosaic prepared from aerial
photographs of refraction effects at Half Moon
Bay, Calif. Note that the waves are breaking
over a submerged reef offshore.

Figure 26 shows waves which have passed

32

through the entrance of Humboldt Bay and into
the bay. Note that waves are breaking inside
the inlet. This breaking is probably the result
of a combination of shoal water and a tidal cur-
rent running opposite to the direction of wave
travel. Very often in the preparation of a refrac-
tion diagram, when waves are carried over a
shoal of limited area, the wave crests appear to
cross each other. That such a condition can
occur is illustrated in the upper right-hand corner
of figure 26 where the waves do cross. That a
shoal area exists at this locality is shown by the
hydrographic chart in figure 27 which covers the
section of the bay appearing in figure 26. Note
that where the waves cross, they augment each
other and breaking results.

The important features of refraction illustrated
by the photos in figures 28-33, inclusive, are
indicated in the caption under each photo.

(MN WOYS i
9LE%_ LYVHO ©
“INBWLYVd3d AAVN

WVYSIVIG NOI

SAVM GNO93S
91450 SIHdVuS
dO AVIYSAO N

YOSYVH VILNAOVId 3

yOs

o9VYIIY

eetr

Figure 23.

33

(M'N WOUS 3AVM GNOO93S O01)
1299 ON LHWHD "391440 SIHdVHOOUGAH
"IN3WLYVd30 AAWN 4O AVIH3AO0 NO a0vW

YOSYVH VILNAOVId ATLL

WVYSVIG NOILOVYSSY JAAVM

a7:

"94€2 LYVHO 40 AvTN3AO 3HL

WOUS O3HuB4SNVHL 3Y3M G-@ ONY V-V SLNOH4 3AVM :310N

SQNO93S 99; 40 GN3 3H1 iv @-a@ 3AVM 40 NOILISOd SI o-@

SQNOD3S 99/ JO GN3 3HL lv V-¥ 3AVM “4O NOILISOd Si v-V

_—— sawooas zi NI 39NvAav.

Figure 24.

34

3NO
NOYGYNOS ALL

ae eee oe
a1v9s
vl 1 “a SN3T 0002) ov

6! ‘12 yaeWaA0N
AVG NOOW 41VH

NOWOVusau SAVM
SIVSOW a30uLNooNn

igure 25.

F

35

Figure 26.—Wave refraction inside Humboldt Bay. Note wave fronts crossing at shoal in upper right-hand corner. At this point
the waves augment each other and break.

36

wea’

=" HUMBOLDT BAY
9

FIGURE 27

Figure 27.

37

1000
SCALE - FT.

Figure 28.—A rial photograph showing refraction of waves at Purisima Point, Calif. The waves shown here are from winds blowing
directly on to the coast. Note that the waves in the upper portion of the picture are almost parallel to the shore line as a result
of refraction. The deep water wave length is about 600 feet; the wave height 6 feet; and the breaker height 9 feet.

38

Figure 29.—Aerial photograph of surf at Oceanside, Calif. Two wave systems are present in this picture. There are small wind
waves coming from the upper right, and a long low swell from the upper left. The swell is almost invisible in deep water, but
peaks up near the shore to form the predominate breakers. Note how the waves break in short segments, where crests of the
wind waves are superimposed on the crest of the swell. The wind waves have a deep water wave length of about 50 feet and a

height of 1 to 2 feet. The swell has a deep water wave length of about 1,000 feet, with a height of 2 to 3 feet. The breaker
height is about 5 feet.

39

Figure 30.—Wave refraction at Moss Landing. Note the ‘‘flat’’ water at the end of the pier where the Monterey Canyon approaches
the shore. Compare complicated refraction pattern with hydrography in figure 7.

40

FT

Jj
_
b ¢
oO
i?)

island at

Note reflection from small

Calif.

,

ind waves at Farallon Island

ion of both swell and w

iffract

ion and d

Refract

31

Figure

lower left

41

1000
SCALE- FT.

Figure 32.—Wave refraction in inlet to Morro Bay. Note reflections from Morro Rock.

42,

es

Figure 33.—Wave refraction at Duxbury reef near Bolinas, Calif. Note waves breaking on reef.

43

APPENDIX

THEORY AND PLOTTING DATA FOR REFRACTION SCALES

The theory involved in the construction of the
scales shown in figure 5 for plotting refraction
diagrams by the wave-advance method is briefly
as follows:

It is desired to plot values of wave advance as a
function of the ratio, d/L,. By proper spacing
of values of d/L,, the upper plotting edge of the
scale can be made a straight line, hence it can be
constructed with considerable accuracy.

Referring to the following sketch, x represents
the base length of the scale and the ordinate at the
right-hand side represents the wave advance in
deep water; that is, the advance is some multiple
n of the deep water wave length, Lo.

} ne IES

For any particular value of the depth-length
ratio, such as dn/L,, the distance from the left-
hand end of the scale to the point where the wave
advance is nL, is, by similar triangles, given by
the relationship,

Oy nLn
X nL,
or
L,,
A= XT (1)

For any chosen length of scale (8 inches for
scales A and B, fig. 5) values of X, for various
assumed values of d/Z, are calculated by the
following procedure:

For shallow water, the wave velocity is

ea! tanh ae (2)
and
L
C=7 (3)

or equation (3) may be written

44

IF IE

pase eS
C—T? L,]5.12

(4)

hence, a combination of equations (2) and (4) gives

L tan h (FF)

mn

Table 4 gives the steps in computing values of
X, by use of equation (1) and (5). Column (1)
shows various values of d/,. From Breakers and
Surf, H. O. No. 234, values of L/L, have been
obtained for the corresponding values of d/Z, and
tabulated in column (2). Column (8) shows
values of d/L which were obtained by dividing
column (1) by column (2). Column (4) is the

(5)

TABLE 4

Computations and plotting data for wave refraction scales

d L d on nom Xn
The Lo L 7, | tanb—> | (inches)
1 2 eS 4 5 6
0. 0020 0. 1120 0. 1079 0. 1125 0. 1120 0. 896
0025 . 1250 . 0200 . 1257 1250 1.000
.0030 . 1368 0219 . 1376 1367 1. 094
0035 . 1478 0237 . 1489 1478 1, 182
0040 . 1578 . 0253 . 1590 1577 1, 262
- 0050 . 1764 0284 . 1784 1765 1. 412
- 0060 . 1930 . 0311 . 1954 1930 1. 544
0070 . 2082 . 0336 . 2111 2082 1. 666
0080 . 2224 0360 . 2262 2224 1.779
.0090 - 2355 0382 . 2400 . 2355 1. 884
-010 . 2479 . 0403 . 2532 . 2479 1, 983
012 . 2712 . 0443 . 2790 2714 2.171
014 . 2922 047 . 3010 . 2922 2.338
. 016 . 3116 0513 . 3223 3116 2. 493
. 018 . 3300 0546 . 3431 3302 2. 642
. 020 . 3469 0576 . 3619 3469 2.775
022 . 3633 0606 . 3808 3634 2.907
. 024 . 3786 0634 . 3984 . 3786 3.029
- 026 3931 . 0661 . 4153 . 3930 3. 144
. 028 . 4072 0688 . 4323 4072 3. 258
.030 . 4199 0714 . 4486 4205 3. 364
.035 . 4518 0775 . 4870 . 4518 3. 614
. 040 - 4803 . 0833 . 5234 . 4803 3. 842
045 . 5065 . 0888 . 5580 . 5065 4. 052
. 050 . 5310 0942 5919 . 5313 4. 250
055 . 5538 0993 . 6239 5538 4. 430
. 060 . 5752 1043 6553 . 5752 4, 602
065 5954 1092 . 6861 5954 © 4.763
070 . 6143 1139 7157 6142 4.914
075 - 6323 1186 . 7452 6323 5, 058
- 080 . 6493 1232 . 7741 6493 5.194
- 085 . 6654 1277 - 8024 . 6654 5. 323
090 . 6808 1322 . 8306 6808 5, 446
095 . 6954 1366 . 8583 . 6954 5. 563
. 100 . 7094 1410 - 8859 . 7094 5. 675
. 110 . 7352 1496 . 9400 . 7352 5, 882
. 120 . 7588 1581 . 9934 . 7588 6.070
. 130 . 7805 1666 1. 0468 . 7806 6. 245
. 140 . 8002 1750 1. 0996 . 8004 6. 403
. 150 . 8183 1833 1, 1517 . 8183 6. 546
. 160 8349 1917 1. 2045 . 8350 6. 680
.170 . 8501 2000 1. 2566 . 8501 6. 801
. 180 . 8639 2084 1. 3094 . 8639 6. 911
. 190 . 8768 2167 1. 3616 8768 7.014
. 200 . 8884 . 2251 1, 4143 8884 7. 107
. 220 . 9089 2421 1, 5212 9089 7.271
240 9259 2592 1, 6286 9259 7.407
. 260 . 9400 . 2766 1. 7379 9400 7. 520
. 280 9516 . 2942 1, 8485 9516 7. 613
. 300 . 9612 . 3121 1. 9610 . 9612 7. 690
. 350 . 9780 . 3579 2. 2488 9780 7. 824
- 400 . 9878 . 4049 2. 5441 9878 7. 902

product of 27 and the values in column (3).
Column (5) is the hyperbolic tangent of the values
in column (4). Column (6) gives values of X, in
inches and is obtained by multiplying values
in column (5) by 8 inches, the base length of
the scale.

In plotting the scales, the value of ordinate at
the right-hand side (where d/L,=0.5) can be made
any convenient values. Scale A was made with
a height of 2 inches; whereas, scale B was made
only 1 inch. This selection of heights seems
suitable for the usual hydrographic charts.

The equations for determining the time interval,
t, between wave crests or the number of wave
lengths, », between crests are determined as
follows:

If the chart scale is in the form ¥ and y repre-
sents the ordinate in inches where d/L,=0.5, then

y nL,

12” §

prea) Soe Soe | YS
12L, 12(6.127") (61.44) T°

(6)

45

The time interval, ¢, between crests is the
distance advanced by the wave divided by the
wave velocity; that is, at d/L,=0.5

lb — tithe
TO eae aa @)
or
d @LEnT (8)
Thus, for scale A (fig. 5), where y=2 inches

(0.0326)S
nee ys)

i (9)
and
__ (0.0326)S
i= eee (10)
For scale B (fig. 5), where y=1 inch
(0.0163)S
n= we (11)
and
_ (0.0163)8
i= T (12)

U.S. GOVERNMENT PRINTING OFFICE: 1948

x
i
oe

Ss