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WAR DEPARTMENT CORPS OF ENGINEERS, U. S. ARMY

A STUDY OF PROGRESSIVE
OSCILLATORY WAVES
IN WATER

TECHNICAL REPORT No. 1

BEACH EROSION BOARD
OFFICE OF THE CHIEF OF ENGINEERS

RE a HT I IE TTT BI FV ED PTE ETE I YS ST A SE EE SE
UNITED STATES GOVERNMENT PRINTING OFFICE - - WASHINGTON : 1941
For sale by the Superintendent of Documents, Washington, D.C. - - Price 10 cents (paper)

WAR DEPARTMENT,
OFFICE OF THE CHIEF OF ENGINEERS,
May 19, 1941.

Approved for publication.
By authority of the Secretary of War:

J. L. SCHLEY,
Major General,
Chief of Engineers.

(11)

WAR DEPARTMENT

OFFICE OF THE CHIEF OF ENGINEERS

BEACH EROSION BOARD

The Beach Erosion Board of the War Department was established
by Section 2 of the River and Harbor Act approved July 3, 1930
(Public, 520, 71st Cong.), to cause investigations and studies to be
made in cooperation with the appropriate agencies of various States
on the Atlantic, Pacific, and Gulf coasts, and on the Great Lakes, and
the Territories, with a view to devising effective means of preventing
erosion of the shores of coastal and lake waters by waves and currents.
The act approved June 26, 1936 (Public, 834, 74th Cong.), authorized
the board to make investigations with a view to determining the most
suitable method of beach protection and restoration of beaches in
different localities; to advise the States, counties, municipalities, or
individuals of the appropriate locations for recreational facilities; and
to publish from time to time such useful data and mformation con-
cerning the protection of the beaches as the board may deem to be
of value to the people of the United States.

As of September 1940, the membership of the Beach Erosion Board
is: Col. Jarvis J. Bain, C. E., senior member; Lt. Col. John F. Conklin,
C. E.; Lt. Col. Charles H. Cunningham, C. E.; Maj. A. C. Lieber, jr.,
C. E., resident member; Dean Thorndike Saville, College of Engineer-
ing, New York University; Gen. Richard K. Hale, Director, Division
of Waterways, Department of Public Works, Boston, Mass.; and
Prof. Morrough P. O’Brien, University of California; 1st Lt. Wiliam
C. Hall is recorder of the board.

(TIL)

TECHNICAL REPORTS

Technical Report No. 1—A Study of Progressive Oscillatory Waves
in Water.

Technical Report No. 2—A Summary of the Theory of Oscillatory
Waves (in preparation).

The authority for publication of this report was granted by an act
for the improvement and protection of the beaches along the shores
of the United States, Public, 834, 74th Congress, approved June
26, 1936.

This paper reports the results of the first of a series of experiments
adopted by the board in 1939. It deals with the basic characteristics
of oscillatory wave motion in water. The experimental work was
under the direction of Lt. William C. Hall, C. E., and the report was
prepared by Dr. Martin A. Mason, Chief of the Research Section,
Beach Erosion Board.

A. C. Lirper, Jr.,
Major, Corps of Engineers,
Resident Member.

(Iv)

TABLE OF CONTENTS

Page
AIS tROlE SATO lS er eene ee earns ets eh es ee NNN 9 (et CT ee sat Vv
BUD) FL it eT CON hep ca ce eee pa a pe as eS IL ener VI

SECTIONS

TL, » JUS eS ROS NBN AON a Rea pe ORL ee AP ar 1
am EAUIT]D OSC meena nen apie ae eae ak I Sel ee eee TE Se ee 2
Sem ELCOCE CHUITC ae ey ay pmeieret a AR phy ape as ep CGD Nee tah ag SIE be Is a 3
ZC pIMent yan da Methodse ssa. see as Sek Ee ae 3
SED) COPANO LET RNA ViCS ene ene eer wee ies Bele hae et PE tes oe ee 6
Gmmolallow-a Water Wiaviess sense see Ue LS Ne es ee oe Sao an Soe eee 18
Chey LESCAUT USSH OY Ds ep Ie ge RS ns AA ly ha ag eA 28
Sum CONC SIONS tye eta me Rea ree RSL U RT Renee eo Aes eRe eae ee 30

Appendix I, Previous Investigations Relative to the Verification of Wave Theory.
Appendix IT.—Bibliography.

LIST OF SYMBOLS

a=A function of the wave amplitude or height (feet).
a=ch/L

C=The velocity of wave propagation (feet per second)
d=Still water depth (feet).

e= Base of natural logarithms.

g=Acceleration of gravity (feet per second per second).
h=Wave amplitude or height (feet).

h' =Depth below still water level (feet).

k=onr/L

L=Wave length (feet).

T= Wave period (seconds).

U=Velocity of mass transport (feet per second).
x=The horizontal semi-axis of a wave orbit (feet).
X = Horizontal axis of reference.

z= The vertical semi-axis of the wave orbit (feet).

Z= Vertical axis of reference.

SUBSCRIPTS

ave= Indicates an average value.
m= Indicates a measured value.
s=Indicates a value at the water surface.
t= Indicates a theoretical value.
z=Indicates a value at depth z.

(Vv)

DEFINITIONS

PROGRESSIVE OscILLATORY WavE—A periodic wave motion of finite amplitude
advancing over a water surface without change of type; the length of wave is
large compared to the wave height.

DrEp-wATER Wave—A wave traveling in a depth of water greater than half the
wave length. This definition, and that following, are used for purposes of
classification, and do not denote a distinct physical separation of character-
istics.

SHALLOW-WATER WavEe—A wave traveling in a depth of water less than half the
wave length.

Sritt WaTEeR LEveL—The level at which water would stand if wave action ceased.

Wave Lenata—The horizontal distance from crest to crest of the wave under
discussion.

Wave Preriop—The time interval between consecutive similar phases of a wave
motion.

Wave Hercut—The vertical distance from top of wave crest to the bottom of
the wave trough.

Wave VeLociry—The rate of advance of the wave crest relative to a fixed point.

Orpit Raprus—(Orbit Semi-axes)—The radius of the circle, or the semi-axes of
the ellipse, which a water particle describes during the passage of an oscillatory
wave.

Mass TrAansport—The translatory movement of a water particle induced by the
passage of an oscillatory wave in deep water.

(VI)

A STUDY OF PROGRESSIVE OSCILLATORY WAVES IN WATER
1. INTRODUCTION

Present knowledge of the motion of oscillatory waves in water con-
sists almost wholly of theoretical studies made by mathematicians
seeking either a solution of the classical hydrodynamical equations
for an incompressible heavy liquid, which satisfies certain conditions
dictated in part by logic and in part by experience, or a solution by
geometrical methods based on observation of natural phenomena.
Certain well-determined rigorous solutions of the problem, as posed,
differing chiefly in their definition of the motion of the water particles
involved have been obtained.

Gerstner’s theory,! which was the first solution obtained, is based
on geometrical considerations, and states that the wave surface is a
trochoid, the paths of the water particles being circles. The diam-
eters of these circles, or particle orbits, decrease with increasing depth
below the surface according to an exponential law. The motion of
the wave is rotational. The Gerstner solution rigorously satisfies the
equations of motion and limiting conditions of the problem regardless
of wave height.

Stokes criticized Gerstner’s solution for requiring a rotation, since
Lagrange had shown that the motions of liquids generated from rest
under the influence of impulsive forces are necessarily irrotational;
but did not otherwise question the exactness of the Gerstner solution.

This criticism led Stokes and others to search for a solution to the
problem of surface waves, which would satisfy all the limiting condi-
tions and be such that the eddy vector, or rotation, would everywhere
be zero. Approximate solutions through a fifth approximation were
obtained by Stokes, who failed, however, to prove the convergency of
the series used in the approximations and consequently the exactness
of his solution. Levi-Civita, following Stokes’ work, succeeded in
showing the series to be convergent and obtained a rigorous solution of
the wave problem for the condition of infinite depth identical to that
of Stokes.

In these latter solutions it is found that the particle orbits are not
closed paths but open, indicating a current (defined as ‘‘mass trans-
port”) in the direction of wave propagation. Also, the velocity of
propagation of the wave is dependent upon its height; whereas Gerst-
ner’s solution indicates independence from the wave height.

1 References to all works cited will be found in appendix II, Bibliography.

(1)

2

There exist, therefore, two rigorous mathematical solutions of the
wave problem for water of infinite depth: that of Gerstner (rotational,
closed particle orbits, velocity of wave independent of height); and
that of Stokes-Levi-Civita (irrotational, open particle orbits, velocity
of wave dependent on wave height).

The validity of the solutions discussed is limited by the condition
of infinite water depth. For the case of finite depth of water, in
which we may consider the depth to be of an order of magnitude less
than the wave length, the solution of the problem is much more
difficult. Laplace, and later Airy, found a rigorous solution of the
problem which is comparable to the solution of Gerstner for infinite
depth. The Laplace-Airy solution defines the surface as an elliptic
trochoid, the particle orbits as closed ellipses, the motion as rotational,
and the wave velocity as independent of the wave height.

Stokes investigated this problem and arrived at a solution to a
third approximation; the particle orbits are open, and the motion is
irrotational. This solution is not, however, rigorous, and it remained
for Struik, using the methods of Levi-Civita, to first find a rigorous
solution of the problem of an irrotational wave in finite depths.
Struik’s solution is similar to Stokes’ third approximation, and shows
that the velocity of propagation depends on the wave height, the
particle orbits are open ellipses, and the motion is irrotational.

The present status of oscillatory wave theory may be summarized,
then, as follows: Rigorous solutions for infinite water depths; solutions
of Gerstner and of Stokes-Levi-Civita (similar to those of Rayleigh);
rigorous solutions for finite water depths; solutions of Laplace-Airy
and of Stokes-Struik.

As noted by Favre, experimental confirmation of none of these
theoretical solutions has been obtained. Observations of wave motion
in the sea are very difficult to obtain with a high degree of accuracy
and while laboratory studies may be made with facility and high
accuracy, there is no confirmation that the waves studied are true
counterparts of ocean waves. Observations of wave phenomena in
nature are essential; but laboratory studies leading to the evaluation
of existing theories may be invaluable as a guide to field observation
programs and technique.

2. PURPOSE

It is the purpose of this study to seek in the laboratory experimental
confirmation of the theories outlined above, by comparison of experi-
mental and theoretical values of various wave characteristics. The
experimental data were obtained with the equipment described im
section 4, and are valid only for the case of uniform water depth. A
short summary of previous experimental work will be found in
appendix I.

3

3. PROCEDURE

The study was divided into separate consideration of deep-water
and shallow-water waves, with the distinction between wave types
made on the basis of the commonly employed criterion cited by
Gaillard; deep-water waves are those propagated in depths greater
than half the wave length, shallow-water waves are those propagated
in depths less than half the wave length. It may be noted that while
this division of wave types is arbitrary, it is sufficient for the purpose
of this study.

Twelve deep-water and twenty shallow-water waves, covering the
available range of wave characteristics were studied.

4, EQUIPMENT AND METHODS

The wave tank.—The concrete wave tank is 85 by 14 feet with a
depth of 4 feet. One side is fitted with six glass windows each 24 by
40 inches. Hach window has etched scales graduated to hundredths
of a foot.

The .wave generator.—The wave generator is a counter-weighted
wooden plunger with the forward face set at 51° with the vertical. It
is driven by a variable speed, three horsepower direct current motor.
Wave heights are varied by setting eccentrics, adjustable by quarter-
inch increments to 7% inches, and plunger connecting arms, adjustable
in 14-imch increments.

Wave absorber—A 6-inch layer of \% to #4 inch commercial gravel
placed at an angle of 12° with the horizontal is constructed in the end
of the tank opposite the wave generator to serve as a wave absorber.
A screen consisting of one thickness of number 16 screen wire laced
to a thickness of 44-inch wire mesh is stretched four inches above and
parallel to the top of the gravel slope. -A wave absorber of similar
design but of greater slope is placed behind the plunger.

Wave height measurement.—Three sets of point and hook gages
were used to determine still water levels and to measure wave heights.
A point gage was adjusted to the top of the wave crests and a hook
gage to the bottom of the troughs; the wave height being the distance
measured between point and hook.

Wave length measurement.—Each of two point gages, one fixed
and the other movable horizontally, were connected in an electrical
circuit with a neon bulb and ground in the tank. The gages were
adjusted to just touch the wave crests, the passage of a crest com-
pleting the electrical circuit, and causing a short flash of the neon
bulb. Adjustment of the movable gage, to secure synchronous
flashing of the neon bulbs, indicated that the points were set on a
wave length or its multiple.

408321—41—2

4

Wave velocity measurement.—Wave velocity was measured by timing
the passage of a wave crest over a measured baseline. Float-operated
mercury switches in electrical circuits flashed neon bulbs on a panel to
trace the progress of wave crests. A two-way switch and a key
permitted selection of a wave crest for measurement. A solenoid in
the electrical system automatically started and stopped a stop-watch
and the wave crest closed the circuit at the beginning and ending of
the course.

Photographs.—A Retina II-f2 camera and a 16 mm. Eastman Ciné
Special motion picture camera comprised the photographic equipment.

Profiles against a 6-inch grid on the wall of the tank were photo-
graphed with both cameras.

Particle motion shown by injecting drops of a fluid mixture having
the same specific gravity as water was photographed at the window
at the center of the tank. Still exposures and motion pictures were
made.

The fluid used was a mixture of xylol and butyl phthalate colored
with zine oxide. Injection was made with a glass tube having a small
orifice at one end and a rubber bulb at the other. A scale on the
tank window served for the measurement of orbit dimensions.

Wave period.—The wave period was computed from the average
period of several trains of waves.

Measured wave length—The movable point gage was adjusted to
obtain synchronous flashes for an even multiple of wave length. The
number of waves measured was noted by visual observation, and the
average length of wave thus measured was recorded as the wave length.

Measured wave velocity —The wave travel was timed over a 30-foot
course. The velocity in feet per second was obtained by dividing
the length in feet by the time of travel in seconds.

Measured wave height—Three combination hook and point gages
were set at 12, 24, and 37 feet from the wave generator. In general,
each point and hook was read to still water level before the beginning
and after the end of the run. During each run, five readings were
made. The average was taken for each gage, and the wave height
recorded was the average for the three positions. Readings are to
thousandths of a foot.

Wave profile—The wave profile against the measured grid on the
tank wall was scaled from the enlarged photograph of the wave. The
ordinates were measured at the tenth points of the wave length and at
additional points a twentieth and a fortieth of the length each side
of the wave crest.

Orbit radius.—The measured value of the orbit radius was taken as
one-fourth the sum of the horizontal and vertical diameters of the
particle orbits obtained as described under ‘‘Photographs.’”’ The
diameters were measured between vertical and horizontal tangents

5

to the photographed curves. When the particle, because of mass
transport or specific gravity effects, did not traverse a closed curve,
the diameter was computed as shown on the curve, Figure 1.

It is assumed that the velocity due to mass transportation or
differences in specific gravity were uniform over the period during
which the orbit trace was obtained. Then the true diameter is the
mean of the distance between the tangents at points 3-1 and 3-5.

DIRECTION OF WAVE TRAVEL

STILL WATER LEVEL

ORBITAL MOTION

Ficure 1.

The same reasoning is followed for the determination of the vertical
diameters.

The depth of the orbit center below still water level is considered
to be the depth of the mid point of the vertical diameter as determined
above.

Mass transport—The particle orbits were used to determine the
existence and magnitude of mass transport. Referring to the figure
of the preceding section, when a curve was open on the horizontal
diameter, the amount by which it failed to close was considered as

6

being due solely to mass transport. This distance (1—5 in the figure),
in true magnitude, was divided by the wave period to obtain the
velocity of mass transport. The direction of mass transport was
determined by reference to the direction of travel of the wave, trans-
port in the direction of wave travel being designated as positive.
Surface mass transport—The existence and magnitude of surface
mass transport was determined by a floating cork ball 1% inches in
diameter. The time for travel of the ball over a measured course of
ten feet was observed and the velocity of surface mass transport

10
btained as —.
obtained as —,

Percentage of wave height above still water level —Measured values of
the proportion of wave height above the still water level were ob-
tained from wave profile photographs. The proportion is expressed
as the ratio of the height of the wave crest above the still water level
to the wave height. For each characteristic wave the percentage
wave height above still water level was measured at three positions
in the wave tank, 12, 24, and 37 feet from the wave generator.

5. DEEP-WATER WAVES

The basic data for the deep-water waves studied are given in
table 1.

The ratio h/Z is employed as a criterion of wave type—ratios
greater than about 0.03 corresponding to storm waves—and as a
parameter in the computation of wave characteristics by the Stokes-
Levi-Civita equations.

d x.
TABLE 1.—Basic data—Deep-water waves (F>05)

1 2 3 4 5 6 U 8 9 10
Co Tm-Lm
Run Rave T'm Cm Lm d d/Im have/Lm CaT mn Sa
Feet Sec. | F.p.s.| Feet Feet Feet Percent
ioe sate) oe eee 0. 276 0. 850 4,39 3. 89 3 0.771 0.0707 3. 73 —4.4
7 EE 237 1.003 5.10 5.14 3 . 584 . 0460 5.17 —0.5
(See eee 230 | 0.860 4.40 3. 83 2.5 . 653 . 0600 3. 79 —1.0
|e ee 206 985 5.00 4, 87 2.5 513 . 0423 4,93 +1.2
ge ee eee . 242 848 4,42 3.79 3 792 . 0638 3. 76 —1.1
1h Sek Se at 4 219 969 4.89 4.76 3 630 . 0460 4.74 —0.4
1 eee Se 189 966 4.81 4.75 2.5 527 - 0398 4.65 —2.1
if ee eee ee . 252 847 4, 23 3. 57 2.5 706 .0705 3. 58 +0.3
LOSS ks ee .196 842 4,31 3. 55 3 845 . 0552 3. 63 +2.2
ee Nee ae eee eS 161 996 5.07 5.02 3 598 - 0321 5.05 +0. 6
DOT ES eee ae ae 167 843 4, 33 3. 60 2.5 695 . 0464 3.65 +1.4
pe aay et Soe In? ES . 296 988 5.05 4, 92 2.5 509 .0601 4,99 +1.4
A:verdgess=2\|2 2 uses | Soe | et Se Et A a RT| SE een AT Mi ee ee —0. 20

The values listed in column 10, table 1, are of interest as showing
the accuracy of measurement. For any type of wave motion the
product of wave velocity and wave period is equal to the wave length.

a

This law is fundamental and any deviation therefrom may be con-
sidered as the result of errors in observation or measurement. In
the present study the maximum deviation is of the order of 4 percent;
the average error being 0.2 percent. In the evaluation of the data
presented hereafter the accuracy of the test as indicated by column 10
should be considered.

The measured values of wave velocity and wave length, for each
wave studied, are compared to the corresponding values computed
from theory, in table 2. Columns 2 and 3 list the wave velocity and
wave length as computed by the Gerstner (trochoidal) theory, where:

C.=5_ (1)

EHO Py tnt (eS: (2)

T,, is the measured period which is assumed to have been correctly
determined. Columns 4 and 5 list the percentage differences between
measured and theoretical values. The averages given should be
considered only in conjunction with similar averages; the comparisons
for each test, when considered in company with values in column 10,
table 1, giving a more accurate picture of the relation. Note, for
example, Run 1, the accuracy of measurement is —4.4 percent, the
comparison of measured and theoretical velocity is +0.7 percent.
Inspection of the sign of the differences shows that if the measured
velocity was 4.4 percent too small, then the value in column 4, table 2,
when C,, is corrected by increasing it 4.4 percent, is of the order of
+5.0 percent.

It will be noted that the agreement between the average measured
values and the average theoretical values derived from Gerstner’s
formulae is of the order of 0.5 to 1.0 percent.

The theoretical values for wave velocity and wave length computed
by the formulae of the Stokes-Levi-Civita theory where

Shes 2h? Apa
or S214 +o --—--) (3)

Ga —— Oe
are given in columns 6 and 7.

Again comparing average measured and theoretical values as above,
it is found that the agreement is of the order of 2.5 to 3.0 percent.
Since this theory requires the existence of mass transport, i. e., a trans-
port of water in the direction of wave travel, then in a closed tank (or
any finite body of water) there must be a return flow of water counter
to the direction of wave travel. It is assumed that this flow is uni-
form over the tank transverse cross-section, and reduction of the theo-
retical values by the velocity of the return flow has been made.

8

TABLE 2.—Deep-water waves (> 0.5) comparison of measured to theoretical
_ velocity and length

2 | Sur 3 4 5 6 7 8 9
1 Gerstner Cr Cn Bear Levi-Civita C= Cy |e
eames tear lc Ct Ly te ta dal eee ee Cc; L':
(Ff In Ch Lit
F. p. 8. Feet Percent | Percent | F.p.s. Feet Percent | Percent
Ee = ae ee seer 4. 36 3. 71 0.7 +4.9 4.39 3.75 0 +3.7
J NE pe pgs ea 5. 14 5.15 —0.8 —0.2 5.14 5.18 -—0.8 —0.8
RSE EAS eiradie wi Sas 4.41 3. 79 —0.2 +1.1 4.42 3. 82 —0.5 +0.3
ee a a ee oe 5.05 4.97 —1.0 —2.0 5.05 4.99 —0.0 —2.4
1G Ye een hee ee Se Ee 4. 35 3. 68 +1.6 +3.0 4. 37 3.72 +1.1 +1.9
144252 path eae eae 4.97 4.81 -1.6 —1.0 4.98 4. 84 —1.8 -1.7
Ny ee ae ee ee ee 4.95 4.78 —2.8 —0.6 4.95 4. 80 —2.8 —1.0
Td age ere PS Ee 4.34 3. 68 —2.8 —3.0 4.37 3.72 —3.2 —4.0
11G eet SE ons ANS ee Pe 4. 32 3. 63 —0.2 —2.2 4. 33 3.65 —0.5 —2.7
QO ees 5. 10 5.08 —0.6 —1.2 5.10 5.09 —0.6 —1.4
7 as ee cp 4, 32 3. 64 +0. 2 —1.1 4.33 3. 66 0 -1.6
Oe eee 5. 06 5.00 —0.2 —1.2 5.08 5. 04 —0.6 —2.4
(A VOTap vera =8 |e eee oe ee Se —0.85 = 07202. ose Fee ee —0.9 —1.0

The measured values for the percentage of wave height above still
water level are tabulated in table 3. The results are presented
graphically in Figure 2, where the solid line represents the theoretical
values derived from Gaillard’s (trochoidal theory) formula:

Zee: rh
z= 0.54 AL (4)
and the dotted line, the values derived from Levi-Civita’s formula:
Zig t oes te GENET On

d
TABLE 3.—Percentage of wave height above still water level—deep-water waves( ¢>0.5)

Wave Wave

Aver- | height Percent Aver- | height Percent

Run | os) age | above ebaxe Run | Posi h age | above above

tion i; wave still Sehgal tion is wave still water

height woier level height water level

eve eve

Is psec se 1 | 0.0756 | 0.293 0. 159 B4s.30)| |i se ee 1 | 0.0392 | 0.186 0. 094 50. 6
1 A Sareea 2] .0666 . 259 . 156 GO 25 plete 2} .0400 . 190 . 098 51.6
1 again mee 3 . 0717 . 279 . 153 54.8 1 eee 3 . 0404 . 192 . 106 55.1
2h ee 1 . 0483 . 248 . 144 58. 1 182eee 1 . 0762 . 272 . 159 58.5
Y esis 2 . 0471 . 242 . 130 53. 8 1S eee 2 . 0698 . 249 . 142 57.0
ae we 3 . 0428 . 220 .124 56. 3 18222 52 3 . 0661 . 236 . 144 61.0
Lee 1 . 0611 . 234 . 130 55. 6 ik? ol Se 1 . 0569 . 203 . 120 58.9
A eae 2 . 0582 2220 . 122 D487 || Oa ae 2 . 0552 . 196 . 106 54.1
cea 3} .0608 . 233 135 BO | LO sent 3 | .0541 . 192 . 104 54. 2
jeaseue 1 . 0425 P21 .116 ba700||||s20eoe a= 1 . 0293 . 147 . 076 61.7
L eee eS 2) .0425 . 207 . 108 EYP FI NPs eee 2 | 20321 . 161 . 085 52.8
be eee 3 . 0400 . 195 . 106 645301) 202222 =" 3 . 0351 .176 . 096 54. 6
ik hese 1 . 0697 . 264 . 161 BLO) 22225252 1 . 0486 .178 . 096 53.9
IR ea oead 2 . 0594 «225 . 132 Stet meee 2 . 0472 .170 . 090 53. 0
13h ss 2 3 . 0622 . 236 . 126 32.05 I )220 25 85— 3 . 0431 - 155 . 086 55. 5
La sheny 1 . 0450 . 214 . 109 G09) ||haoneea as 1 . 0620 . 305 .179 58.7
are Re 2) .0475 . 226 .110 ABET ede a2 oa 2) .0602 . 296 . 169 bi1
14cei eS 3 | .0458 . 218 . 120 A) {| Pah 3 | .0583 . 287 . 155 54.0

_ NotE.—Position 1 is 12 feet from wave generator. Position 2 is 24 feet from wave generator. Position 3
is 37 feet from wave generator.

9

The results of this comparison are not conclusive. For values of
$< ~0.05 the measured values agree fairly well with the values

required by either theory.

PERCENTAGE OF WAVE HEIGHT ABOVE STILL WATER
DEEP-WATER WAVES

(GAILLARD)

h
nests ty (LEVI-civiTA)

PERCENTAGE ABOVE STILL WATER LEVEL

Figure 2.

The measured wave profiles are shown in non-dimensional form on
Figures 3 to 5. Plotted on the same graphs are the wave forms given
by Gerstner’s theory:

X,—R6—,r sin 0
(6)
Z,=R-—,r cos 6
and by Levi-Civita’s theory:
i WE an OLY
Z;=4 Cos Sal k+54@ k ) cos 2k.X,+
+(30 ash? + eek’) 18 BkX,— 30° og Ae (7)

+ gt cos 5kX,

: 584
eel ele 23 4h5
where a=5 pak h T2988" h

10

Re

PAE ETE CAE
N JN \

N PSU

ee ae 4
ie

WAVE PROFILES
DEEP WATER WAVES

4
5
re
ra)
‘
5
iu
=
:
un
w
x
2
6

a
wW
rs
a
rd
WwW
oO
|
|

——--— OBSERVED

FicureE 3.

11

WAVE PROFILES
DEEP WATER WAVES

STOKES-LEVI- CIVITA

~---OSSERVED
—-—GERSTNER

Figure 4.

408321—41——3

12

STOKES-LEVI- CIVITA

---- OBSERVED
—-— GERSTNER

2)
re)
=
ie
oO
oO
ae
ud
>
<
=

DEEP WATER WAVES

Figure 5,

13

The actual wave profiles do not agree with either theoretical profile.
The irregular profile of these waves is believed to be due to phenomena
of reflection at the wave absorber, and to certain mechanical defici-
encies in the wave generator. The elimination of the irregularities
will require further study; the purposes of this investigation are not
believed to require their elimination, though desirable.

The dimensions of the paths described by individual water particles
are given in table 4. The theoretical values of the orbit radii are the
same for the Gerstner and Stokes-Levi-Civita theories, their value
being given by

R,=ae—™ (8)

where #,=theoretical orbit radius at depth h’ below still water level.
The last column of Table 4 lists the percentage differences between
measured and theoretical values, the positive sign indicating that the
measured value is larger than the theoretical. The data are presented
graphically in Figures 6 and 7.

The comparison of measured and theoretical values of surface mass
transport is shown in table 5. The theoretical values are given by
the formula: :

U,, =e AE (9)
derived by Stokes. It will be recalled that Gerstner’s theory does not
admit the existence of mass transport.

ORBIT RADII— DEEP-WATER

== 8)

Ol,
saree 4 K
=1334 NI Hid3d

Oo
m
v
+
= al
z
n
m
m
4

al
5 I
Y

H
MEASURED ASURED VALUES=©0
THEORETICAL " EORETICAL “ =—

-l
RADI! IN FEET

Figure 6.

14

ORBIT RADII— DEEP-WATER

RUN {7

4 =1334 NI H1id30
_¢*1334 NI Hid30

’

H

MEASURED VALUES= ®
THEORETICAL = =——

.t
RADII IN FEET

FIGURE 7.

TaBLe 4.—Orbit radii—Deep-water waves (E> 0.5)

Measured - ;
cahtorletily wien emeneenaht =e witty Theoretica gh
Point Depth radius ie
Horizontal} Vertical Radius Ri Rr
diameter diameter Ru
0. 69 0. 080 0.078 0.0397 0.0454 —12.5
0. 96 055 - 055 . 0275 0292 -5.7
aaa 047 . 047 - 0235 0226 —4.0
SONS 028 . 032 . 0150 0168 —10.7
1.50 025 .025 -0125 -0125 0
0. 357 146 . 146 -073 0773 —5.6
0. 600 097 .099 049 . 0520 —5.8
0. 820 067 . 068 . 034 . 0366 -—7.1
1.10 049 049 . 0245 . 0232 +5.6
1PSH 033 037 . 0180 . 0164 4+9.7
1.78 020) flees ce eee - 0100 - 0077 +30.0
0. 47 106 117 - 0557 - 0544 +2.4
(UC) gl ek een See 077 0385 0438 -—1.2
(et) i | ae ee eee 048 - 024 0263 -0.9
1.15 037 032 .0172 0177 —2.8
0. 51 105° jest SS 5e ase 0525 0561 —6.4
0. 64 (18 en | eee eee, ee 0462 0473 —2.3
0.90 06553) 25s eee 0327 0335 —2.4
1.10 O45 Sits 2 222 See 0225 0258 -128
1.40 O35) ae ee 0175 0182 —3.8
0.18 159 . 150 077 0749 +2.8
0. 365 .121 115 -059 . 0587 —0.5
0.415 Siti .107 -055 -0549 +0.2
0. 56 - 090 - 085 043 0453 -5.1
0.7 063 060 031 . 0333 —6.9
1.05 048 045 023 0237 —2.9
1.19 041 -039 020 0198 +1.0
1. 26 038 . 034 .018 0180 0
0. 20 163 aes) 079 .073 +8.2
0. 49 114 110 056 0497 12.7
0.79 070 . 063 - 0332 - 0335 —0.9

15

TaBLE 4.—Orbit radii—Deep-water waves (E> 0.5) —Continued

Measured
Theoretical} pop '
Point Depth radius —_-
Horizontal! Vertical Radius Ri Rs
diameter | diameter Ru
0. 98 0. 053 0.052 0. 0262 0.0261 +0.6
1.19 039 039 0195 0198 —1.6
1.42 029 024 0142 0145 —2.1
0.16 163 151 0785 - 0769 +2.1
0. 29 135 126 -065 . 0647 +0.4
0. 79 070 064 .0335 . 0336 —0.3
0.98 053 046 . 0247 0261 —5.4
1.23 035 033 0170 0187 —9.1
0.48 118 090 0480 0541 -—11.3
0. 70 063 056 0297 0368 —19.4
0. 87 042 043 0215 0273 —21.3
0. 96 038 041 0195 0232 —16.0
1.00 035 035 0175 0216 —18.6
0. 29 116 120 .0590 0602 —2.0
0. 36 100 106 0515 0534 —3.5
0. 55 068 074 .0355 0388 —8.5
0. 79 048 058 0265 0242 +9.5
0. 89 036 046 -0205 0203 +1.0
0.39 090 114 0510 0494 +3.2
0. 49 082 104 047 0483 +8.5
0. 84 060 078 0345 0290 +19.0
1.09 036 038 -0185 0204 —9.3
0. 61 062 068 0325 0287 +13.2
0. 80 046 048 0235 0205 +14.6
1.05 030 . 030 -0150 0133 +12.8
0.115 237 . 228 -116 128 —9.4
0.217 198 . 199 099 112 —11.6
0. 346 174 173 - 087 0950 —8.4
0. 428 152 155 077 0856 —10.3
0. 460 142 139 -070 0823 —14.6
0. 560 128 129 . 064 0708 —9.6
0. 596 115 115 -057 0691 —17.4
0. 669 114 113 . 056 0629 —11.1
0. 766 097 099 049 0573 —14.0
0. 870 088 085 - 043 0486 —11.5
0. 929 081 082 041 0451 —9.1
RN Sasa Ie are ek ate aE eRe siRegeetne |. 22 8y eee SIS = Ce SNS 2 Se Sue —2. 96
TaBLE 5.—Surface values— Mass transport
Measured Theoretical | Um—Us
Run Averees Bae transport d transport Ui
Un U; percent
eee ARMs Shoe eRe occ L SiS 37.7 10 0. 265 3 0.175 +51. 4
RE a ee ENS Soe Se SoS 63.9 10 156 3 117 +33. 4
CO ci OS Ta Re ANP Te 52.0 10 . 192 255) lle +12.3
Die ea ee eee So SS 102.8 10 097 2.5 087 +11.5
1b Se a ee one eee 40.1 10 249 3 169 +47.4
1 ae Tame rapes eae Ae 66.3 10 5 3 . 093 +62. 4
(2 RE SP a Se oe eee 115. 2 10 - O87 2.5 - 083 +4.6
1 ee ee ee 64.4 10 . 155 2.5 - 167 —7.2
sO Oe ee ee ae 106. 4 10 094 3 Sables —16.2
OO ee ee Ene cokers SO Sk aioe! 341.8 10 . 029 3 044 —34.1
ap TRE Sea es eS eer ree 22183 10 044 2.5 - 083 —47.0
Op ie Te ae he 105.8 10 094 2.5 . 158 —40.5

16

TaBLE 6.— Mass transport

picasuted = =

: ranspor ‘m A

Point Depth | per period | (ft./sec.) a (tt./sec.) | Um-Us

(feet)

0. 69 0.0 0.0 3 +0. 0008 —0. 0008
0. 96 —.010 —.0118 3 —. 0126 +. 0008
1.11 —.017 —.020 3 —. 0163 —.0037
1, 25 —.021 —. 0248 3 —. 0184 —. 0064
1.50 —.019 —.0224 3 —. 0205 —.0019
0. 357 +. 053 +. 0624 3 +. 0457 -+. 0167
0.60 +. 039 +. 0459 3 +. 0089 +. 0370
0. 82 —.018 —.0212 3 —.0070 —. 0142
1.10 —.019 —. 0224 3 —.0161 —. 0063
1.31 0 0 3 —.0191 +. 0191
1.78 —.019 —. 0224 3 —.0215 —.0009
0. 47 +. 0204 +. 024 3 -+.0195 +. 0045
0. 62 +. 0076 +. 009 3 +. 0048 +. 0042
0. 91 —.0051 —.006 3 —.0090 +. 003
1.15 0 0 3 —.0137 +. 0137
1.78 —. 0136 —.016 3 —.0171 +. 0011
1.93 —.015 —.0172 3 —.0173 +. 0001
1.61 —.011 —.0130 3 —. 0168 +. 0038
1.33 —.0053 —. 0063 3 —. 0154 +. 0091
2n15 —.006 —.0071 3 —.0137 +. 0066
1.60 —.0141 —.0146 3 —.0115 —.0031
1.65 —.0130 —. 0134 3 —.0117 —.0017
1525 —.021 —.0217 3 —. 0092 —.0125
1. 44 —.008 —. 0083 3 —. 0103 +. 0020
0.99 +. 0125 +. 0129 3 —.0055 +. 0184
1.14 —.0041 —.0042 3 —.0080 +. 0038
0. 65 +. 006 +. 0062 3 +. 0055 +. 0007
0.82 +. 0016 +. 0017 3 —.0012 +. 0029
0. 180 +. 037 +. 0383 2.5 +. 0288 +. 0095
0.315 +. 0120 +. 0124 2.5 +. 0125 —.0001
0. 415 +. 0120 +. 0124 2.5 +. 0052 +. 0072
0. 560 0 0 2.5 —.0015 +. 0015
0.79 0 0 2.5 —.0071 +. 0071
1.05 —. 0050 —. 0052 2.5 —. 0097 +. 0045
1.19 —. 0099 —.0103 2.5 —.0105 +. 0002
1. 26 —.0170 —.0176 2.5 —.0107 —. 0069
1.18 +. 001 +. 001 2.5 —.0105 +. 0115
0. 95 —.003 —.003 2.5 —.0091 +. 0061
0.65 +. 024 +. 025 2.5 —.0045 +. 0295
0.69 +. 018 +.019 2.5 —. 0054 +. 0244
1.30 —.0173 —.0204 2.5 —.0215 +. 0011
1.14 —.0213 —.0252 2.5 —.0199 —.0053
1.02 —.0087 —.0103 2.5 —.0178 +.0075
0. 86 —.0053 —. 0063 2.5 —.0135 +.007.

. 82 —.0093 —.0110 2.5 —.0120 +.0010
71 —.0077 —.0091 2.5 —.0066 —.0025
56 +.0107 +. 0126 2.5 +. 0053 +.0073
60 +.0227 -+.0268 2.5 +.0120 +.0148
40 +.0170 +.0201 2.5 +.0269 —.0068
27 +.0500 +.0591 2.5 +.0565 +. 0026
16 +.053 +. 063 3 +. 0629 +.0001
68 —.017 —.0202 3 —. 0006 —.0196
86 —.0073 —.0086 3 —.0063 —.0023
77 —.0077 —.0091 3 —.0038 —.0053
91 —.006 —.0071 3 —.0070 —.0001
92 —.001 —.001 3 —.0017 +.0007
83 +.001 +.001 3 —.0004 +.0014

.70 —.0007 —.0007 3 +.0021 —.0028

. 63 —.013 —.013 3 +. 0037 —.0167

52 0 3 +.0071 —.0071
46 ++. 0027 +. 0027 3 +. 0094 —.0067
38 +. 0077 +.0077 3 +.0130 —.0053
30 +.0027 +. 0027 3 -+.0173 —.0146
17 +.023 +.0273 2.5 +. 0398 —.0125
22 +. 021 +.0249 2.5 +.0316 —.0067
30 +.017 +.0201 2.5 +.0218 —.0017
39 +. 0007 +. 0008 2.5 +.0129 —.0121
48 +. 0013 +.0015 2.5 +.0068 —.0053
68 —.0011 —.0013 2.5 —.002 +.0007
77 —.005 —.0059 2.5 —.0042 —.0017

. 78 —.015 —.0178 2.5 —.0044 —.0134

. 86 —.0076 —.0090 2.5 —.0059 —.0031

1.03 0 0 2.5 —.0079 +.0079
1.09 0 0 2.5 —.0084 +. 0084
0.15 +.029 +. 0293 2.5 +.1038 —.0745
0. 24 +. 033 +. 0334 2.5 +.0697 —.0363
0. 44 —.0057 —.0058 2.5 +.0305 —.0363
0. 55 —.0013 —.0013 2.5 —.0160 +. 0147
0. 82 —. 0063 —.0064 2.6 —.0058 —.0006

17

TABLE 6.— Mass transport—Continued

wessured 7 Oo
ny ranspor ‘m $
Point Depth | per period | (ft./sec.) d (ft/sec.) | Um—Us
(feet)
Dba Ob en oe ete coe 0. 76 —0.0020 —0.0020 2.5 —0. 0022 +0. 0002
Dba pee east ee Sok oe . 89 —.0010 —.0010 2.5 —.00 +. 0086
OPS a a a ee See 1.12 -. —.0051 2.5 —.0179 +.0128
2a 0 meen re owe Pee SS 0.115 +. 028 +. 0282 2.5 -+. 1062 —.0780
25-10 een ees eee oe eee 217 +.012 +. 0121 2.5 -+.0750 —.0629
Pees oe eS eee eee 346 +. 020 +. 0203 2.5 +. 0465 —.0262
ORS Be Se ee ee eee 428 +. 009 +. 0091 2.5 +. 0323 —.0232
alee nes oo NR Se Boa ae 460 +.014 +. 0142 2.5 +.0274 —.0132
PRICE oe SL ee ee eee eee 560 —.010 —.0101 2.5 +. 0148 —.0249
PASI Ufa os Se ee ees 596 +. 011 +.0111 2.5 +.0112 —.0001
Dam G Meee ee pas eh SE EEE 669 - 000 000 2.5 +. 0044 —.0044
Domi peer ne eee 2 766 +.005 +.051 2.5 —.0026 +. 0536
2a — lS eeene See See Se Se 870 +.012 +.0121 2.5 —.0086 ++. 0207
ORS ee ee ee 929 +. 006 +.0061 2.5 —.0113 +.0174
PAW OTA OIC IMOTON CO seas ee eer oe Dl ee cee ee Pe Se Set Poe —0.0021

Table 6 lists comparative values of mass transport at various depths
for each of the waves studied. The measured values were obtained
as described in section 4. The theoretical values are given by Stokes’
theory as:

ka?C

U=h’av’Ce*’ — re (10)

Since mass transport in a closed tank is modified by a return flow
which results in zero and negative measured values of mass transport
in the bottom section of the tank, percentage comparisons are not
feasible.

The data are presented graphically in Figures 8 and 9.

MASS TRANSPORT

=b2qzeer2kh’_ kate
U=k?atce ==

°

g g
uv u
a 4
x x=
z Le 5
a

a m
m mn
=) 4

|_| | |) MEASURED VALUES
| | THEORETICAL
aia iP

° +.05 +.10 IS . Be (o) +.05 +10
VELOCITY IN FEET PER SECCND VELOCITY IN FEET PER SECOND

=0| 1 | MEASURED VA
= THEORETICAL

FIGURE 8.

18

MASS TRANSPORT

1333 Ni Hid3za

=

BESS RR aReR AON M fel shat ohh
_| | MEASURED VAL’ MEASURED VALUES =
THEORETICAL " = THEORETICAL " =
; ; J : t
| |
+.05 +.10 +15
VELOCITY IN FEET PER SECOND

+++

|
i Jaa! Te

FIGURE 9.

6. SHALLOW-WATER WAVES

The basic data for the shallow-water waves studied are given in
table 7.

TaBLe 7.—Basic data—shallow-water waves ($<05)

1 2 3 4 5 6 7 8 9 10 il
Run Rave T'm Cm Lm d ft ze hare b./as CmT'm ce = ==
ft. sec. | ft./sec. ft. The Te ft percent
1. 220 6. 25 7. 52 3 0. 399 0.0242 | 0.987 7.62 +1.3
1. 205 6.05 6. 97 2.5 - 359 . 0221 - 980 7.29 +4.3
1. 432 6.08 8. 97 2 . 223 0177 . 886 8.71 —3.1
1. 196 5.62 6.81 2 294 . 0313 952 6. 72 +1.3
0. 987 4.97 4.89 2 - 409 . 0573 988 4.91 +0. 4
1. 436 6. 20 9.15 2 .219 0265 880 8. 90 —2.7
1. 231 5.77 7. 24 2 276 . 0340 940 7.10 -1.9
1.003 4.84 5.08 2 393 . 0419 986 4.86 —4.3
1. 200 5.78 7. 20 3 417 . 0356 989 6. 94 —3.6
1. 188 5. 56 6. 96 2.5 ~ 309 . 0333 980 6.61 —5.0
1, 232 6. 10 7.64 3 - 393 . 0406 986 7. 51 -1.3
1. 251 6. 15 7.65 2.5 327 - 0329 969 7.69 +0.5
0. 990 4.95 4.83 2 .414 - 0325 - 989 4.90 +1.4
1. 222 5.79 (En 2 . 281 . 0353 . 944 7.08 —0.6
1. 496 6. 51 9. 53 2 210 . 0245 866 9.73 +2.1
1. 496 5. 90 8.77 1.5 171 . 0192 772 8.82 +0.6
1. 468 5.77 8. 62 1.5 .174 . 0281 800 8. 47 —1.7
1.911 6. 49 12. 11 1.5 .124 . 0119 654 12. 40 +2.4
2. 080 5. 37 11.09 1.0 - 090 . 0080 . 512 11.16 +0.6
1. 506 5.09 7.86 1.0 .127 - 0228 - 665 7.66 —2.5

The ratio b,/a, column 9, is the ratio of the vertical to horizontal
amplitude of motion of the water particles in the wave surface.

19

The remarks previously made concerning the accuracy of measure-
ment are applicable also to these shallow-water wave tests.

The measured values of wave velocity and wave length, for each
wave studied, are compared to the corresponding values computed from
theory in table 8. Columns 2 and 3 list the wave velocity and wave
length as computed by the Laplace-Airy (trochoidal) theory, where:

—oo senilh = (11)

and:
L tae (OF I; m

T,, is the measured period which is assumed to have been correctly
determined. Columns 4 and 5 list the percentage differences between
measured and theoretical values. It will be noted that the agreement
between the measured values and the theoretical values derived from
the Laplace-Airy formulae is not as close as for the deep-water
waves, varying as much as about 6.5 percent.

TaBLe 8.—Shallow-water waves G <0.5 )—comparison of measured to theoretical
velocity and length

1 2 | 3 4 5 6 w 8 9

Gerstner Cm—C: | Lm—Lt Stokes-Struik Cm—C’: | Lm—L's

Run Pee EL Se REE Sop jer 8 |e enn L's

Ct In Cc’, L’;
ft./sec. ft. percent | percent | ft./sec. ft. percent | percent

6. 16 7. 53 +1.5 —0.1 6.19 7. 56 +1.0 —0.5
6. 04 7. 29 +0.6 —4.1 6. 06 7.31 —0.2 —4.7
6. 49 9. 30 —6.3 —3.5 6. 51 9. 34 —6.6 —4.0
5. 81 6. 96 —3.3 —2.2 5. 87 7. 04 —4.3 —3.3
4. 97 4.93 0 —0.8 5.14 5.10 —3.3 —4.1
6. 45 9. 29 —3.9 —1.5 6. 51 9. 38 —4.8 —2.5
5.91 7.30 =—2.4 —0.8 5. 99 7.40 —3.7 —2.2
5.05 5. 09 —3.8 —0.2 5.13 5.17 —5.7 —1.7
6. 06 7. 30 —4.6 —1.4 6. 14 7.39 —5.9 —2.6
5.95 7.09 —6.5 —1.8 6. 00 7.15 —7.3 —2.7
6.19 7. 56 —1.4 +1.0 6. 30 7.79 —3.2 —1.9
6.19 7.77 —0.6 —1.5 6. 25 7. 84 —1.6 —2.4
5. 01 4.97 —1.2 —2.8 5.06 5. 02 —2.2 —3.8
5. 94 7. 28 —2.5 —2.2 5.97 7. 32 —3.0 =2:7
6.61 gon —1.5 —3.8 6.68 | 10.02 —2.5 4.9
6. 05 9. 06 —2.5 —3.2 6. 10 9. 14 —3.3 —3.0
5. 98 8.81 —3.5 —2.1 6.09 8.97 —5.3 —3.9
6.39 | 12.22 +1.5 —0.9 6.41 | 12.27 +1.2 —1.3
6.44 | 11.35 —1.3 —2.3 6.50 | 11.46 —2.4 —3.2
6.11 7.71 —0.4 +2.0 5. 10 7.73 —0.2 +1.7
BEE ae ee eee ee Pe SS —2.1 fa ee ae) Poe re —3.2 —2.7

The theoretical values for wave velocity and wave length computed
by the formulae of the Stokes-Struik theory, where

2nd 2rd 8ad 8rd 4nd 4nd
pee Cl ie tamen Ls elte L {8 see T) +12 522
C eioma riper dict a anata Oe ABE | 2)
ppt i aah
and: JOY sO" Il,

are given in columns 6 and 7.

20

TABLE 9.—Percentage of wave height above still water level, shallow-water waves
(2<05)

Wave height | _ Percent

ses Average : wave height
Run Position 4 wave height aboversty above still
water level
1 pe ee ee eee ee eee a 1 0. 0250 0. 188 0.101 53.7
OS a eee ae ee ee 2 0246 - 185 . 098 53.0
pees 5 ee een eee eee 3 0217 . 163 088 54.0
(ee ee ee en tee PS 2 1 0229 . 160 086 53.7
(RAS ee ee eae ee ee 2 0221 . 154 082 53. 2
Geese ea oe eee eee 3 - 0221 . 154 076 49.4
seo Se ee ASSES Se eB NBS FOS 1 0191 -171 088 51.5
1 Oe eee ee es Ee eee ee eA 2 0175 . 157 083 52.9
( (SS OES ess 3 Saree Oey ee ae ae 3 0167 . 150 081 54.0
Gees ates 8 ae Be ee ee ee 1 0331 . 225 118 52.4
8 eae 2 03808 .210 113 53.8
8 = 3 0301 . 205 118 57.7
9 33 1 0593 . 290 . 169 58.3
9 Bt 2 0568 . 278 165 59.3
9 == 3 0556 . 272 157 57.8
10_ ee 1 0275 RY ty} 131 52.0
10_ ne 2 0262 . 240 125 52.1
10_ ae 3 0258 . 236 .129 54.7
ib wel 1 0363 . 263 142 54.0
Pi 2a 2 0334 - 242 136 56. 1
Te = 3 0320 . 232 - 132 56.9
ibe = 1 0431 . 219 - 108 49.3
12. 5 2 0439 . 223 124 55.6
12_ Sa 3 0392 . 199 102 51.2
byes ae ee ses a = 1 0368 . 265 131 49.4
1s SPE a SP as os ag ek yg ret Er! 2 0357 . 257 - 132 61.3
1 Ue eee Ee ee eee ee 3 0342 . 246 BP ( 51.7
1A eR ES She Eee ee. Beas See oe ee it 0358 . 249 125 50. 2
NG See oe ee ee ee 2 0332 .2ol 123 53. 2
Ge ee ae 3 0339 . 236 144 51.2
Ea ee ee ee Ae et 1 0412 .315 170 53.9
Pee So OS Be Se ae oe et 2 0397 . 303 155 51.2
3 Na eater? Mireceine |, Senald ae Se ee ee 3 0406 . 310 168 54. 2
0 ee a ee Se ee eee 1 0329 - 202 137 54.4
i, [os ge ea Sok rk ee St Pe Pe Se ee ea 2 0328 . 251 131 52.2
Oa es fla ne ee 2 Se ee Sk 3 0328 . 251 144 57.4
7) eRe ee en ee a ee 1 0338 . 163 090 55. 2
ee ee eee Sere eee 2 0331 - 160 084 52.5
it ears ee Se ee 3 0313 151 078 51.7
26h en oo 3 eh Be eS a A 1 0367 . 261 149 56.9
OG kta. 2 ee ee ee 2 0355 . 253 134 53.0
Lee ee ee EN ES Ses ee YS eee ee 3 0337 . 240 133 55.4
Deen a a SE SE 1 0237 . 226 133 58.8
De Se ee ee 2 0236 225 124 55.1
Be ee ee a 3 0255 . 243 132 64.3
43 a ee ee eee oe <3 Se tees 1 0191 . 167 084 50.3
og ee es eee 2 Ne 2 0173 . 152 076 50.0
Woe ccc sah She 2 kN co oss 3 0226 . 198 093 47.0
DOE Se Lae Se Ree OM oe Ed 1 0315 . 272 151 55.5
31 lige Pea Sas, SR OT Seip pot 3 Dew Se ee 2 0289 . 249 124 49.8
OU ae SIS aed oS eee ee 2 3 0238 . 205 110 53.6
DOS Ses 2 ee Ns 2 eR 1 0101 . 123 062 50.4
eee eS Se PE) ee eee 2 0115 . 139 088 63.3
Reece 5) Se ae ee ee ee 3 0147 .178 110 61.8
Le 8 ee ae os ee A bh ee 1 0081 . 090 059 65.5
Le earn et a Be Ee a eee 2 ee 2 0073 . 081 052 64.2
eee eee a eae ee 3 0084 . 093 060 64.5
DoE Se Mg on TS at ee 1 0262 . 206 111 53.9
an eat ee eed Se 2 0214 - 168 . 085 50.6
7 ee ae ae Sen pa ee 3 0209 164 .101 61.6

Again comparing measured and theoretical values as above, it is
found that the agreement is on the whole good though individual
comparisons show differences amounting to about 4.5 percent.

The measured values for percentage of wave height above still
water level are tabulated in table 9. The results are presented
graphically in Figure 10. The theoretical values are derived from the

21

Laplace-Airy formula, similar to that of the Gerstner theory (Kq. 4),
and the Stokes-Struik formula:

4nd 4nd 4nd 4nd
40.544 baa) at alts: a i) (13)
eure E

With few exceptions the measured percentage of wave height above
still water level is several percent higher than that indicated by either
the Laplace-Airy or Stokes-Struik theories; agreeing most nearly with
the former theory.

brates! THEORETICAL LAPLACE- AIRY STOKES -STRUIK
ll AIRY |STOKES *. s f

a

ul a
x «
= p=)
< 2
w w
= =

50 60 50 60
THEORETICAL THEORETICAL

PERCENTAGE WAVE HEIGHT ABOVE STILL WATER LEVEL
SHALLOW-WATER WAVES

Ficure 10.

Some of the measured wave profiles are shown in nondimensional
form on Figures 11 and 12. Plotted on the same graphs are the wave
forms given by the Laplace-Airy theory:

X,—R6—x sin 0
Z,=2 COs 0 (14)

and by the Stokes-Struik theory:

2rd. 2nd 4rd 4nd
a
Z,=«a cos kX,— (Gated Ga ene cos 2k.X,+ (15)
el~e L ce

ved — Und See isd tea —‘s
eb en

TTY
wal

LAPLACE - AIRY
—-— STOKES- STRUIK
---- OBSERVED

fi

22

~--,

eS
eae) a = ee Pe

Fieure 11.

WAVE PROFILES
SHALLOW-WATER WAVES

h 3
Where a=s— sa (e*)

z 0.38
LAPLACE- AIRY WAVE PROFILES

—-— STOKES- STRUIK P, 2
SS Gai SHALLOW-WATER WAVES

Figure 12.

The relation between the measured and theoretical average semi-
axes of the paths described by individual water particles are given in
table 10. The theoretical values of the orbit radii are the same for
the Laplace-Airy and Stokes-Struik theories, their values being
given by

d ee cosh *(d—h’)
horizontal semi-axis: =z,=a——3,7—_ (16)
sinh
? Rete be sinh= (d—h’)
vertical semi-axis: = 2,=a———,,7—
sinh

where x, and z,—theoretical orbit sem-iaxis at depth h’ below still
water. The positive sign indicates that the measured value is larger
than the theoretical. Some of the representative data are presented
graphically in Figure 13.

24

d
TABLE 10.—Average orbit semi-azes, shallow-water waves ( #<0.5)

Average Average
d
Run ‘Z | Horizontal | Vertical sok Z | Horizontal | Vertical
Xm—X: Zm—Zt Xm—Xt Zm—Ze
Xt Zt Xt Zt
6} See eases 0. 090 —10.7 Ss dit fl Pb Se 0. 327 —1.1 —3.4
hi ee ees . 124 —8.7 = 6287 | |" 6x sees zee sess 359 —10.3 —34.4
32. meee eee wee ee . 127 —3. 3 +13. 5 16 { . 359 —1. 7 +3. V6
7. es meee -171 —16.4 Cy fc Te | tea tat nr . 359 —3.2 +8.9
pak eee Somer .174 —0.6 Cant el | fe) eee Ses, Se . 393 —7.6 —1.9
fee Be - 210 +1.0 re eet |e) a . 393 —6.3 —5.6
1 (ee Sees - 219 +17 Se) Ha al | Hat ee - 409 —3.8 —5.1
| Poe ee ae - 223 176 = 25.6) || 202225222 oe .414 +2.7 +1.8
ib ee Seer - 276 —Te2 SOR hbase eee .417 +1.3 —0.6
(iene eee . 281 +2.0 +1.7
m==measured horizontal semi-axis
X.=theoretical horizontal semi-axis.
7m=Ineasured vertical semi-axis.
Z,.=theoretical vertical semi-axis.
d
TaBLeE 11.—Mass transport, shallow-water waves L< 0.5
Point Depth feet | dfeet | Urft./sec. | Um ft./sec. | Um-Us
dae ee Sil ete Bee
7 a eee se Saas eee ape er EEE EE SCE 0.016 3 +0. 077 +0. 040 —0.037
Pe eae ae ee - 138 |- = +. 062 —.002 —.064
Be a eR see ee a ee - 106 +. 064 +. 002 —.062
a a . 278 +. 045 +.003 —.042
a - 408 +. 031 +. 009 —.022
Ue Bee eee ae eee Sas sete Ree eases - 689 +. 012 —.015 —.027
Wan nnn nn enna nnn nnn nnn n= - 495 +. 023 +. 002 —.621
(ee Se ee ee ee . 672 +. 013 . 000 — 019
a ee - 876 +. 003 —.018 —,.021
eee eae ee ee erence 1.089 —.004 +. 008 +.012
ee En eee = —.006 +. 012 +.018
EN ee 0. 190 2.5 +. 033 +. 005 —.028
, UW) eee eae +. 035 +. 022 —.013
8. 2-2-2 -2---- 2 -- == === == === === === === = TGs | acess +. 029 +.013 —.016
ee eo ee seo 2 po soceed ceeceeeseesaces Nl |e res +.018 —.014 —.032
[oe so eee (iM | Bee Spee +. 008 -+.002 —.006
Goan oc eee deere soca secs a seceses ASHi ee ean esas +. 014 +. 006 —.008
Uf Bae = ee re ee Si Oa | eee eel +. 001 —.008 —.009
Se a ee Pa 20 eae —. 006 +.007 +.001
Dosen ctenssee dtee seco sacen es eeeaases 2A a ee —. 004 +.014 +.018
7: one Savane Sele ee anon Oe ee Sm oes 0. 146 2.0 +. 027 +..009 —.018
He poe cee a oeeinc eines Sea eee tec Sssesseeseess= 74435 eee +.018 +.009 —.009
Be 32 ee eee Se ae ae ae eee i ee +.013 —.006 —.019
ee CE eee ee +. 007 +.014 +.007
Got ssa se ee en oe eee A386) 22-5358 +.009 —.003 —.012
Use ea RES eee See es Se Senses OS5it22-5-sesee +. 002 . 000 —.002
Wea 18) | peceecosae —.001 +.004 +.005
eee 1 O18} Se =e —.004 +.003 +. 007
a a a ee 0.035 2.0 +. 049 —.005 —.054
Fe Ree ae Een ae Rn Se OST ECS ERS PON | een +. 029 —.011 —.040
be ee Se ee ee CC pea ee +.015 - 000 —.015
Choc Sono e eons eee ee a ee Seema 4 +. 006 —.002 —.008
Dewon ens scones a reece es ea ea Serer Seses aes tii) Eee —.002 +.002 +. 004
(ee ee aie eS ee S25) Soce ae —.004 +. 007 +.011
] See ee ee oe 660) |2- 52522224 +. 007 —.005 —.012
Deedes Gora e ee eeen een eon eee eee LI 1S eee —.012 +.002 +.014
MH ese sereossess seat asos scceesteesassess 0.030 2.0 +. 026 +.003 —.023
, ye) +. 022 +.012 —.010
Oo ee ea a ae oe aan amen 1 a eee +.018 +.017 —.001
Se Stee ae nas aoe ee eee eee aS S19) eeaacoes +. 013 +. 031 +. 018
Davo nssstaseselel se sas sananecenaoSsessssns 5 ay a eee eae +.012 . 000 —.012
Gb so 326s sso a as ee ese eee B74; ee Soe +. 008 . 000 —. 008
( ee nen ee eee ee bO8 |-=----2528 +. 005 —.004 —.009
Steet so s5tth ee eee eee. il eee —.002 +. 005 +. 007
ee HG YES eee ee —.006 +. 008 +.014
Ue a ees ae See aa Sean 1,208 [oe —. 005 —.015 —.010

205

TABLE 11.—Mass transport, shallow-water waves G < 0.5 )—Continued

Un- U:

Point Depth feet | d feet U; ft./sec. | Um ft./sec. ft/sec

The comparison of measured and theoretical values of mass trans-
port, at various depths below the water surface, is shown for several
representative waves in table 11. Asin the case of deep-water waves,
the trochoidal theory does not admit the existence of mass transport;
the theoretical value a by the Stokes-Struik theory is:

U,=Ka?0* 2nd 7 Sven Mn Daca, (17)
- oe = ee tanh-7

The data are presented graphically on Figure 14.

26

SHALLOW WATER WAVES

7)
ul
x
<
we
=
uJ
7)
=
a
o
fe)

FicureE 13.

27

15
-0.04 fe) +0.05 +010
FEET PER SECOND

RUN 3}
ie}

to) +0.05
FEET PER SECOND

Lo
3.0 -0.04 ° +0.05 +0.10
-0.04 C) +005 +010 ; FEET PER SECOND

FEET PER SECOND

RUN 24 RUN 32
° 0

2.0 “0, +0.05 +0.10
=0.04 ° +005 FEET PER SECOND
FEET PER SECOND

RUN 28
i)

Bk © WW

25
- 0.04 i) +0.05
FEET PER SECOND

RUN 25
1°)
te) +0.05
FEET PER SECOND
RUN 29

5
+0.05 -0.04 1°) +0.05
FEET PER SECOND FEET PER SECOND

MASS TRANSPORT
SHALLOW WATER WAVES

Figure 14.

28
7. DISCUSSION

Previous to any discussion of the results obtained in this study
certain remarks should be made concerning their limitations.

Primarily, it should be noted that the generated wave requires a
certain distance of travel after its generation to reach a stable form.
It was noted during the course of the experiments that the measured
values of wave velocity, wave length, and wave height, changed with
time and position in the tank. For example, an average decrease in
wave height of 0.011 foot was noted over a space of 12 feet immediately
preceding the center observation window of the wave tank; while over
the succeeding 13 feet the average decrease was only of the order of
0.001 foot. Further observations indicated that at a distance of 2
feet from the plunger, the shape of the particle orbits was essentially
as indicated by theory, but that the forward travel of the water
particles was small and practically constant from surface to bottom.

In the generation of the wave it seems reasonable to assume that
since the water is not displaced according to the laws governing the
particle motions then a certain number of cycles, or oscillations, of
the water surface will be required after displacement of the water by
the generating plunger, before the water particles take on the more or
less stable movement indicated by both theory and observation. The
variation of wave height should be a sufficient criterion as to the
attainment of a stable form by the generated wave. Observations of
many waves indicate that in the present wave tank and with the
present method of generation, a distance equivalent to about 6 to
10 wave lengths is required for the assumption of a stable form by the
wave. These experiments were by no means definitive and the
results cited must be regarded as merely indicative of a condition,
rather than as quantitative values accurately defining the condition.

The distance from the wave generator to the section of the wave
tank selected for making the observations for this study is about 24
feet. It follows from the above discussion that insofar as this effect
is concerned the most accurate results are those for waves whose
lengths were in the neighborhood of 4 feet or less.

It is believed that the magnitude of the error resulting from observa-
tion of the wave characteristics before a stable form is attained is, in
general, small. However, for the determination of mass transport,
which is a secondary effect, the error introduced by this condition may
be appreciable. The quantitative determinations of mass transport
reported herein therefore may not be the true magnitudes of the
effect.

<a : : ae
The ratio az has been adopted in wave experimentation as a criterion

to differentiate deep- and shallow-water waves. The reasoning lead-

29

ing to the adoption is based on the fact that above a certain value
(about 0.5) of the ratio, the value of the term tanh =a appearing in
the Gerstner and Laplace-Airy equations of wave motion approaches
very closely to unity. It should, however, be noted that the dis-
tinction of deep- or shallow-water waves made on this basis is arbitrary
and, as far as the Stokes’ theories are concerned, may have practically
no significance. In fact, Stokes defines the deep-water wave as one
for which “the depth of the fluid is very great compared with the
length of a wave’’, when we may without sensible error suppose the
depth to be infinite.

In reality, as an examination of the formulae in Section 6 will show,

the ratio 7 is effective as a parameter in Stokes’ theory until large

values of the ratio are reached. If the comparison of shallow-water
wave theory with experimental results shows Stokes’ theory to be
most nearly applicable to the waves studied, it would appear, there-
fore, that the present arbitrary distinction between deep- and shallow-
water waves should be revised to be more in accord with the principles
on which Stokes makes the distinction.

Considering first the results of section 5, the comparison of wave
lengths and wave velocities (all comparisons are understood to be
between measured and theoretical values, unless specifically stated
otherwise) indicates agreement of experiment with both of the theories
studied within the limits of experimental error, the closer agreement
being with the Gerstner theory.

The comparison of the proportions of wave height above still water
level indicates agreement with both theories within the limits of the

experimental error, for values of E less than about 0.05. Above this

value the measured percentages tend to be higher than those predicted
by either theory.

It should be noted in this connection, and also with respect to the
wave profiles, that the experimental values are susceptible to con-
siderable error by reason of a phenomenon of reflection from the wave
absorber. The first waves generated from water at rest had smooth,
flowing profiles. As soon as these first waves reached the absorber it
was noticed that small, short waves appeared to be superimposed on
the generated waves, until after a short time the initial smooth wave
surface became a surface of more or less irregular outline. This
condition is shown clearly in figures 3 to 5. Furthermore the fact that
the generated waves had not attained a stable form before measure-
ment probably had an appreciable effect on the shape measurements.

With respect to the wave profiles there is little evidence to show
agreement with either of the theories investigated.

30

As noted previously, Gerstner’s theory does not provide for mass
transport of the water in the direction of wave travel, the paths of the
water particles being, in the case of deep water, circles. According to
the Stokes-Levi-Civita theory the paths are open; but are basically
circles modified by the effect of mass transport. The magnitude of
the motion is the same in either case, except as modified by mass
transport. The results listed in tables 5 to 7 show that the curves are
not closed, but open, indicating that mass transport does exist. The
measured values of the orbit radii are generally in fair agreement with
theory, with a tendency for the experimental values to be smaller than
the theoretical being shown.

The results of the measurements of mass transport are not conclusive
except insofar as they confirm the existence of mass transport. Quan-
titative verification of the theory could not be made with the equip-
ment now available. The importance of mass transport in natural
beach processes should not be underestimated. It is, for example, a
possible explanation of rip tides, since on any extended ocean coast
there may exist a localized seaward current; 1. e. a rip tide, in com-
pensation for the shoreward current of mass transport.

For the case of shallow-water waves (section 6) the comparison of
wave length and wave velocity indicates agreement within the limits
of experimental error with both of the theories considered, the closer
agreement being with the Stokes-Struik theory.

The comparison of proportion of wave height above still water level
is inconclusive, verification of either theory within the limits of experi-
mental error being indicated.

The wave profiles show noticeably better agreement with theory
than do the deep-water wave results. Within the limits of error
agreement with either theory is indicated.

The comparison of orbit semi-axis values shows, as for the deep-
water waves, a tendency toward smaller measured values than are
indicated by theory. However, the existence of mass transport is
clearly shown. The magnitude of the mass transport effect could
not be quantitatively verified.

8. CONCLUSIONS

1. All of the characteristics of the oscillatory wave in deep or shallow
water required by the irrotational wave theories have been reproduced
in the wave tank, with the exception of the wave profile.

2. It has been shown that mass transport does exist, as indicated
by the theories of Stokes, but the theoretical magnitudes have not
been quantitatively verified.

3. Since mass transport has been shown to exist, it follows that the
Gerstner and Laplace-Airy wave theories, which do not admit its
existence, are not applicable to the waves studied.

dl

4. The magnitude of the difference in value of the wave character-
istics, other than mass transport, defined by the rotational or irrota-
tional theories is smaller than the experimental error associated with
the tests described herein.

5. Certain experience in wave experimentation has been gained by
this essentially exploratory study which leads to the following general
statements.

a. Prior to further experimentation for the purpose of verifying
wave theory, studies should be made of the generation of irrotational
waves by mechanical means, and of the necessary length of wave
travel after generation required for the wave to achieve a stable, or
permanent, form.

b. The techniques of wave measurement in the laboratory should
be improved, a maximum error in experimentation of 0.25 percent
being desirable when comparisons similar to those of this study are
to be made.

c. Provision should be made in future experiments to determine
directly the existence or non-existence of rotation in the wave motion
by study of the particle motion.

AppENDIx I

PREVIOUS INVESTIGATIONS RELATIVE TO THE
VERIFICATION OF WAVE THEORY

This appendix lists and discusses briefly the available data regarding
the verification of wave characteristics predicted by theory. The
complete references noted are listed in appendix II.

This study does not include wave generation, pressures, breaking,
or movement of materials, hence these subjects are not discussed.

Primary elements —(Velocity, length, period)—Gaillard (1904)
found the measured velocity to be 3 percent greater for 85 observa-
tions than the Gerstner theoretical velocity for deep water. In
shallow water he found the measured velocity to average 0.2 percent
greater than the Laplace-Airy theoretical velocity for 84 observations
at St. Augustine, Fla., and 631 observations on Lake Superior.

Thorade (1931) lists the following table summarizing a large number
of comparisons of measured velocity with velocity computed from
length and period by Gerstner’s equations for a large number of
observations.

Cc Per- Cc Per-

Author Year Ocean Gee leant) Cs lect Remarks
WARIS se) So 22 sso tsa s8 1871 | All oceans_______ 96 | +2 96 | +3] 5 groups, totaling about
4,000 observations.
devBenazess2-= = eee 1874 | Atlantic___-.._-- UDO sea 1A ee eat | (Se 25 groups, swell.
{ 109 SR ee | eee 14 groups, wind waves.
H. M.S. ‘‘Gazelle’’_._| 1874-76 | All oceans___---- 100 | +2/)] 101] +5 | 4 groups, violent winds.
Abercromby--_-_------ 1888) || seacities = =-s-=—— 106 +3 114 | +7 | 6 groups, violent winds.
Schottes: Stas 1893 | Atlantic and |f 99] +2 99 | +3 | 7 groups, wind waves.
Indian. \ 104} +1] 109} +8 8 groups, swells.
Gassenmayr-____------ 1896 | Atlantic___---__-- { 90; +2 83 +4 | 20 groups, wind waves.
99 +7 96 +3 | 8 groups, swells.
PAV CLAS CS. oan |e ee eee heer es ee oe 100 |} +1 100 | +2

In 1929, at Long Branch and Seaside Heights, N. J., the Beach
Erosion Board found that for 206 observations in shallow water the
measured velocity averaged 1.5 percent greater than the Laplace-
Airy theoretical velocity.

A large number of studies have indicated that oscillatory waves
generated in tanks follow the theoretical relationships for velocity,
length, and period. DeCaligny (1843) and Hagen (1861) were
pioneers in this field. Larras, Stucky and Bonnard, Bagnold and
others have used this relationship as a check in connection with

(32)

33

their experiments. Mitchim (1939) found that for 28 deep-water
runs his measured velocities were 1 percent too large according to
Gerstner’s equation and 0.3 percent too small for the third approxi-
mation of Rayleigh.

Wave profile—Wave profiles have been studied both in nature and
in the wave tank. Gaillard’s work contains four profiles photographed
in the Duluth Canal and compared to the theoretical shapes. Cornish
noted trochoid-like shapes in his observations. Kohlschutter and
Schumacher on the ‘‘Meteor” investigations (1928) and Weinblum,
Schnadel, and Block on the ‘‘San Francisco”’ expedition photographed
waves corresponding closely to trochoids.

In wave tanks, Meyer-Peter, Larras, Waters, and others, have
checked their wave shapes against the theory with good agreement.
R. D. Meyer noted that in shallow water the wave front was steeper
and the back slope flatter than the trochoid.

Dimensionless ratios —The height-length ratio of waves has re-
ceived considerable attention. Paris found that the ratio in deep

L
water =~=39 corresponded to a light sea, 21 to a rough sea, and 19 to

h
a heavy sea. Schott’s observations verified these findings. Gaillard

L
found that the i ratio varied from 9.1 to 15.0 for 235 observations in

shallow water; Cornish observed ratios as low as 13 during storms.

Larras used the length-height ratio as a check on his work on
vertical jetties, keeping the ratio between 20 and 11 for shallow-water
conditions. Mitchim noted this ratio in applying the third approxi-
mation of Rayleigh to his results, using the limits of 12 and 26 for his
studies in deep water.

The depth-length ratio has been generally accepted as the criterion
of deep- and shallow-water waves. Almost all investigators have used
this check in determining the applicable formula.

The percentage of wave height above still water level is of particular
significance in studies of wave pressure against structures. Gaillard
made a total of 834 observations at St. Augustine and Duluth (1890-—
1902), and the Beach Erosion Board made 365 observations at Long
Branch, N. J., in 1929-30.

In laboratories, the following studies are known to this office:
Larras, France, 1937; K. C. Reynolds, Massachusetts Institute of
Technology, 1937; F. K. Klauck, Massachusetts Institute of Tech-
nology, 1937; C. H. Waters, University of California, 1938; and the
Beach Erosion Board, 1939, made observations on the percentage of
wave height above still water level in conjunction with other studies.
All these observations, both in nature and in the wave tank, made
in shallow water under a variety of bottom conditions, showed close
general agreement with the Laplace-Airy theory.

34

Orbit radii.—Aimé (1839) observed upright ellipses off the coast of
Algeria. De Caligny (1843) and Hagen (1861) found that water
particles in shallow water traveled in paths resembling but distinctly
different from the ordinary ellipse.

Mitchim at the University of California in 1939 made 114 observa-
tions of deep-water waves and found an average difference of 8.2
percent between laboratory wave and theory.

Mass transport—Mitchim in his 1939 study found a qualitative
verification of the theory of mass transport for 142 observations in
the University of California laboratory wave tank.

APPENDIX II

BIBLIOGRAPHY

Abercromby, Ralph—Observation on the Height, Length, and Velocity of Ocean
Waves, Philosophical Mag. XXV, 1888.

Aimé—On Wave Motion. (Sur le mouvement des vagues.) C. R. IX, Paris.
1839.

Recherches expérimentales sur le mouvement des vagues. Ann. de Phys. et
de Chim., Paris (31 V. P. 417-27). 1840.

Airy, G. B.—On Tides and Waves. Encyclopaedia Metropolitana. 1848.

Bagnold, Major R. A.—Interim Report on Wave Pressure Research. Journal of
the Institution of Civil Engineers, No. 7, 1938-9, June 1939, London.

Benazé, Pére de—Study of Rolling in an agitated sea. (Etude du Roulis sur mer
agitée.) Brest, 1871.

Theory of Waves. (Théorie dela Houle.) Revue Maritime et Coloniale, 1874.

Block, W.—See Weinblum, G.

Bonnard, D.—See Stucky, A.

Caligny, Anatole F. H. de—Experiments on the Free Wave. (Expériences sur
la génération des ondes liquides dites courantes.) Comptes Rendus LII,
Paris. 1861.

Various Experiments on Waves at Sea and in Canals. (Expériences diverses
sur les ondes en mer et dans les canaux, etc.) Liouv. Jour. de Math. pures et
appliquées, 2, XI, Paris. 1866.

Experiments on the motion of fluid molecules in free waves, in regard to the
method of action in the advance of ships. (Expériences sur les mouvements
des molecules liquides des ondes courantes, considérées dans leur mode d’action
sur la Marche des navires.) Comptes Rendus, LXXXVII. 1878.

Theoretical and experimental researches on the oscillations of water, etc.
(Recherches théoriques et expérimentales sur les oscillations de l’eau et les
machines hydrauliques & collonnes liquides oscillantes.) Versailles, 1883.

Experiments designed to reconcile the theories on the internal movements of
waves in open or closed curves. (Expériences ayant pour but de concilier les
hypothéses sur les mouvements intérieurs des flots dans les courbes ouvertes
et dans les courbes fermées.) Comptes Rendus XVI, Paris, p. 381-387. See
also de Caligny, CR LII, 1861, 1809-1311, LXX XVII, 1878, p. 1019-1023.
LXXXVIII, 1879, p. 67-70, and Liouv. Journ. Math. XIII, Paris, 1848, p.
91-110, 2 Series XI, Paris, 1866, p. 225-266. :

Experiments on a new wave in liquids. (Expériences sur une nouvelle espéce
d’ondes liquides.) Liouv. Jour. Math. XIII, Paris. 1848.

Cauchy, A. L.—Theory of the propagation of waves on the surface of a heavy
fluid of infinite depth. (Théorie de la propagation des ondes 4 la surface d’un
fluid pésant d’une profondeur indéfinie.) Mem. pres. p. div. Savants 4 l’Ac.
Roy. des Sc. de l’Inst. de France. Sc. Math. et Phys. I, Paris, 1827.

(35)

36

Chatley, Herbert—The Breaking of Waves against Vertical Sea Walls. Article
by Herbert Chatley, Researches of J. Larras. The Dock and Harbour Author-
ity, May, 1938.

Cornish, Vaughan—On the Dimensions of Deep Sea Waves, and their Relations
to Meteorological and Geographical Conditions. Geographical Jour., London,
1904, XIII.

Waves of the Sea and other Water Waves, T. Fisher Unwin, London, 1910.
Ocean Waves and Kindred Physical Phenomena, Cambridge, 1934.

Favre, H.—The Problem of Waves. (Le Probléme des Vagues.) Schweizerische
Bauzeitung, v. 108, No. 1 and 2, July 4, 1936. Mathematical theory of waves
and wave action. Laws of movement of water waves of great depth.

Gaillard, D. D.—Wave Action in Relation to Engineering Structures, Corps of
Engineers, U.S. Army, Prof. Paper No. 31, Washington, 1904. Reprinted by
the Engineer School in 1935.

Gassenmayr, O.— Wave measurement in the Atlantic Ocean. (Wellenmessungen
im Atlantischen Ozean.) Mitt. a. d. Geb. Seewesens, XXIV, Pola. 1896.

Gerstner, Franz V.—Theory of Waves, ete. (Theorie der Wellen) Abh. Kgl.
Bohm. Ges. Wiss. Prag, 1802; also Gilberts Annalen der Physik XXXII, 1809,
p. 412. Reprint in W. Webers Werken, Bd. V. Berlin, 18938, p. 249 ff.)

Hagen, Gotthilf H. L.—Handbook on Marine Construction. (Handbuch der
Wasserbaukunst.) Part 3, Das Meer. I, Berlin, 1863.

Theory of Ocean Waves. (Theorie der Meereswellen.) Pogg. Ann. d. Phys.
CVII.

On Waves in Water of Constant Depth. (Uber Wellen auf Gewassern von
gleichmassiger Tiefe.) Abh. K. Ak. Wiss. Berlin, 1862, Math. Abh.

Hsu, R. 8. and Reynolds, K. C.—An experimental investigation of the effects
of the approaching sea depth and the dimensions of the rock mound foundation
on the wave action on vertical breakwaters. Mass. Inst. of Tech., 1939.

Klauck, Frederick R.—Effect of Shape of Sea Wall on Beach Erosion. Thesis,
Mass. Inst. of Tech., 1938.

Kohlschutter, E——Wave and coastal photographs. (Wellen- und Kustenaufnah-
men Research expedition SMS PLANET, 1906-07, III, Ozeanographie, Berlin,
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od

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See also Hsu, R. S.

38

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39

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