_. .~. unJ.

DEPARTMENT OF THE ARMY ';0, .. CORPS OF ENGINEER .. l ,,= .

BEACH EROSION BOARD

OFFICE OF THE CHIEF OF ENGINEERS II

ORBIT AL VELOCITY ASSOCIA TED WITH WAVE ACT ION

NEA R THE BREAKER ZONE

TECHNICAL MEMORANDUM NO. 79

ORBITAL VELOCITY ASSOC IATED WITH WAVE AC TIO N

NEAR THE BREAKER ZONE

TECHNICAL MEMORANDUM NO. 79

BEACH EROSION BOARD

CORPS OF ENGINEERS

MARCH 1956

---------

POREWORD

One of the facts of water wave mechani cs about which very lit tle is known is the water part icle motion in shal low water . Yet this orbita l part icle motion must account to a lar ge degree for the maintaining (if not the i nitiation) of material in suspension in the nearshore zone. It probably also account s for a certain amount of the lit toral movement , particular ly in an onshore-offshore direction, and, by differential velocities in the two directions, the size sorting of sand in the near shore area. This repor t presents the result s of a field study of the maximum orbital velocities observed under wave action of vary-ing character istics and attempts to relate them to certain theoretical wave parameter s.

This repor t was prepared at the Scripps Institution of Oceanography of the University of California in pursuance of contract DA-49-055-eng-3 with t he Beach Erosion Board which provides, in par t, for the study of wave motion in shallow water. The au thor s of the report. Douglas L. Inman and Noriyuki Nasu are member s of the staff of t hat institution.

Views and conclusions stated in the report are not necessarily tho se of the Beach Erosion Board.

'lnis stu 1 ".. • . \\.

TABLE OP roNTENTS

List of Figures & Tables ----------------------------------

List of Symbols -------------------------------------------

Abstract --------------------------------------------------

Introduction ----------------------------------------------

Method ----------------------------------------------------

Orbital Ve loc ity Meter --------------------------------Cal ibrat ion --------- --------------------------- -----Errors in Measuremen t of Orbi t al Velocity ------------Electronics -------------------------------~---------

Wave Measurements -------------------------------------Wave Pe r iod -----------------------------------------Wave He ight -----------------------------------------Depth of Water --------------------------------------

Ideal Waves --------------------------------- --------------

Airy Waves --------------------------------------------St okes' Waves -----------------------------------------Solitary Wave Theory ----------------------------------

Analysis of Data ------------------------------------------

Wave Height and Period - -------------------------------Orbital Velocity --------------------------------------

Distribut ion of Maximum Onshore and Off shore Ve loci ties ----------------------------------------

Compar ison of Measured and Ideal Velocit ies ---------Conclusions -------------------------------------------

Acknowledgments -------------------------------------------

Literature Cited ------- -----------------------------------APPENDICES

Appendix lA-IF Tabulation of Wave and Orbital Veloci ty Measurement s

Appendix I I Physical Characteristics of the Orbita l Veloc ity Meter

a

Pa.ge

b

c & d

1

1

2

6 6 9

10 10 10 11 11

13

15 15 16

18

27 28

28 33 41

41

42

Figure

1.

2.

3-5.

6.

7-8.

9-10.

11-16.

17-24.

25-26.

27.

28-33.

34.

35-40.

Table

---- -- --------

LIST OF FIGURES

Notation and definition of symbol s ---------~------

Schematic diagram ---------------------------------Handling and operation of orbital velocity meter

Calibration curve for the orbital velocity meter

Calibration of the wave pressure record -----------

Determination of still water depth ----------------

Samples records of wave pressure and orbital

velocity ---------------------------------------Relation of wave height to wave period ------------

Distribution of maximum orbital velocities --------

Graph of the function U = ! NC for solitary waves-s

Comparison of orbital ve loc ity with solitary

theory -----------------------------------------Relation of orbital velocity to wave period -------

Comparison of orbital velocity with solitary and Airy-Stokes theory -----------------------------

LIST OF TABLES

3

4

4-5

8

12

14

19-21

22-25

29

30

34-36

37

38-40

1. Summary of wave properties - ----------------------- 26 2a-2b. Statistics of the distribut ions of horizontal

orbital velocities ----------------------------- 31 3. Threshold velocity of the orbital meter system ---- 32

b

LIST Of SYMBOLS

In general a prefix to a symbol indicates the method used in ob-taining the data; 0 refers to direct visual observation, a to Airy and Stokes theory, s to solitary theory, etc. Suffixes indicate the part of the wave to which the data pertains; c refers to the crest, 1 and 2 to the troughs preceding and following the crest, etc. Most symbols are defi.ned schematically in figure 1.

a - Prefix indicating data based on equations for Airy or Stokes waves ; also the coefficient of virtual mass of the orbital velocity meter (equation 1)

C - Ve loc ity of propagation of the wave crest

d - Still water depth (equation 3, figures 9 and 10)

g

h c

- Accelerat ion of gravity 2 32.2 ft/sec

- Depth of water under wave cr est

- Depth of water under trough preceding wave crest

- Depth of water under trough following wave crest

= ~ (hI + h2 )

- Wave height measured from preceding trough to crest

- Wave height measured from crest to following trough

= 1 (HI + H2 )

- Significant wave height; average of the highest one-third of t he waves. The number of waves to be averaged was based on the average period of the ten highest waves of relatively simple profile occurring during ~ 20 minute interval.

HIIlO - Average height of the highest one-tenth of the waves

L

III

0 '

- Wave length

- Slope of regression l ine

- Punct ions in solit ary wave theory defined by equations 12 and 13

- Prefix indicating data obtained directly from visual ob-servations of water level on staff

c

s

t

T

T n

u

u c

---------

_ Prefix indicating data based on solitary wave equations

- Time. usually measured from wave crest

- Wave period

- Neighbor period; one-half of the elapsed t ime bet ween t he passage of the preceding wave crest and the following wave crest

- Significant wave period; average period of the hi ghest one-t hird of the waves

- Horizontal component of orbi tal velocity

- Maximum onshore component of prbital ve l ocity. measured near the bottom during passage of wave crest

U l , u2- Maximum off shore component of orbi_tal velocity , measured near the bott om during passage of preceding and fo l lowi ng wave troughs respectively

U = 1 U c I + i I ul + u2 1 ; absolute sum of maximum crest and a:ver age trough orbital velocities

Ul = I ucl +1 ull; absolute sum of maximwn orbital velocitie s under wave crest and preceding trough

U = a

2 n H T

1

°nh 21T d Sl - L

absolute sum of maximum crest and

t r ough velocit ie s comput~d at the bottom for Airy and Stokes waves

U D ~ NC; maximum orbi tal velocity computed at the bottom for a S solitary wa ve

Wl

, w2

- Maximum up and down components of orbital velocity preced-ing and following passage of wave crest respective ly

x - Hor i zontal coordinates measured from w~ve crest and positive in direction of propagation

z - Vertical coordinate measured positively from the bottom upwards

r - Relative wave height, HIb

d

ORBITAL VBLOCITY AS~CIATBD WI11f WAVB ACI'ION NEAR 'DiE BRBAXER ZONE*

by D. L. lORan and Horiyuki Nasu

University of California Scripps Institution of Oceanography

La J olla , California

ABSTRACT

The orbital veloci ty associated with ocean surface waves in shal low water was aeasured f or various wave conditions at La Jolla, California, as part of a program of study of sand trans-port and beach erosion. The measuremeDts were .. de near the bottom and just seaward of the breaker zone in water depths ranging froa about five to fifteen feet and for wave heights as great as seven and one-half feet. The current measuring de-vice consisted essentially of a cylindrical rod fixed rigidly at one end like a cant i lever, and the system was so arranged that t he orbital ve loci ty could be interpreted froa t he bending of the rod caused by the force exerted by the moving water. Orbital velocity and wave pressure were recorded simultaneously for time inter vals of about twenty minutes.

The observed aaximum horizontal velocities compare favorably with velocities predicted fr om solitary wave theory for most waveiwith simple prof iles when the ratio of the wave height to water depth is greater than about 0.4. The agreement with theory is somewhat better for longer period waves, and in some ca ses is still quite good in regions where the ratio of wave he ight to water depth is less than about 0.2. On the average, the onshore veloci ty associated with the passage of a wave crest was greater in magnitude and of shorter duration t han the off shore velocity unde r t he wave trough. The differences i n crest and trough velocities varied from wave to wave, but in general correlated with the shape of the waves.

I N'rnODUCTION

There has been much interest recently in wave action in sha llow water and in the orbital or particle mot i on a ssociated with such waves. Although an understandi ng of the mechanics of wave motion is essential in the study of sand transportation, forces on, structures and r elated phenomena, no adequate theory has been developed to describe the properti es of waves in shallow water . Some progress, from a practical standpoint, has come from adaptation of the fora and mechanics of certain ideal waves such as the solitary and Stokes' waves to limited r egions nearshore. However, such application does Dot allow for all progressive changes as waves travel from deep to shoal water, nor for the inherent variability of wave proper ties in nature .

* Contribution from the Scripps Institution of Oceanography, New series No. 841 .

Most experimental investigations of orbital velocity and other wave properties have been restr i cted to smal l scale mode l studies in wave tanks where properties such as wave period and height usually remain constant. Thus the laboratory waves are essentially i deal waves and lend themselves to comparison with theory, but like the theory lack many of the com-plexities of natural waves.

Recent laboratory studies of wave motion in shallow water are des-cribed by Hamada ( 1951), Iversen (1952), Daily and Stepban ( 1952), and Ippen and Kulin ( 1955). A brie f study of the orbital ve l ocity associated with ocean waves i s given in t he Interim Report of the Beach Erosion Board for 1933. Thi s s tudy was based on motion pict ures of the exposed portion of a _staff connected to a subsurface float, and its results are in good agreement with solitary wave theory as adapt ed to the surf zone by Munk (1949a).

An instr ument was designed and built at the Scripps Institution of Oceanography to measure the orbital velocities associated with ocean sur face waves nearshore. The instrument was placed near the bot tom and just seaward of the breaker zone in water depths ranging from about five to fifteen feet. Fif teen series of measurements, each with a duration of approximately twenty minutes, were obtained for various wave conditions. Six of these were sufficiently compl ete to warrant a detailed description in t his report.

MB'nfOD

The procedure used in the measurement of the orbital velocities and related wave parameters is described in some detail here because of the many variables involved and because of the complexities of ocean waves. The notation used is defined in figure 1 and in the preceding list of symbols. The general plan of operation and the field procedures are illustrated in figures 2 through 5.

The current meter used for measuring the orbital velocitieS was a drag type in which the force exerted by tbe moving water on a rigidly mounted rod was interpreted in terms of the velocity of the water . A pressure type wave meter was used with the current meter. and the two meters mounted on a tripod designed so that the orbital current meter was relatively uninfluenced by the field of flow around the tripod (figure 4). The tripod and meters were lowered into the water from the Scripps Institution of Oceanography pier by a crane and placed a distance of 15 feet from the pier to ainimize the influence of the pier pilinls (figure 3) . The tripod feet had weights and stabilizing rods to reduce motion of the unit while on the sand bottom. The tripod was oriented from the pier by three guy lines so that the rod element in the current meter was parallel to the crests of the waves. Since the objective of the investisation was to measure orbital currents near the bottom. the current ~ter was mounted 0.82 feet (25 cm.) above the bottom. This height was selected because theory indicates little change of velocity with depth in this

2

VISUAL OBSERVATIONS

j DIRECTION

" PROPA

SHOlE RECORDING ~

.'.':-: .•. . ..-"-1: '

WAVE PRESSlJRE

VERTICAt VELOCITY

HORIZONTAL VeLOCI TY

Figure 2. Schematic diagr

FiGUre 4. De ceils of the tripod supporting tne orbit~l velocity rr;"ter and the ,;ave me t er. t·cet er box (rigt;t background) houses the volt~ge re€,-ulator.

VISCOUS AND TURBULENT FORCES ON THE

SPHERE AND ROD PRODUCE RESISTANCE CHANGES IN THE

T ,

Figure 5. Detail of the orbital velocity meter.

5

-----~.

region (Munk, 1949a, plate 10), and it was thought in general to be above the zone of high concentrations of suspended sand and bot tom boundary effects. The bottom sediments in this r egion consist of well-sorted fine sand with a median diameter of about 180 microns. In t he region where the orbital velocities were measured the sand bottom had an average slope of about 1 in 35. The bottom sloped seaward from this area at the rate of 1 in 45 out to the 50-foot cont our where the slope increased.

The output of the two meters, after proper modulation and amplifi-cat ion, was fed into a six element galvanometer type oscillograph and the horizontal and vertical components of orbita l velocity, and wave pressure were recorded simultaneously as a function of time on light sensitive tape. A block diagram giving the essent ial s of the inst rumenta-t ion procedure is shown in figure 2.

ORBITAL VELOCITY METER

Basically the instrument consisted of a smal l flexible beryllium-copper rod with a sphere mounted on one e~d and the other mounted riSidly to a support (figure 5). Pour strain gauges wer e cemented at 90 inter vals arounl the circumference of the supported end of the rod and connected so that the opposite pa irs form two adjacent legs of a Wheat stone bridge (appendix II). Vi scous and turbulent forces on the rod and sphere produce stra i ns which change the resistance of the strain gauge element s, which in t urn unbalance the bridge and produce an electrical potential which is interpreted in terms of the orbital velocity. The dimensions and physical constants of the meter and the tripod are listed in append ix II.

By using two Wheatst one br idge circui ts and letting opposite pairs of strain gauges form two legs of each bridge , two components of velocity were obtained. In this study the meter was always used with the r od horizontal and paral lel to t he wave crests. Thus the velocities measured consisted of the vert ical component and a horizontal component in the dir ection of wave propagat ion.

Calibration: When the forces acting on the orbital meter are re-solved and considered to be acting on t he center of the sphere, they can be expressed in the fol lowing manner:

bu 2 f = a bt + bu (1)

wher e f is the total force considered as acting upon the center of the sphere, u is the velocity of the wa ter relative to the meter , "a" is proportional to the virtual mass ooeffic ient, and lib" is proportional to the drag coefficient. Within the ranges of velocity and acceleration f ound in the field, the values of "a" and lib" were found experimentally to be about 80 and 5.2 cgs units respectively.

6

,Both the sphe re and rod dimensions are such tha t their drag coeffi-,cient s ar e near ly constant f or ve l ocities from 0.2 to over 15 ft ./sec., thus making the strain proportional to the square of the velocity for steady state uniform fl ow within these ranges. The meter was calibrated for steady flow at the Hydro court, University of California. Berkeley, for velocitie s ranging from 1/3 to 3 ft.lsec. The results of the l aboratory calibration were in good agreement with values calculated from the physical constants and dimensions of t he meter (appendix II).

Thereafter. the meter was calibrated in the field by applying forces to the end of the rod ( center of the sphere) wh ich were equivalent to the f orces resulting fr om a given steady flow velocity around the meter. In the f ield t he meter was placed in a large basin of still water so that the calibration \'lOuld be made under buoyant forces similar to those dur ing the measurement of orbital ve locities. A typica l calibration curve obtained in this manner for vert ical and horizontal velocities is shown in figure 6. For th is case the up and down cal ibrations were approximately the same, but the onshore and offshore components differed slightly. Because the meter response is proportional to the squar e of the ve loc ity it is insensitive to low veloc ities but gives relatively good discr i mination for high velocities. A measure of the lower threshold velocity is given in t able 3. whe re the velocity required to give a 1 mm. displacement from the null posit i ons on the orbital r ecord is listed for each series. ( See page 32 f or Table 3.)

In addition to establishing the null or rest point of the meter during the field calibrations, the null point was also obta ined when the instrument was in the ocean by placing a plast ic sleeve over the rod and sphere. This was done to insure applicat ion of the calibrat i on to operating conditions where the wat er pressure and temperature dif fered. Consideration was given to temperature compensation in the design of the instrument by mounting the two stationary resistance elements of the Wheatstone bridge in the waterproofed brass housing at the support end of the rod. Thus all legs of the bridge could be maintained at the same temperature.

The effects of accelerative force s on the orbital current meter were measured experimentally in the laboratory by subjecting the instrument to simple harmoni c motion both in air and in water . The inertial Characteristics of the moving instr ument were obtained by assuming the e ffect s of flulidrag in air to be negligible. Then since the value s of f, b, u. and ~ f rom equation (1) were known, the value of Ita" was calculated from t e exper imental data. The value of "a" was found to vary with the period of oscillation, and to obtain an upper limiting value of about 80 gram em. (appendix II ) .

As an approximation to f ield conditions , the experimental r esult s of the calibrations were applied to theoretical velocity dist ributions. Two case s were consitlered, one assuming the ve locity distr ibution to be

7

50

40

30

20

t E 10 E 9 > e 1-"

7

Z 6 w ~ 5 w U 4

that for waves with sinusoidal prof ile as in the case of waves of smal l steepness , and the other that the velocity distribution was of the solitary wave type descr ibed by Munk (1949a).

In both of these theor etical cases it was found that the magnitude of the acce lerative terms in the record of orbital ve l ocity in the vicinity of the crest s and troughs of the wave would be negligible com-pared to the magnitude of the drag or velocity effects when the wave periods are greater than f ive seconds . The relatively small magnitude of accelerative terms in these theoretical cases suggests that accelera-tive terms are a lso small in the case of actual waves outside of the breaker zone. However, it seems unlikely that the accelerative effects can be neglected in the region between the crest and trough of the wave where mini mum velocit ie s and maximum acce ler ations occur. For this reason computations involving orbital velocities, and comparisons of observed velocities with theoretical velocit ies were restricted to velocities measured at t he crest and trough of the wave where the maximum horizontal component of force occurs. This decis ion to tabulate only crest and trough measurement from the records seriously restricted the scope of the study , since the continuous profile of force as a function of time could not be interpreted in terms of velocity; however, it appreciably lessened the number of a ssumptions which are inherent in this type of interpretation.

Thus, the record from the orbital me t er is that of the forces ex-erted on the meter by the Motion of the water, and where the record has been interpreted in terms of velocity , the assumpt ion is made that the drag force on the meter is prpportional to the square of the veloci ty.

Er rors in Measur ement of Orbital Velocity: The interpretation of maximum orbital velocity from the record of the orbital meter i s predicated on the following assumptions: a) the meter response was proport ional to the square of the velocity, b) the instrument was level, c) the axis of the rod in t he meter was parallel to the crest of the waves, and e) the null point established during calibration is the point of zero velocity.

As discussed pr eviously~ the assumption that the meter response is proportional to the square of the ve locity is probably just ified for maximum velocity readings. Errors of thi s type are probably less than 1% and would tend to give high readings. The mete r level was probably main-tained within 10 degrees of horizontal at all t imes~ so that the measured horizontal velocity wQuld be decreased by less than 2%. However, a 10 degree tilt would result in an excess reading on the vertical velocity component equivalent to 17% of the total horizontal velocity. Since the verticai velocity components are expected to be small near the bottom, this may r esult in a large error in interpret ing vertical velocities. Inspection of the records (f igures 11-16) suggests that the meter was not always 1evel t and for this reason detailed analysis of the vertical velocity component has not been attempted.

9

The error in alignment between the wave meter and the wave crest was probably no greater than 10 degrees for l ong crested waves and not over 15 for short crested waves. This would re sult in an error of 2 to 4~ in orbital velocity.

Collectively these errors should r esult in the meter giving a lower horizontal velocity than the true one, but are probably all small compared to the general error inherent in calibrating and zeroing the meter in t he field.

Electronics: A conventional carrier type amplifier , consis ting of a voltage amplifier followed by a power amplif ier, was used with the orbital velocity meter (figure 2). Since the electric leads were l ong, it was not f easible to use alternating currents to energize the strain gauge elements. Therefore, the strain gauges were powered by direct current and the direct current signal output f rom the Wheat stone bridge was in-verted by means of a chopper or modulator so that it could be more r eadily amplified. Por most of the work a vibr ating type Stevens-Arnold #222 chopper was found to serve quite satisfactorily as a modulator. Adjustable biasing potentials were incl uded so that the meter could be zeroed in water.

The output of the power amplifier was fed to a demodulator and fil ter system which furni shed a direct current potential to the oscillograph recording elements. A General Electric six-element type PM-10 oscillo-graph was used to make a simultaneous r ecord of the orbital met~r output as well as wave pressure and a time pulse. The galvanometer elements, which .record on photosensitive paper, have essentially uniform response up to 3,000 cycles per second and thus do not introduce any appreciable time errors in the measurements.

WAVE MEASUREMENTS

A continuous r ecord of wave pressure was obtained f rom a thermopile wave meter mounted on the tripod with the orbital CtErent meter. The thermopile wave meter operates on the principle of the polytropic gas law, and was used in this study because i t was small , light-weight, and had simple cable requi rements (Isaacs and Wiegel, 1950) . The meter was calibrated during each operation by comparing t he meter output with visual measurements of wave height obtained from a calibrated staff mounted directly above the instrument (figures 2 and 3) .

Wave Period: The record from the wave meter was used direct ly as a time indIcator for the passage of wave crests and for obtaining t he per iod between crests. Because of the complexity of the waves it was necessary to define and obtain several types of wave period from the pressure record. The first period, T, was used only for evaluating wave height from the record as described in the following section. It is def ined as the time interval of the ri se and fall of water associ ated

10

with the passage of the wave crest and was measured on the record from the minimum pressure just preceding the crest to that immediately follow-ing the crest ( figure 1).

Since waves near the breaker zone are not sinusoidal and each crest tends to be somewhat independent of its neighbors . the essential time element should give the separation of each crest from preceding and following disturbances. Consequently , a "neighbor period" , T , for a given wave crest was defined as one-half the t ime interval fram the preceding to the following wave crests. Any disturbance less in height than one-fourth of the average of the heights of the crest under con-sideration and the preceding crest was omitted from consideration. The neighbor period was used in all calculations where the period of i ndividual waves was concerned.

Por general characterization ot ~ne group aspect of the waves during each observation, the significant wave period , T1/3 , was obtained by averaging the period of the highest one-third of toe waves during each series of observations. The number of waves to be averaged waS based on the average period of the ten highest waves of relatively simple profile occurring during a 20-minute interval . General features of the waves during the observations are l isted in table 1, and graphs of wave height versus period for each series in figures 17 through 24.

Wave Height : Although the met er operates on the pr inciple of the polytropic gas law, the wave pressure could not be obtained directly from the record because the meter is not completely adiabatic and there is a lag in the silver-constantine thermopile . Since these discrepancies in the meter response are a f unction of period, the wave record was corrected and made proportional to pressure by a correction factor ob-tained from the calibration of the meter .

Because the interpretation of wave height from shallow water pressure records is not c l ear, the meter output was calibrated for wave height by obtaining the least squares best fit ( regression line) of the corrected meter response as a function of the observed wave height ( figures 7 and 8) . In general the wave height used in this study was the average of the heights from the wave crest to the preceding and following troughs, al-though each was used separately in some phases of the study.

Each observational series included several hundred waves , and approxi-mately 20% of these waves were visually measured on the graduated staff above the orbital meter tripod and used for calibrating the wave recorder. In the final analysis of most data , wave height was based on this calibration. Nomenclature and the method of computation are given in the l ist of symbols and figure 1.

Depth of Water: An average depth f or the water during e~ch observa-t ion was interpreted from the visual measurements of water level on the calibrated staff mounted above the orbital meter tripod. As mentioned

II

N

Figure 7.

70

SERIES NO. 7 .

60

i 50

€ ...:-:x: ~ 40 W :x: w >

20 a:: ~ IJ'J Z

10

3 4 5 6

VISUAL WAVE HEIGHT, oHI FT._

Calibration of t he ,.ave pressure record from the regress ion line of instr~~ent wave height as a function of heigh t from "/av~ stiff. The instrument ,,,aVe he is;ht is the ci1art displc. cement of the pressure record cor-rected for the fre quency response of the instrument.

70

SERIES NO. 15

60

t 50 ~ ...:-:x: 40 a:: 20 ~ IJ'J Z

10

2 3 4 5 6

VISUAL WAVE HEIGHT, oH FT. ---.

Figure 8. Calibration of the ,!ave pressure record for

series 15 (refer to figure 7).

previously, the water depth under the wave crest, h , and the preceding and fol lowing troughs, oh , and oh2' were systemat~cilly. measured for about 20% of t he waves. ~rom these measurements an average wave height, H, was obtained for each wave crest :

o

( 2)

The prefix "0" indicates that the dat a are obtained from visual measure-.ents of water level on the wave staf f and not from an interpretation of the wave pressure record.

The still water depth, d, was defined as the depth axis intercept of the least squares orthogonal regression line (Hald, 1952, P. 601) of hand H. The equation for this line can be expressed as, o c 0

h =m H +d o c 0 ( 3)

where h is the distance of the wave crest above the bottom, m is the slope 8fCthe regression l ine and d, is the h axis intercept. Thus d is an approximation to the average water depth as the wave height approaches zero. Graphs of hand H are shown in figures 9 and 10, and the values of d and m for eaghcserie~ are listed in table 1. ( see page 26 for Table 1J

If equation (3) is written in the form

m = h o c

H o

d

it is evident that the slope of the line, m, is the ratio of that portion of the wave crest which extends above the average depth to the total wave height . The value of m was found to range from 0.76 to 0.86. This means that for a given wave height, measured from the wave trough to wave crest , approximately 76 to 86% of the height is above water level and the remain-ing portion, extending to the wave trough , is below.

IDEAL WAVES

In deep water, waves tend to be sinusoidal and each wave is an inter-dependent part of a broader pattern or group. The motion of such waves is principally dependent on their length , t o a l esser extent on their height, and is independent of depth. The wave concepts of Airy and Stoke apply best to waves in deep water where wave length is the essential descriptive parameter.

In shallow water and especially near the breaker zone , the sinusoidal or continuous frequency characteristics commonly associated with waves in deep water are lost and individual waves tend to retain their identity. Here, the length of the wave is less significant, and the depth of water

13

i u ~ o

14

SERI ES NO. 14

13

12

11

:I: 9 ~ a.. w o ~

t3 e tr u

___ STILL WATER DEPTH

7

6

50~----~----~2----~3----~4----~5~--~6~--~7 WAV E HEIGHT. oH FT, _

Figure 9. Determinat i on of still water depth from the orthogonal regression line of the wave height and the depth of wat er under t he wave creat. The slope of t he line, m, defined by equa"ion(3),ia a measure of the proportion of t he wave height which extends above still water depth. Depth, d = 7.3 ft; sl ope, m • 0.86.

14

SERIES NO. 15

13

12

11

i G:10 " --'b I ~ a.. w 0

~

and he i ght of wave become control ling factors. In prof ile t here is a tendency for t he waves to consist of isolated crests separated by r elatively flat troughs. Por this reason the solit ary wave concept of Scott-Russell has been applied t o waves near the surf zone , and has met with some degree of success. Br ief summaries of the essential rel ations for Airy, Stokes, and solitary waves are given below.

AIRY WAVES

The equation of classical hydrodynamics gives the wave phase velocity e as a function of wave length L and depth of water d:

tanh ~ d L

(4)

where g is the acceleration of gravity. The instantaneous horizontal and vert i cal components of orbit al veloc ity u and ware:

2 11

!L!L cosh -L- z 211 (x - et ) u = T . h 2 11 cos L un ~ d

(5)

11 H sinh ..?....!! z 211 L

(x - et) w = T . h 2 11 sin -d L S1n L'" ( 6)

where T = LIe is the wave per iod, H is the wave height , z is ~he vertical coordinate measured f rom the bottom up\tards, and x and t are the horizontal distance and time respectively measured from the wave crest.

The above general theory, attributed to Sir George Airy in 1845 (Lamb, 1945 , p . 368) , is strictly applicable to waves of infinitely small height in water of any depth. The wave profile i s sinusoidal and the particle orbits are closed circles in deep water and ellipses in shallow water. Inspection of equations 5 and 6 shows that the orbital velocities vary as sine functions with time, and that the magnitudes of the maximum onshore and offshore velocities are equal. The requirement of small wave height and the resulting symmetry of orbital velocity are not in agreement with ocean surface waves nea.r the breaker zone.

STOKES' WAVES

of the theories for waves of finite height , that of Sir George Stokes (1847) has met with most success. The Stokes' wave has flatter troughs and steeper crests than the Airy wave. The particle orbits are not closed, put lead to a slight net transpor t in the direction of wave propagation.

15

The horizontal and vertical components of orbital velocity for Stokes' waves are, to the second approximation:*

(7)

cosh 211 ,lH2 411

11H -z 211 3 ~osh --r- z L (x-Ct) 411 u = cos - + - cos L (x-Ct) T sinh ~d L 4 LT (sinh .?!!- d) 4 L L

1!! z 411 (8)

'11H sinh 211 3 112H2 sinh L z 411 L (x-Ct) sin w = - s in - + -- -(x-Ct) T sinh 211d L 4 LT (sinh .?JL d) 4 L L L

Inspection of equation (7) shows t hat the f irst term, which is identical to tbe complete expression for the horizontal orbital velocity of the Airv wave (equation 5), is positive under the wave crest and negative under the trough, while the second term is positive under both crest and trough and negat ive at 1/4 and 3/4 wave' lengths from the crest . The effect of this term is to increase the magnitude but shorten the duration of the orbital velocity under the crest, and decrease the magnitude but lengthen the duration of the offshore velocity under the trough. This asymmetry of the orbital velocity profile with time is more nearly l ike the solitary wave and is in keeping with observations of shallow water waves in nature. It should be noted, however, that the sum of the absolute values of maximum velocities under the wave crest and trough has the same magnitude for both Airy and Stokes waves. Thus along the bottom the sum of crest and trough velocities for both waves is given by :

U = a

u crest

z = 0 + u trough

z = 0 2

11H T

1

sinh ...1!... d L

( 9)

This expression for the orbital velocity is in a convenient form f or comparison with f ie ld observations because the effects of other velocities that may be present at the t ime of measurement are minimized so long as their period is long compared to that of the waves.

SOLITARY WAVE THEORY

Munk (1949a) has summarized useful relationships derived from solitary wave theory and made them accessible to numerical examples by plotting various dimensionless relationships. These relationships were used to compute theoretical velocities for comparison with those measured in the field.

Pollowing Munk (1949a), McCowan's approximation for the horizontal orbital velocity, u, and the associated vertical velocity, w, can be

16

*These equations result from the partial derivatives of the second order velocity potential given by Stoke s (1847, eq. 18), and are identical to

equations 3 and 6 of Morison and Crooke (1953). Personal commuoOication from R. A. fuchs. University of California, Berkeley.

expressed as: z M ...!... u 1 + cos M h cosh h = N (10) c (cos M z + cosh M .!)2

h h

sin M z sin M x h h w

= N (11) C (cos M z + cosh M .! )2 h h

Where C = V g(H + h) is the velocity of propagation of the wave crest, H is the wave height, h is the depth of water under the wave trough, M and N are f unctions of the re l at ive wave height, Y = H/h, and are defined by the equations

Y =

N =

N M

2 J

tan

. 2 S1n

1 2

Directly beneath the wave crest both x and ware zero and equations (10) and (11) reduce to

= NC

z 1 + cos M h

At the bottom z is zero and the velocity beneath the crest becomes

u = (u) s x = 0

z = 0 = 1 2 NC

(12)

(13)

(14)

(15)

Values of U for various water depths and wave heights are graphed in figure 27. s

According to solitary theory , water particles are essentially at rest at distances greater than about x = 10h in front of the wave crest. As the crest approaches they move forward and upward and attain their maximum veloci ty at the instant the crest passes. Following the passage of the crest the velocity of the part icles decreases and they move downward, eventually reaching a new rest posi tion , a dist ance r in advance of their former rest posit ion. Thus, when sol i tary waves

17

travel through st ill water, partic les undergo relatively f lat traj ectories with motion only in the direct i on of wave propagation. The velocity ~iven by equat ion ( 15). U, is therefore the maximum velocity that a par ticle would att ain atSthe bottom during the passage of a solitary wave in still water .

Closed or partially closed trajectories and orbital motion more nearly resembling those for shallow water waves in nature are obtained from the theory by superposing some sor t of return flow on the solitary wave (Munk 1949a). If we co.nsider a uniform velo.city v opposed to the direction of propagation o.f a train of solitary waves, the maximum velocity at the time of crest passage will be U-v. If the solitary waves are far apart , the velo.city under the trgugh will be v, and opposite in direction to that unde r the crest. In this case the sum of the absolute values of maximum velocities under the wave crest and trough would be U. Thus U and U are comparable in that both represent the absolute sum~ of crestSand trSugh velocity for their respective types of wave mo.tion.

ANALYSIS OP DATA

The complex character of wave motion in nature made systematic analysis of the records difficult. It was possible to obtain frequency distributions of various properties such as wave height and period and some aspects of the orbital velocity. However , in order to. compare the measured values of velocities with wave theory it was necessary to. neglect group characteristics and to co.nsider each wave individually, including only those wave s which had relatively simple pro.files as in figure 11. The distinctio.n between simple and complex wave profiles was necessarily arbitrary. Examples of simple waves are listed in the captions for the sample wave records. In general, compounded or double crested waves, as illustrated by the first wave in figure 16 and the last wave in figure 15, were excluded from comparison with theory , but were included in frequency distributions which were used to characterize "the wave and orbital spectrum. Thus all waves were used to plo.t wave height-perio.d diagrams, f igures 17 through 23, and histograms showing the distributions of maximum orbital ve l o.cities , figures 25 and 26.

The simple waves were used for co.mparison with wave theory, figure 28 through 39, and a l ist of the pertinent measurements for each wave is given in appendix I .

Two techniques were used to compare the simple waves graphically with theory. The waves were compared with solitary wave theory by plotting the ratio of the observed horizontal velocity, U, to. the predicted velocity U (ordinate) as a function of the relative wave height H/h. The pro.x~ity of plo.tted points to. a value of unity for the orbital velocity ratio is a measure of the agreement between measured and pre-dicted velocities. Co.mparisons of this type are shown in Pigures 28 through 33.

18

CREST WAVE PRESSURE

I I I I I I I I I I ! I • t I \ I I I I J I I I I I I I I I I I I I I TIME IN SECONDS--

SERIES NO.7

DEPT H 9. 1 FT.

Pigure 11. Sample of the record of wave pressure and orbital velocity , series NO.7. Waves number 70 through 73

""1 4 -II:: o % .. 3 · z

~ 0 2 ~ 0

II:: 3 t;: "'12 14 ~

figure 12.

are shown here . All four waves were considered to have relatively simple profiles; wave and orbital velocity values are tabulated in appendix lB.

HORIZONTAL VELOCITY

f'\ J \ .

TIME IN SECONDS - SERIES NO . 9

DEPTH 5.5 FT.

A portion of the record of wave pressure and component of orbital velocity, series No.9. number 243 through 246 are shown. The first considered to have simple profiles and their are listed in appendix IC.

19

horizontal Waves

two 'were properties

~

t: ~ 2 '" iii x 3-", I

011, > I

0

'" VI I " 0 I-' I ..

II' ~

TIME IN SECONDS-SERIES NO . 12

DEPTH 7.2 FT.

Pigure 13. Sample of the record of wave pressure and orbital velocity, series No. 12, showing wave numbers 23 through 31. The first three and the last three were considered to have simple profiles and their properties are listed in appendix ID.

.... x ..

6

~2

'" > .. :J '

o 4

1)3 ';0

::: 2

~ 0 ... 2

I) 3 o

4

CREST (\ ~ ~ '1 II I ~ , ) , I I I

J I "

\ I \ ( \ f, \ """,SSV", N '\J l~ ,~ ~t V ~~. v:,,,eA~ V"oe,TY _" JIH 2 l~ -V WI· [YI\ $, ~ I l!

j J HORIZONTAL VELOCITY I \ 0

_ '- A ./"\... ./'\.. ) . __ 2

JJI"'~I.~.Jj ""=",~""",~",=,,, TIME IN SECONDS ---

SERIES NO. 14 DEPTH 7.3 FT.

Figure 14. Sample of the record of wave pressure and orbital ve locity, ser i es No . 14, 'showing waves number 40 through 47. The first two, which were measured just before breaking, were considered to have relatively simple pro-f i les and t heir properties are listed in appendix IE. The dashed line s indicat ing the crests of these two waves were based on visual measurements of wave height because the pressure record was off scale.

20

I I I II I I It I I I I I I I I I I I \ I I l •• I I I I • I I 1,1 I "1'1 I II I I I I I I I I l, I I I I I , • I I I

T IME IN SECONDS_ SERIES NO.15 DEPTH 8.5 FT.

figure 1S. Sample of the record of wave pressure and orbital velocity, series No. 15, wave numbe~s 3 through 13. Wave numbers 4, 5, 6 , 8, and 11 were considered to have simple profiles ; their properties ar~ listed in appendix If .

! 13 oL J I I I I I I I I I I I I I ~ I I, \ " , .\ I \ ,I, \ I , , I \ I I I I I I , I I J J , i" I J I , J \

TIME IN SECONDS _ SERIE S NO 15

DEPTH 8. 5 FT.

figure 16. Sample of the record of wave pressure and orbital velocity, series No. IS , showing wave nUllbers 168 through 174. The second through fifth waves were considered to have simple profiles and are tabulated in appendix IP.

21

T

8

SERIES NO. 7 ALL WAVE'S

· · C · , · I · • · I : , . • · . I · · · · . I

4 6 8 10 12 14 0~------~2~------~--------~--------~--------~--------~------~

8

PE RIOD. Tn SEC.-

Figure 17. The re lation of wave height to wave period for individual waves, series No. 7. This wave height-period dia.gram shows the wave height, H1, aeasured from preceding trough to crest, plotted against wave period.

SE RIES NO. 9 ALL WAV ES

I •• 1

2 .. - ~ .' . . 1.- · :·i·~··· ., . o .. . "..e .... .

..I- •••• • • .. -..I • • a:."~"f&.N •• ~

o o . : •• --. we . - • . . .. .- ..... .. o .. ..,,,..1 . .

0~------~2~------~4--------~6--------~8--~-----1~0--------~12--------~14

PERIOD. T. SEC. -..

Figure 18. Wave height-period diagram for series No. 9 showing the wave height, H, averaged from crest to preceding and following trough, plotted against wave period.

22

i t

e

6

, SERIES NO. 12 ALL WAVES

.. .

.. . . • • • I- • • . . . . -1:-:;': ... . . ..... . . . . . . _, ..... . .. . ... '. . .... ..

I ' ••• '., •.• 1 • ] . , ... . .-'. . • • r..-t!..te •• • U I. ..--.J:'. .'.. e.

· · .. , i'· I'··.. '. · · 1.-' • e. .: •• • • e • •

• • •

o L---------~2~--------4~~------~6~--------~e--------~10n--------,,~2--------~,4 PERIOD, T" SEC.----'

Figure 19. Wave height-period diagram for series No. 12.

e

SERIES NO. 13 ALL WAVES

6

t ..,: I&-

~ "':4 X ~ ILl X . . .. ILl

~ ~ 2

• - ._. -• . - .. ..

2 4 6 8 10 12 14 0~------4-------~------~------~--------,~------~------T. PER IOD, Tn SEC ~

Figure 20. Wave height-period diagram for series No . 13.

23

1 Ii: =t ..... %: (!)

UJ %:

UJ ;

8

6

4

2

0

8

2

SERIES NO. 14 ALL WAVES

• •

• • • • - • • • •• •

• • .. • • • • .. •• • • • • • •

• • • • •

•

2 4 6 8 10

PERIOD, Tn SEC.~

Figure 21. Wave height-per iod diagram for

SERIE S NO. 15 ALL WAVES

o 00

o

o 0, .. .. , .. .. . • • • t. • ••

.. " .... .. I I.. .. .. ". ".. : I .. :

.... "" -: "... '1" .." .. " ." .. " ., ........ i " .. .. .. ,.:r. .... . ... \ ",= ' .. " .. i"· : .. . :: ... " ........ " .. " .. .." .. .. .... .. ... ": .. .".

": .. . .. ": .... " .. .. •• ". -·"·1" .... o

o 0

series No.

•

•

12 14

14.

O~--------~2----------~4------~--~6~---------8~--------~10~--------~12~--------~'4

PER IOD, Tn SEC . ~

Figure 22. ~lave hei ght-period diagram for all waves in series No .. 15, using the average wave height H. Compare with figures 23 and 24.

24

(

,

i I-' \.I..

::z::

..... :x: ~ IIJ :x: IIJ > ~

---

8

i 6 I-' \.I..

-£

~ 4 :x: ~ IIJ :x: w >

TABLE 1. SUImllary of wave properties .

?J . y St i l l

2J Relativf Wave

Peri od Wave Hei ght Wat er ~vave Number of I'laves - - - Depth Rati& He:Lght

Duratior Descripti on Analyzed Tl/ 3 Hl / 3 HIIIO H d t l /3 max Series or ~'laves Total " . , Gec, f • ~ ....£6. f m sec. - - o1Llp_e -- -7 937 e 11, lAO 1 6 7.8 4.1 5.0 6.2 9.1 0.B5 0.4B

period superimpo ed

9 OB3 5 se h t est ed 2 6 4 5.4 2.0 2. 5 3.1 5.5 .76 0.4 swe1 , 9 sec secon-dary

12 35B 5} sec swell 249 IB5 5.5 2.7 3.2 4 .1 7. 2 0.79 0.41

13 1253 complex 5 and 12 203 ~ - 7i _3J , 2 7.2 0.B3 0. 5.3 seo swell

tl14 63a 1 sec swel l, shorter 213 6r}1 9.8 4. 5 5.5 7.0 7.3 0.86 0. 67 period secondary with tendency to r ide preceding crest

15 1730 7~ sec swell, tends 280 105 6.9 3.B 4.5 5.3 B.5 0.a5 0.48 to be short crested

Y The number f waves to be averaged for T J H J and H a was determined by averaging the periods of large we defined wave t~ains. The perl~rt obl~~ed in t~ts way approximated t he value listed for t he significant b'ave period, Tl/3. - _

11 Faulty wave pressure record. T 13 and Hl/3 estimated from visual measurements. Ther e were few waves with aim 1 rofiles.

~ Porti ns of wave pressure rec rd off seal. Ap r ximat ly 2/3 of the w ves had 6 . 5/ Pro ortio f the wave cr st above till water level. See quati 3. £I Relative wave h i ght f the significant av s.

The ab olute sum of maximum horizontal crest and trough velocities was obtai from the field data by the following relations :

u = u + i c (16)

(17 )

The fir st represents a total absolute maximum velocity from the wave cr t to the average of the preceding and following troughs, and the latte fro the wave crest to the preceding trough. To obtain the value of t e corresponding predicted velocity U and u

1' the wave

heights e e computed accordingly, and the abs~lute vefocity obtained·from the grap 0 equation 15 (figure 27).

A g obtained and sup r solitary depend on Airy-Stoke 3, 5 , and applicable mated by

WAVE HEIG

eral comparison of measured velocities with ideal waves was plotting the orbital velocity as a function of wave height,

posing the theoretical curves for Airy-Stokes waves and ves (figures 35-39). Since the solitary theory doe not r quency , it is represented by a single curve, while the

lation requires a curve for each wave period. Curves for ) - second periods were drawn for each series; the period most for the various wave heights in each series can be approxi-spection of the wave height-period diagrams.

ND PERIOD

The r ~tionships between wave height and period, a.s shown on the height-per diagrams (figures 17-24), are similar to those obtained for ocean ves by Putz ( 1951, 1952), although there is a more pronounced tendency in hese waves for the high waves to have longer periods. This tendency fo ~ wave height to increase with period is present in the two wave seri s, 7 and 14, which have the longer significant period. Those observation where the periods were shorter , such as series 9, 12, and 15, and tho e where complex wave trains were present , as in series 13, show no te _ncy for increase in period with wave height .

As mentioned in the discussion following equation (3 ), the slope of the l in When the depth of water beneath the crest is plotted as a function of wave height is a measure of the proportion of the total wave hei h which occurs above still water depth . For the six series of waves lyzed the slope, m , ranged from 0.76 to 0.86 (table 1), i ndicat " g that 76% to 86% of the wave height was above still water level hile 24~ to 14% was below this level. These findings are in general agreement with model studies of waves near the breaker zone (Wiegel, 1950; Iversen, 1952). Comparison of the slope with the signifi-cant per i d, Tl/3 ' ( table 1) shows that greater values of m occur for

27

----

series having longer periods and for those series which were measured closer to the breaker zone (high values of the relative wave height, r1/3). Thus series 14 has the longest significant period , the highest

va be of YIL3 and the largest value of m, wherea~ series 9 has the shortest sign1ficant period. the lowest value of '1/3' and the lowest value of m. The relative influence of period and depth individua1~y on the value of m cannot be ascertained from this data~

ORBITAL VELOC ITY

--- ---

In general the graph of the horizontal component of orbital velocity asa function of time resembles that of the wave pressure (figures 11-16) . The_data indicate that the horizontal velocity is more dependent on the

. rate of change in level of the water surface during the passage of a wave train than on the actual height of the wave. This is illustrated by the fourth wave crest in figure 13 ; the wave is almost as high as those preceding it , but shows a more gradual rise in level from the preceding trough to the crest Which results in a very low crest velocity. The following wave also has a gentle slope preceding the crest and a low crest velocity, whereas the succeeding trough velocity which is associated with a rapid change in water l evel is exceptionally high. The asymmetry in wave profile and the associated asymmetry in the horizontal crest and trough velocity of waves very near the point of breaking is i llustrated by the first two waves of figure 14 which were measured just before they broke .

Distribution of Maximum Onshore and Offshore Velocities: Histograms showing the frequency of occurrence of various values of orbital velocity under wave crest and trough are presented in figures 25 and 26, and the statistical constants such as mean, standard deviation, and root-mean~ square values of the crest and trough velocities for each series are listed in tables 2a and 2b. The data are computed and presented in two ways; the first gives the distribution of the maximum orbital velocities under each wave crest and trough (figures 25, table 2a), and the second gives the distribution of maximum onshor e and offshore orbital veloci t i es between zero crossings of the horizontal velocity record (figure 26, table 2b) . The f irst method counts as a wave crest each disturbance of the water surface which results in a maximum peak on the horizontal orbital velocity record. The second method considers only the maximum ve locities which occur between changes in direction (zero crossings) of the horizontal velocity.

These two~hods of presentation differ because the water in the surf zone is in a continual state of f luctuat i on (Munk , 1949b; Shepard and Inman, 1950 , f igure 11) , and because there is apparently some tendency in shal low water f or large waves to " trap" smaller waves, result ing in double crested waves and the accompanying double peak in onshore velocity . Double crested waves such as the fifth wave in figure 14 and the last wave in f igure 15 result in trough velocities between the cres ts which are in an onshore direction. For these reasons

28

N to

~ lO c . ..

, "

9

12

. . . \.I) rur fltc

OFFSHOl'I!

Figure 25.

DISTRIBUTION OF MAXI"'U'" ORBITAL VELOCITIES UNDER WAVE CRESTS AND TROUGHS

DEPT" I ''''' 140 Cf.WII!t£&UfIV£ .V£8

OEPtH Ill!! FT ~C6 COJtIISCC\.TTI"''[ wavES

ot""'H 1.2,.r 2.4. toHst.eU'TfYr; "''''U

, '"

, '"

14

15

O(,"TI1 't 2 ..-t. Z0 3 COliIsfC\,ITIYI! .... Vi·S

P~l"T'" 7 :3 n 21' CONsrCUT l'lft .,,,"'-[.5

DEPOTH ... , 1'r '280 cOItS!CUTrV[ .AYE'!

u.. ,nTI t[c; ONSMOIllE

Histograms of the distribution of maximum horizontal orbital ve l ocities under each wave crest (shaded) and t rough ( unshaded). Each disturbance of the water sur face which results in a maximum peak on the horizontal velocity record is considered as a separate wave crest .

, "

TIItOUGH VELOCITY

u ..... d 9

12

OISTRIBUTION OF .... XIMU .. ONSHORE AND OFFSHORE ORBITAL V£LOC1TI£S

8HWEEN ZERO CROSSINGS

D(PTI1 11.1 n.

U. H£T / stc ONStCOfU

SEIUU 110

13

14

15

,, : rr

. . "" fttl/HC .. ...,.,

Pigure 26. Histograms of the distr ibution of maximum horizontal orbital velocities occurring between zero crossings ( changes in direc-tion) of hor izontal vel oc i ty. Onshore orbital velocities under the wave crest a r e shaded; of fshore velocit ies under the wave trough are unshaded.

2 ~~-----+~~-----~------~~~-------r-------~

SOLITARY WAVE THEORY

u • J.. NC • 2

DEPTH h FEET

Figure 27. Isolines of waVe height, H, as a function of the maximum horizontal orbital velocity at the bottom, ~U, and the depth of .vater under the wave trough, h, rrom equation 15. sU is the absolute sum of the maxi-mum velocities under the wave crest and trough.

30

TABLE 2a. Statistics of the distribution of m.a.ximum orbital velocities under wave crests and troughs. Histograms for these velocities are plotted in fieure 25.

Onshore l·'ax:i.mum Hori20ontal Offshore Haximwn Horizontal Velocity, ft/sec '!elocity, ft/sec

Serie · ~wn 11~an Standard* Root-me an- Haximum I·lean Standard Root-mean-

Uc Deviation square Velocity ul Deviation square Veloci ty

7 3.0 0.9 3 . 2 4.9 0.7 0.8 1.1 3.3

9 2.6 0.8 2.7 4.2 0.1 1.4 1.4 4.3

12 1.6 0.7 1.7 3.1 0.8 0.8 1.1 2 .4

13 1.6 1.0 1.9 4.6 1.8 0.8 2.0 3.9

14 2.0 1.0 2.3 4.7 1.6 0.8 1.7 3.7

15 2.7 0.6 2.8 5.1 0.6 1.4 1.5 3.8

I I ,

ABLE 2b. Statistics of the distribution of maximum horizontal onshore and offshore orbital velocities etween zero crOSSings. Histograms of these velocities are plotted in figure 26.

Onshore :·[aximum Horizontal Offshore liaximum Horizontal Velocity, ft/sec 'lelocity, ft/sec

Ratio eries

Standari' Ua/_ :·~~an Root-mean- Maximum He an Standard Root-mean- l·faximum uc Deviation square Velocit.y til Deviation square Velocity

ul

7 3.1 0.7 3.2 4.9 0.9 0.7 1.2 3.3 3.4

9 2.7 0.8 2.8 4.2 0.7 0.9 1.2 4.3 3.9

12 1.6 0.7 1.7 3.1 1.0 0.6 1.1 2.4 1.6

13 1.6 1.0 1.9 4.6 1.8 0.8 2.0 3.9 0.9

14 2.0 1.0 2.3 4.7 1.6 0.8 1.8 3.7 1.3

15 2.6 0.8 2.7 5.1 1.4 0.8 1.6 3.8 1.9

Average 2.2

*

31

TABLE 3. Threshold ve locity of the orbital velocity meter .system for each ser ies of observat ions. The threshold velocity is arbitrarily defined as that steady flow required to produce a 1 mm di splacement from the null position on the orbital velocity ~ecord (see figure 6).

Horizontal Vertical Velocity Velocity

Series rt/sec rtLsec

7 1.0 0.6 •

9 1.0 0.5

12 0.5 0.4

13 0.6 0.4

14 0.7 0. 5

15 0.9 0.6

32

the first method of summation, which includes each velocity peak, results in the occurrence of some trough velocities on the onsbore or crest side of the histogram (figure 25).

The maximum velocity under the wave crest was always onshore or in the direction of wave propagation , and for all of the wave series analyzed with the exception of series 2 and 13 the mean onshore maximum velocity exceeded the offshore velocity . The two series out of the fifteen for which the mean of the maximum trough velocities was greater than the -'mean of the crest velocities were both characterized by irregular and confused wave trains, in which there was a high incidence of wave crests with gentle forward slopes and steep back slopes. However, even in these cases the highest velocities measured were in an onshore direction.

Coaparison of Measured and Ideal Velocities: The maximum orbital velocities of the more uniform or simple waves were compared with values computed from equations for the solitary and Airy-Stokes waves. For purposes of comparison the absolute sum of the observed maximum crest and trough velocities as defined by equation (l6) and (l7) were used 'so that longer periOd fluctuations in the surf zone could be neglected. Inspection of the data (figures 28-40), shows that the scatter of plotted points is fairly large f or all series, and that the scatter increases with decrease in relative wave height. The increase in spread of plotted points for low waves results in part from decreased sensitivity of the orbital velocity meter at low velocities. However, it is apparent from the plotted data that the velocities measured in the field are on the average in better agreement with the solitary wave than with the Airy-Stokes relations, although portions of some series are not in good agreement with either.

Series 7 and 14, which have longer significant periods of approxi-mately 8 and 10 seconds, are in good agreement with solitary theory (figures 28 and 31). Series 12, Which consisted of a relatively uniform swell with a short significant period of 5! seconds, had orbital velocities which averaged about 20% lower than velocities predicted froa the solitary wave equations (figure 30) . The observation with the shortest significant period, series 9, showed the greatest scatter in plotted points, but on the average also gave lower values than predicted ( figure 29). However, this series was complex; the waves showed a tendency towards short crestedness and included wave trains with a primary period of about 5 seconds and a secondary period of 9 seconds (figure 18). The other short crested group , series 15, is in f air agreement with solitary theory for waves with relative heights greater than about 0.4 , but shows a progressive departure from predicted values for lower waves (figures 32 and 33) . This series also shows a tendency for the shorter period waves to be associated with lower values of orbital velocity (figure 34). All of the series of observations had significant periods longer than the minimum allowable period suggested by Munk ( 1949a, plate 2) for applying solitary wave theory to the su%f zone .

33

1.8 SERIES NO. 7 DEPTH 9 .1 FT.

1.6

14 .. i 12 .. .. :S1:i g 1 ~----------~~~~~----r-----~--~-t1~·-.-.----~----------------------~ '" 0: >- .8 !::: .., o a:d .ti > -' ~ 4 ii ~

0.1 0 4 0 5

RE L ATIVE WAVE HEIGHT 1!! --+

Ul • hAc! + Iu,l

,U,· tNc

06 0 7

Figure 28. Comparison of the maximum horizontal orbital velocity measured near the bottom with the velocity predicted by. solit ary wave theory. Ul is the absolute sum of the observed crest and preoeding trough velocities as i llustrated in t he inset ; sUI i s obtained f rom f i gure 27 by using appropriate val ues of Hl and ~1 ' These data are for selected waves of 8im~. profIle and are tabulated in appendix IB.

1.8

SERIES NO. 9 DEPTH 5 .5 FT.

1.6

1.'1

t 1.3 ::>I~

Q

08

~ 1.0~----------------~------,---~~----~------~--~--------------~ 0:

>-~

U .8 g w > -' .6 := m ~ ~

.2

0 1 0 .2

. . .

U - lu .. l+ rIU,+ U21

sU ·tNC

0 .3 0.4 0.5 0.6

RELATIVE WAVE HEIGHT ~ __

Figure 29. Compar ison of the maximum horizontal orbi~al velocity measured near the bottom with the velocity predicted by solitary wave theory. U is the abso-lute sum of the observed crest and the average of the preceding and following trough velocities as illustrated in the inset; sU is obtained from figure , 27 ';,y using appropriate values of H and h. These data ar~ for selected waves of simple profile and are tabulated in appendix IC.

0.7

1.8

1.6

1. 4

.2

0.1

1.ec-

14t-

1.2 -

SERIES NO. 12 DEPTH 12 FT.

0 .2

Figure 30.

SERIES NO. 14

. . : -: I • I :. I

: I II

, . · • I •• , I · .

.. I • .. . . . :

t · .. ~. --rf---"

~ Uc ~ I

0 .3 0.4 0 .5 H

RELAT IVE WAVE HEIGHT h -

U • IUeI+ t lu, + Uzi sU • t NC

0.6

Comparison of the maximum horizontal orbital velocity measured near the bottom with the velocity predicted by solitary wave theory. These data are for selected waves of simple profile and are tabulated in appendix ID.

DEPTH 7. 3FT

0 .7

-

-

-

-

i ~------~~--------------------~----------~----~.~~------~----~------~ :J1~1 .01

Q I-

~ .8 -

>-l-

II 61--9 ~

01 I I I I I

02 03 04 0 5 06 0.7

RELATIVE WAVE HEIGHT ~ __

Figure 31. Comparison of the maximum horizontal orbital velocity measured near the bott om with the velocity predicted by solitary wave theory . These data are for selected waves of simple profile and are tabu-l ated in appendix IE.

35

-

-

-

-

0.8

1.B

1.6

~I~ I. O!

o

!:i

SERIES NO. 15 DEPTH 8.5 FT.

~ 1 .O~------------------------------.---Lo~~----~~~~----~~---------------1 >-!:: t>

g .8 w >

~ .6 -l-t:> '3 '" > ..J

~ m a: 0

Fieure 32.

1.8

. . :. ... ••• ! . ....

1 • t \" .. ~ -.f:-----'" ~ lie ~ I

Uz

U 'Iucl + flU,'" uzi sU ,iNC

0 .3 0 .4 0 .5 0 .6

REL ATIVE WAVE HEIGHT * --Comparison of t he maximum hor izontal orbital velocity measured near the bottom ~nth the velocity predicted by solitary wave theory. These data are for selected waves of simple profile and are tabu-lated in appendix If.

0 .1

SERIES NO. 15 DF.PTH 6.5 FT

1.6

1.4

1.2

1.0 I : ! . I ' .

.8

.6 ..

.4

0 .1 0 .2 0 .3 0 .4

RELATIVE WAVE

I . t

!oJ II: 0 :x: '" z 0

w II: 0 :x: '" ... ... 0 •

0 .5

HEIGHT 2:!!..-. h,

U1• IUel + lUll

SU" ~ NC

0 .6 0 .7

Figure 33. Comparison of the maxim~~ horizontal orbital velocity measured near t he bottom with the velocity predicted by solitary wave theory. These data are based on velocities under the wave cr est and preced-ing trough; compare with figure 32 which is for·the same wave seri€s, but is based on the crest and the . average of the preceding and follo~nng trough velocities.

36

1.6 -

SERIES NO. 15 DEPTH 8 .5 FT. 1.4

t 1.2 !

• • J~ • •

1o .... ---------------- .. . .. •• •• ..:.~---------------._., o . • .1.·· . . ~ .: I., ...... .. - .- • • • ~ ~.

• >- .8 • :. • • t: •• ... .

o

~ ~ .6

~ o iii ~ :4

.2

o

o o

• o

• •• • • o

I ·0 0 •

o

o. . o

I I I I I I OL---~-~2~-~---4~-~~-~6--~---B~-~~-~10~-~--~1~2--~-~14

PERIOD , T. SEC. ~

Pigure 34. Relation between the ~atio of observed to predicted orbital velocity and the period of individual waves, series No. 15.

37

t 8

0 w 7

U IJJ III

"

8

t;: 6 ::l

~ 5 (,)

o ..J

~4 ..J

;! iii 3 a: o

2

8

SERIES NO. 12 DEPTH 7.2 FT

H· 0 .78h

2 3 4

WAVE HEIGHT H FT._

Figure 37. Relation of rnrudmum horizontal orbital velocit y to wave height, td th curves based on iry-Stol;es and solitary Nave t heory superimposed for reference. These data are for sel ected waves of simple profile and are t abulated in appendix I D.

i SERIES NO.14 DEPTH 7.3 FT. u 1 IJJ III

" ~ "- 6 ;:)

~ 5 (,)

o ..J

~ 4 ..J ;! iii 3 II: o

2

2 3

WAVE HEIGHT oH FT.--

U • l"el ~ tIUl+~1

sU = tNC U - ll!!!. -'-

Q - TSII'IH~

6

Figure 38. Relat i on of maximum horizontal orbital velocity to wave height, wit h curve s based on Airy-Stokes and solitar y wave t heory superimposed f or r eference. These dat:-. ar e for sel ected \'-laves of simple profile and are tabul~ted i n appendix IE.

39

1

1

B

t (,) 7 W III "-t:: 6

~ 5 (,)

o -J !;!: 4

m 3 Ir o

2

B

SERIES NO. 15 DEPTH B .5 FT.

I •

. :

2 3 4

WAVE HEIGHT H FT _

t lie I

U .1u.:I+t IUI+Uzl aU · t Nc JJ. ..41!lL __ I_

T SINH Z:d

5 6

Figure 39. R.el ation of maximum horizontal orbital velocity t o \,lave height , with curves based on Airy-St okes and solitary wave theory superimposed for reference. CO!Jpare with figure 40 'Whi ch is based on t he absolute sum of t he crest and pr eceding trough velocities . Thes e data are fo r selected waves of simple profile and are tabulated in appendix IF.

f SERIES NO. 15 DEPTH 8.5 FT. (,) III VI "-t-' LJ..

:5 >-~

o o -J W > 4 -J

~ iii :3 a: o

2

2 3

UI • 1,,

•

It made l ittle difference in comparing orbital ve locit ies with theory whether the observed velocities were measured from crest to pre-ceding trough, crest t o f ollowing trough, or crest to the average of the preceding and following trough velocit ie s . The scatter of pl ot ted point s was less in the latter case because the averagi ng process is a lso a smoothing process. Comparison of the different methods of presentation for series 15 are shown in figures 32, 33 , 39 and 40.

roNCLUSIONS

The analysis shows that the observed maximum horizontal orbita l velocities in general compare more favorably with velocit ies predicted from solitary wave equations than from the equat i ons of Airy and Stokes. The observed velocitie s were in better agreement with solitary wave theor y when: 1) the wave profile was not complex, 2) the effective wave period was longer than about 6 seconds , and 3) the relative wave he i ght was greater than about 0.2. While agreement with theory was s omewhat better for the longer period waves, in general it was still quite good for most s~ple waves with relative wave height s greater than about 0. 4.

ACKJIl)WLEDGMENTS

Many people on the staf f of the Scripps Inst itut i on of Oceanography and elsewhere have given free ly of their time and energy during the development of the orbital velocity meter and the succeeding f ield and laboratory investigations. The basic de sign of the orbital velocity meter follows the principle originally suggested by Carl Eckart. The construction of the meter and the design of the e lectronic components were under the supervision of James M. Snodgrass. Assistance in the field work and analysis of data was given by Jean Short, Earl Murray , and Ruth Young, and assistance with illustrations was given by Robert C. Winset t and James R. Moriarty. Valuabl e suggestions and guidance during the course of the study were cont r ibuted by Robert S. Ar thur, Walter Munk, John I saacs , J. D. Frautschy, and Francis X. Byrnes.

41

- - -------

LITERATURB CITED

Airy, Sir George . 1845 . " Tides and Waves", Encyclopaedia Metropo1itana,

Beach Erosion Board. 1933. Interim Report , Office, Chief of Engineers, U. S. Army.

Daily, J. W., and S. C. Stephan , Jr . 1952. "The Sol itary Wave". Hydrodynamics Lab., Mass. Inst. of Tech., Tech. Report No.8.

Hald , A. 1952. Statistica~ Theory with Engineering Applications. John Wiley & Sons, New York

Hamada, Tokuichi. 1951 . "Breaker s and Beach Erosion" . Transportation Technical Research Institute, Tokyo, Report No . 1

Ippen, A. T. and Gershon Kulin . istics of the Solitary Wave". ~. , Tech. Report No . 15.

1955. "Shoaling and Breaking Character-Hydrodynamics Lab. t Mass, Inst. of

Isaacs, J. D., and R. L. Wiegel . 1950. "The Thermopile Wave Meter ," Trans. Amer. Geoph. Unio~ Vol. 31, PP. 711-716

Iversen, H. W. 1952. "Laboratory S tudy of Breakers" . Gravity Waves, National Bureau of Standards, Circular 521 , PP. 9-32.

Lamb, H. 1945. Hydrodynamics 6th ed. Dover Publications . New York

Morison, J. R. , a.nd R. C. Crooke. 1953. "The Mechanics of Deep Wa ter Shallow Water, and Breaking Waves. " Beach Erosion Board, Corps of Engineers, Tech. Memo. No. 40, 14 PP.

Munk, W. H. 1949a. Sur f Problems. " 424.

"The Solitary Wave Theory and its Application to Ann. New York Acad. Sci., Vol . 51, Art. 3 , PP. ~6-

Munk, W. H. 1949b. "Sur f Beats" . Trans. Amer. Geoph. Union, Vol. 30, PP. 849-854

Putz, R. R. 1951. "Joint Variation of Wave Height and Wave Period For Ocean Swell" . Univ. of California, Inst. of Engrg. Res., Tech. Rpt. HE-1l6-328 .

Putz, R. R. 1952. "Statistical Distributions for Ocean Waves" . Trans. Arner. Geoph. Union, Vol. 33 , PP. 685-692

Shepard, p. p. , and D. L. Inman. 1950. "Nearshore Water Circulation Related to Bot tom Topography and Wave Refraction" . Trans. Amer . Geoph. Union, Vol . 31, PP. 196-212.

Stokes, G. G. 1847. "On the Theory of Oscillatory Waves" . Trans. Cambridge Philosophical Society. Vol . VIII , pp . 441-455 .

42

Wiegel, R. L., and J. D. Isaacs. 1948. First Report on the Mark V (Thermopile Wave Meter), Univ. of Calif.! DeR~ Engrg.! Tech. Rpt. HE-1l6-28 7

Wiegel, R. L. 1950. "Experimental Study of Surface Waves in Shoaling Water". Trans. Amer. Geoph. Union, Vol. 31 , PP. 377-385.

43

APP~NDIX IA - IF

Wave Properties and Horizontal Orbital Ve loci ty Measurement s

for

Selected Waves of Simple Prof ile

IA Serie s 6

IB Series 7

IC Series 9

ID Ser ies 12

IE Series 14

IF Ser ie s 15

APPENDIX IA - SERIES NO. 6.

WAVE HEIGHT TROUGH DEPTH PERIOD MAXIMUM HORIZONTALY

ORBI TAL W,T..GCITY

WAvi}! H hl h Tn Crest I Trou ,,1

Hl U c ft~~ec ft~ec NUMBER ft ft it it sec ft7~ec

10 6.6 7.6 10.1 9.9 10.0 4.8 2.8 4.2

11 8.6 6.9 9.8 10.0 10.0 3.7 4.2 2.9

18 5.6 6.6 10.2 10.1 9.0 4.6 2.0 3.5

19 7.7 7.1 9.9 10.0 10.0 3.7 3.5 3.4

11 Wave number refers to the chronological sequence of wave crests within the series. Only waves with relatively simple profiles are listed.

at Orbital veloci ties under the wave crest are positive in the direction of wave propagation (onshore) . Trough velocities are considered positive when the motion is in an offshore direction. Negative notation for trough velocity indicates current flowing onshore.

I-I

I

APPENDIX IB SERIES BO. 7

WAVE HEIGHT TROUGH DEPlB PBRIOD MAXIMUM HORIZONTAL Y

WAVEY ORBI'l'AL VELOCITY

Crest Trough NUMBER

HI H hI h Tn Uc u1 u2 rt rt rt rt sec rt/sec tt/sec rt/sec

1 4.3 8.4 6.5 4.2 1.3 1.9

2 5.5 8.2 7.0 4.7 1.9 1.8 . 3 2.8 8.6 6.0 2.9 1.8 0.8

4 3.0 8.6 6.0 3.6 0.8 2.3

7 2.2 8.7 8.0 2.9 0.8 0.4

8 1.8 8.7 6.0 3.3 0.4 1.5

9 3.0 8.6 7.0 3.6 1.5 1.5

10 3.0 8.6 8.0 2.8 1.5 0.4

13 1.6 8.8 7.C 3.0 -0.4 0.4

14 4.0 8.4 8.0 4.2 0.4 -1.2

16 1.8 8.7 5.0 I

2.9 0.4 -1.2

17 2.6 8.6 7.0 4.3 -1.2 0.4 i

19 6.2 8.1 8.0 4.9 2.2 1.9

22 4.0 8.4 7.0 4.7 0.8 1.8

23 4.8 8.3 6.5 4.5 1.8 2.0

24 4.2 8.4 8.0 4.4 2.0 0.4 I

27 3.8 8.4 6.0 4.6 0.8 1.6

28 2.4 8.6 6.5 2.3 1.6 -1.6

30 1.4 8.8 6.0 2.2 0.4 -1.6

1- 2

APP. IB - 2

VAVE H H h h T u u u NUMBER 1 1 n c 1 2

32 1.2 8.8 7.0 2.8 -1.6 1.8

33 4.8 8.3 7.5 4.5 1.8 2.9

34 3.9 8.4 6. 5 2.6 2.9 0.4

35 2.1 8.7 6. 5 3.0 0.4 0.8

36 3.6 8. 5 S.o 4.3 0.8 0.4

JIJ 1.6 8.8 4.0 1.8 0.4 -2.0

4l .46 8.9 3.5 2.3 -2.0 0.4

42 1.4 8.8 4.5 1.8 0.4 -1.6

43 1.0 8.8 6.0 2.7 -1.6 1.3

44 3.4 8.5 6.0 4.3 0.8 1.6

45 2.6 8.6 4.0 2.2 1.6 1.6

46 2.2 8.7 5.5 3.6 -1.6 1.8

47 5.8 8.1 9.5 4.7 1.8 1.5

50 3.0 8.6 6.0 3.3 1.1 1.5

51 3.1 8.6 6.5 3.2 1.6 1.1

52 2.6 8.6 6.5 3.2 0.4 -1.6

53 1.0 8.8 6.0 2.2 -1.6 0.4

54 1.9 8.7 7.5 2.4 0.4 0.4

55 2.0 8.7 9.5 2.3 , 0.4 1.8 I

57 3.0 8.6 5.5 3.2 2.2 0.4

58 2.2 8.7 6.0 3.7 0.4 0.4

59 1.6 8.8 6.5 2.9 0.4 0.4 60 3.3 8.5 5.5 3.9 0.4 0.8

1-3

APP. IB. - 3

WAVE H H h h T u u u NUMBER 1 1 n c 1 2

61 0.9 9.0 5.5 2.0 0.8 0.4

63 1.0 8.8 - 2.2 0.4 -1.4 64 0.5 8.9 8.0 2. 2 -1.4 0.8

66 1.5 8.8 5.5 2.4 0.4 -1.4

67 3.2 8.5 6.5 4.3 -0.4 0.4

68 4.8 8.2 9.5 4.8 0.8 1.8

70 2.3 8.7 7.0 3.7 0.4 0.4

71 3.0 8.6 7.0 3.6 0.4 0.4

72 2.6 8.6 7.0 3.3 0.4 0.8

73 , 4.1 8.4 7.5 4.6 0.8 1.8

75 2.2 8.7 6.0 3.1 0.8 0.4

76 0.7 8.9 6.5 2.1 0.4 0.4

77 1.0 8.8 6.0 2.0 0.4 0.4

78 1.0 8.8 4.0 2.1 0.4 0.4

79 2.4 8.6 4.0 3. 2 0.4 0.8

80 0.5 8.9 5.5 2.3 0.8 0.8

82 2.4 8.6 6.5 3.4 0.4 0.8

83 1.8 8.7 8. 5 3.3 0.8 0.8

84 3.0 8.6 , 9.0 4.1 0.8 1.8

85 3.2 8. 5 5.5 3.3 1.8 -2.0

87 3.3 8. 5 8.5 3.8 0.8 0.4

89 2.2 8.7 9.5 2.6 0.8 0.4

90 1.9 8.7 7.0 2.6 0.4 1.1

1-4

APP. ~ - 4

WAVE H H h h T 11 u u HUMBER 1 1 n c 1 2

91 3.1 8.6 7.0 4.2 1.1 0.8

92 4.2 8.4 7.5 4.6 0.8 0.8

93 4.9 8.2 9.0 4.9 1.6 0.4

94 3.3 8.5 9.5 4.7 0.4 0.4

95 1.9 8.7 9.0 2.4 0.4 0.4

96 2.0 8.7 --- 3.2 0.4 2.2 I, 97 1.2 8.8 --- 1.8 0.4 0.4 98 1.8 8.7 5.0 2.3 0.4 3.1

102 3.9 8.4 8.5 4.4 0.8 2.4

103 3.6 8.5 8.0 3.7 ?4 0.4

104 2.1 8.7 8.0 3.0 0.4 0.4

105 2.0 8.7 7.5 3.6 0.4 0.4

106 4.3 8.4 7.0 4.4 0.4 3.3

108 1.6 8.7 5.0 2.9 0.8 1.6

llO 3.3 8.5 7.5 4.0 0.8 0.4

I II 3.8 8.4 8. 5 4.1 0.4 0.4

112 2.4 8.6 8.5 3.3 0.4 0.4

ll3 1.5 8.8 5.5 2.0 0.4 0.4

114 2.0 8.7 5.5 2.4 0.4 0.8

115 2.6 8.6 6.0 3.1 0.8 0.4

116 1.5 8.8 6.0 2.9 0.4 1.6

118 2.6 8:6 11.0 2.7 1.3 0.4

120 1.0 8.8 7.5 2.3 -1.6 0.8

121 2 .. 2 8.7 6.5 3.0 0.8 0.4

1-5

APP. IB - 5

VAVE H H h h T u u u NUMBER 1 1 n c 1 2

122 1.4 8.8 5.0 2.2 0.4 0.4

123 1.6 8.8 4.5 1.9 0.4 0.8

124 2.2 8.7 6.0 2.3 0.8 1.1

125 0.7 8.9 5. 5 1.7 1.1 -1.6

126 2.3 8.6 4.0 3.0 0.4 1.9

127 1.8 8.7 5.5 2.3 1.8 1.3

129 2.4 8.6 5.0 2.4 1.5 0.4

130 1.8 8.7 5.5 2.1 0.4 0. 4

131 2.2 8.7 7.0 2.6 0.4 0.4

132 2. 0 8.7 8.0 3.2 0.4 1.3

135 2.5 8.6 5.5 2.9 1.6 -1.6

137 2.6 8.6 9.5 2.8 0.8 0. 4

143 4.2 8.4 6.0 4.5 1.8 2.7

144 5.5 8.2 7.0 4.9 2.7 1.8

145 4.2 8.4 7.5 4.2 1.8 1.3

146 4. 2 8.4' 7.0 3.7 1.3 0.4

147 1.6 8.7 7.0 2.8 0.4 0.8

I-6

APPENDIX Ie SERIES NO. 9

WAVE HEIGHT TROUGH DEPTH PERIOD MAXOOJM HORIZONTAL67

WAVEli ORBITAL VELOCITY

NUMBER Crest Troll, he H H h h T u u u 1 1 n c 1 2

rt rt rt rt sec rt/sec rt/ sec rt/sec

5 1.7 1.7 5.1 5.1 8.6 3.6 0 2.3

8 1.6 1.6 5.1 5.1 8.5 2.6 0 0

11 1.8 2.3 5.1 5.0 4.2 2.5 2.6 2.7

16 2.2 1.7 5.0 5.1 5.0 4.0 0 0

20 2.4 2. 4 4.9 4.9 4.8 4.2 2.4 0

30 1.5 1.7 5.1 5.1 4.0 3.0 -2.0 0

41 1.8 2.0 5.1 5.0 5.4 3.0 0 2.9

/JJ 1.9 1.7 5.0 5.1 5.1 2.8 2.3 -1.8

47 1.3 1.7 5.2 5.1 5.9 2.8 -1.8 0

54 1.5 2. 2 5.1 5.0 7.7 4.0 -1.9 2.5

61 --- 1.1 --- 5. 2 4.6 2.9 -2 • .3 0 64 1.8 1.1 5.1 5. 2 .3.4 3.1 0 -1.9

66 1.4 1.2 5.2 5. 2 5.1 2.9 - -0.9 79 1.9 1.9 5.0 5.1 7.5 3.8 1.0 0

86 1.7 1.6 5.1 5.1 7.5 3.6 -0.9 0

88 1.7 1.8 5.1 5.1 4.3 3.0 0 0

99 2.0 1.4 5.0 5.2 3.8 3.0 0 -2.2

102 1.8 1.5 5.1 5. 2 3 9 3.8 -1.8 -0.9

104 1.3 1.1 5.2 5.2 4 .. 2 2.8 -1.8 -2.2

1-7

APP. 10 -2

WAVE H H h h T u u u NUMBER 1 1 n c 1 2

108 2.3 1.8 4.0 5.1 5.7 3.3 3.0 -0.9

112 1.0 1.5 5.3 5.2 5.2 2.9 -2.3 4.2

118 --- 2.0 --- 5.0 5.2 3.4 -2.4 0 119 2. 5 2.6 4.9 4.9 5.0 4.0 0 0

120 2.2 2.2 5.0 5.0 5.7 3.9 0 0

125 --- 1.6 --- 5.1 3.7 3.0 -2.4 1.8 129 2.1 2.1 5.0 5.0 5.1 4.1 0 -1.9

130 1.8 1.4 5.1 5.2 4-9 3.4 -1.9 -2.2

141 1.3 1.5 5.2 5.2 8.0 2.5 0 0

147 1.4 1.5 5.2 5.2 4.1 2.4 1.0

150 3.4 3.1 4.7 4.8 6.3 4.1 0 0

152 2.3 2.1 5.0 5.0 4.2 3.2 0 0

158 2.7 2.8 4.8 4.9 5.9 3.7 0 0

159 2.3 2.0 5.0 5.0 6.2 2.9 0 0

162 2.2 2.5 I 5.0 4.9 4. 5 3.2 0 2.2

163 2.3 2.7 5.0 4.9 5.5 3.5 0 2.4

172 --- 2.6 --- 4.9 4.9 3.1 1.0 0 174 1.1 1.1 5.2 5.2 4.7 2.1 1.0 0

177 1.9 1.9 5.0 5.1 4.7 3.2 0 0 I

183 1.6 1.5 5.1 5.2 5.8 2.4 0 1.0

184 3.1 3.0 4.8 4.8 5.6 3.7 0 0

187 2.9 2.5 4.7 4.9 7.7 3.5 2.2 0

191 2.9 2.4 4.8 4.9 4.5 3.9 0 0

192 2.1 1.1 5.0 5.2 6.0 2.6 0 0

193 1.4 1.2 5.2 . 5.2 4.7 2.6 0 0

1-8

UP. Ie - 3

VAVE H H h h '1' u u u NUMBER 1 1 n Q 1 2

198 1.3 1.7 5.2 5.1 5.1 2.2 1.0 0

203 -- 2.2 - 5.0 4.6 3.4 0 0 204 1.5 1.5 5.1 5.2 5.1 2.3 0 0

205 2.1 1.7 5.0 5.1 4.2 2.8 0 1.0

2fJ9 1.5 1.7 5.1 5.1 5.6 2.2 0 0

212 1.5 1.4 5.1 5.2 5.3 2.9 0 0

216 1.3 1.6 ;.2 5.1 4.0 2.2 1.0 0

221 3.1 3.1 4.8 4.8 5.9 3.8 0 1.0

226 2.2 2.1 5.0 5.0 5.6 2.9 0 1.0

227 2.6 2.5 4.9 4.9 4.9 3.9 1.0 0

228 2.8 2.8 4.8 4.9 5.9 3.7 0 0

I 229 1.6 1.3 5.1 5.2 5.2 2.2 0 0

230 1.5 1.4 5.1 5.2 4.4 3.7 0 0

232 1.0 1.1 5.3 5.2 5.5 2.2 0 0

233 2.6 2.1 4.8 5.0 4.5 3.5 0 0

2J6 -- 1.8 --- 5.1 5.3 3.3 0 0 243 2.8 2.6 4.7 4.9 5.5 3.8 1.0 1.0

244 1.7 2.1 5.1 5.0 5.2 3.0 1.0 2.6

247 1.3 1.3 5.2 5.2 4.3 2.3 0 1.0

248 1.8 1.6 5.0 5.1 3.4 3.1 . 1.0 0

I- 9

APPENDIX ID SERms HO. 12

WAVE HEIGHT TROUGH DEPrH PERIOD MAXIMuM HORIZONT!L?7 VAVFJI

ORBITAL VELOCITY

Crest Trough NUMBER H H h h T u u u 1 1 n c 1 2

ft ft ft ft sec rt/sec ft/sec it/sec

1 2.1 2.4 6.8 6.6 5.2 1.4 1.0 1.4

2 2.2 2.4 6.7 6.6 4.9 1.8 1.4 1.3

3 2.9 2.4 6.6 6.6 7.4 2.8 1.3 0.5

4 2.2 2.4 6.7 6.6 8.0 2.0 0.8 1.9

5 2.3 1.9 6.7 6.7 6.5 1.4 1.9 0.8

11 1.2 0.9 7.0 7.0 5.6 1.3 1.3 0

14 1.6 1.5 6.9 6.8 4.1 1.6 -1.2 0.5

15 1.4 1.3 6.9 6.9 6.6 1.1 0.5 0.9

18 2.3 2.1 6.7 6.7 5.7 1.9 1.6 0.9

19 1.5 1.9 6.9 6.7 5.9 1.2 0.9 1.6

23 1.0 1.2 7.0 6.9 5.2 1.0 0.3 1.2

24 2.6 3.0 6.7 6.5 4.9 2.5 1.2 2.0

25 3.0 3.1 6.6 6.5 6.7 2.6 2.0 1.7

29 2.7 2.8 6.6 6.5 6.1 2.5 1.9 1.2

30 1.3 1.9 6.9 6.7 6.3 1.6 1.2 2.0

31 2.0 1.7 6.8 6.8 4.8 1.4 2.0 0.5

32 1.8 0.8 6.8 7.0 4.6 0.9 0.5 0.5

33 1.3 1.1 6.9 6.9 4.0 1.2 0.5 -0.6

35 1.5 1.3 6.9 6.9 4.7 0.9 1.7 0.8

1- 10

APP. l.D - 2

WAVE H H h h T u u u NUMBER 1 1 n c 1 2

36 1.0 1.2 7.0 6.9 4.9 1.2 0.8 1.0

37 1.9 1.4 6.8 6.9 6.7 1.7 1.0 0.6

41 1.4 1.5 6.9 6.8 6.2 1.1 0.8 1.2

42 I.e 2.0 6.8 6.7 5.9 1.7 1.2 1.2 43 2.3 2.7 6.7 6.8 5.0 1.8 1.2 2.2

44 3.7 3.4 6.4 6.5 4.9 3.0 2.2 1.2

47 2.4 1.6 6.7 6.8 5.3 1.9 1.7 -0.6

4S 1.8 1.0 6.8 7.0 6.2 1.3 -0.6 1.2

49 2.4 2.3 6.7 6.6 6.2 1.9 1.2 1.5

50 2.2 2.0 6.7 6.7 4.9 1.6 1.5 0.8

52 1.4 2.1 6.9 6.7 4.0 1.9 0 1.8

53 3.0 2.6 6.6 6.6 5.7 2.2 1.8 1.2

54 1.2 1.2 6.9 6.9 6.1 1.2 1.2 0.3

55 0.7 0.8 7.0 7.0 5.2 1.0 0.3 0.7

56 1.5 1.1 6.9 6.9 8.0 1.6 0.7 0.3

57 0.8 0.9 7.0 7.0 7.0 1.3 0.3 0.6

58 1.2 1.0 7.0 7.0 4.6 1.2 0.6 0.8

59 1.6 1.4 6.9 6.9 4.8 1.2 0.8 0.8

60 --- 1.5 --- 6.8 40 0 1.2 0.8 1.8 61 1.4 3.0 6.9 6.5 4.5 2.7 1.8 1.4

62 2.7 2.3 6.6 6.6 5.0 3.0 1.4 0.3

63 --- 0.5 --- 7.1 4.2 0.9 0.3 0.3 64 1.0 1.2 7.0 6.9 4.6 1.4 0.3 0.8

65 1.7 2.0 6.8 6.7 6.4 2.3 0.8 1.5

I -II

APP. ID - 3

WAVE H H h h T u u u NUMBER 1 1 n c 1 2

66 1.7 1.6 6.8 6.8 5.5 1.3 1.5 0.7

67 1.2 1.0 7.0 7.0 6.0 1.4 0.7 1.2

68 1.5 1.7 6.9 6.8 7.0 1.5 1.2 1.7

69 2.9 2.6 6.6 6.6 5.7 2.7 1.7 0.8

70 1.7 1.6 6.8 6.8 5.2 1.6 0.8 0

73 1.0 1.2 7.0 6.9 4.0 1.0 0.3 1.2

76 1.3 1.8 6.9 6.8 5.0 2.0 0 1.2

77 1.6 1.5 6.9 6.8 6.0 1.2 1.2 1.2

78 2.6 3.0 6.7 6., 5.4 2.8 1.2 2.1

81 1.3 1.2 6.9 6.9 5.1 0.9 1.2 0.3

82 1.2 1.3 7.0 6.9 4.9 1.2 0.3 0.5

83 1.0 1.1 7.0 6.9 4.1 1.0 0.5 0.5

85 1.6 1.7 6.9 6.8 4.0 2.0 0.8 0

87 2.0 1.8 6.8 6.8 5.5 1.7 1.4 1.6

88 2.7 2.0 6.6 6.7 6.9 2.5 1.6 0.3

89 1.0 1.3 . 7.0 6.9 5.9 1.3 0.3 1.3

90 2.4 3.0 6.7 6.5 5.0 3.0 1.3 2.4

91 3.4 2.7 6.5 6.6 5.3 2.6 2.4 0.8

92 1.6 1.5 6.9 6.8 4.6 1.2 0.8 0.3

93 --- 0.6 --- 7.1 3.8 0.8 0.3 0 95 1.7 1.7 6.8 6.8 4.4 1.6 0.8 0.8

% 1.0 1.0 7.0 7. 0 4.6 0.6 0.8 1.3 97 2.5 2.6 6.7 6.6 4.5 2.5 1.3 0.5 98 1.7 2.1 6.8 6.7 5.6 2.2 0.5 2.0

1-12

APP. ID - 4

WAVE H H h h T u u u NUMBER 1 1 n c 1 2

99 2.7 2.3 6.6 6.6 6.7 1.9 2.0 0

104 1.9 2.4 6.$ 6.6 5.0 2.0 0.9 1.2

105 1.5 1.2 6.9 6.9 5.0 1.2 1.2 0

106 0.9 1.2 7.0 6.9 4.8 1.5 0 1.0

107 1.4 1.1 6.9 6.9 5.6 1.1 1.0 0

10$ 1.4 1.4 6.9 6.9 5.0 1.0 0 0.5

109 1.6 2.0 6 c· . , 6.7 3.7 1.5 0.5 0.7 liO 2.9 2.6 6.6 6.6 5.1 2.0 0.7 1.4

lil 1.5 1.6 6.9 6.8 5.8 1.2 1.4 0.5

li2 2.3 2.1 6.7 6.7 5.8 2.6 0.5 1.3

li3 2.3 2.0 6.7 6.7 5.5 1.6 1.3 0.5

li4 0.8 0.8 7.0 7.0 6.9 1.2 0.5 0.9

117 1.5 1.8 6.9 6.8 5.4 1.5 0.7 2.3

118 2. 4 2.2 6.7 6.7 5.5 2.0 2.3 0.5

119 0.4 0.5 7.1 7.1 4.8 1.0 0.5 0.8

120 2.4 2.4 6.7 6.6 6.0 3.0 0.8 2.1

121 2.4 2.4 6.7 6.6 7.0 2.1 ~.1 1.4

122 1.6 1.5 6.9 e.8 6.2 1.8 1.4 0.3

123 0.8 0.6 7.0 7.1 5.3 1.0 0.3 -0.6

124 1.0 1.1 7.0 7.0 4.7 1.2 -0.6 1.4

128 2.7 2.8 6.6 6.5 5.2 2.0 1.2 2.0

129 2.8 2.9 6.6 6.5 4.9 2.1 2.0 0.9

133 2.5 2.8 6.7 6.5 5.7 2.2 1.4 1.5

134 2.1 2.1 6.8 6.7 4.2 1.5 1.5 0.7

135 1.8 2.2 6.8 6.7 4.6 1.3 0.7 1.7

1-13

APP. ID - ·5

WAVE H H h h T u u u NUMBER 1 1 n c 1 2

136 2.7 2.8 6.6 6.5 4.8 2.0 1.7 1.5

139 1.5 2.0 6.9 6.7 6.1 2.0 0.7 1.2

140 2.8 2.1 6.6 6.7 4. 5 2.2 1.2 0

142 2.9 3.5 -6.6 6.4 6.0 2.7 2.0 2.1

143 3.6 2.5 6.4 6.6 6.3 2.4 2.1 1.4

144 3.0 3.5 6.6 6.4 6.4 3.0 1.4 2.1

145 2.8 2.9 6.6 6.5 5.8 2.6 2.1 1.8

148 1.1 1.1 7.0 6.9 5.5 1.3 0.3 0.3

150 1.0 1.4 7.0 6.9 4.1 1.6 -0.6 1.6

151 I 2.7 2.6 6.6 6.6 5.0 2.7 1.6 1.7

152 3.2 3.3 6.5 6.4 5.4 2.8 1.7 1.8

153 3.0 2.5 6.6 6.6 5.6 2.3 1.8 -0.7

156 1.3 1.0 6.9 7.0 6.4 1.5 0.8 -0.3

157 0.4 0.8 7.1 7.0 6.2 1.0 -0.3 1.2

158 2.3 2.5 6.7 6.6 6.2 2.5 1.2 0.9

161 1.9 1.8 6.8 6.8 4.0 1.4 1.0 -0.6

162 1.2 1.5 6.9 6.8 4.6 1.6 -0.6 1.4

163 2.2 2.3 6.7 6.6 5.0 2.0 1.4 0.3

164 1.0 0.8 7.0 7.0 3.9 0.9 0.3 0.7

165 2.6 2.7 6.6 6.6 5.6 3.0 0.7 1.9

166 1.9 2.5 6.8 6.6 6.4 1.7 1.9 0.8

167 1.3 1.7 6.9 6.8 5.9 1.8 0.8 1.1

168 1.4 1.6 6.9 6.8 6.1 1.3 1.1 1.5

169 1.3 1.3 6.9 6.9 5.3 1.2 1.5 0.9 I ,

1-14

AW. m - 6

WAVE H H h h T u u u HUMBER 1 1 n c 1 2

170 I.) 0.8 6.9 7.0 5.6 1.) 0.8 -0.6

172 2.2 2.4 6.7 6.6 .3.9 2.6 0.5 0.9

174 1.6 2.0 6.9 6.7 5.0 2.2 1.1 0

175 1.56 1.7 6.9 6.8 4.7 2.2 0 2.4

176 1.8 2.0 6.8 6.7 5.8 2.6 2.4 -0.7

179 2.0 2.1 6.8 6.7 5.8 1.8 0 • .3 0.3

182 1.2 1.4 6.9 6.9 6.3 1.7 0 • .3 0.3

187 1.8 1.7 6.8 6.8 6.5 1.6 0 • .3 0.9

188 2.0 2.1 6.8 6.7 5.8 1.3 0.9 0.7

191 0.7 0.7 7.0 7.0 6 • .3 0.9 1 • .3 0

196 0.9 0.7 7.0 7.0 6.0 0.9 0 0

197 1.7 1.9 6.8 6.7 5.3 1.9 0 1.2

198 2.6 2.5 6.7 6.6 5.6 2.0 1.2 1.7

199 2.5 2.7 6.7 6.6 5.5 2.3 1.7 0.9

200 1.2 1 • .3 6.9 6.9 5.2 1.2 0.9 0.8

201 1.8 1.4 6.8 6.9 6.6 1.6 0.8 0

204 2.1 2. 0 6.8 6.7 3.8 1.6 1.6 -0.3

205 2.0 2.6 6.8 6.6 3.8 2 .. 6 -0 • .3 1.3

206 2.2 2.1 6.8 6.7 4.5 1.5 1 • .3 1.0

207 1.6 1.4 6.9 6.9 4.5 1.2 1.0 -0.7

210 3.2 .3.5 6.5 6.4