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SERIES 74 ISSUE 6
CfSYuüTEiFENCINCEmNG HESEARCK BERKELEY, CALiFCRNIA
•tu*"• — T— • »—»••!—••——— • •
WAVE RESEARCH LABORATORY
LABORATORY STUOIES OF DEPTH DETERMINATION
OF THE WAVE VELOCITY METHOD
BY
OSVALO SIBUL
SEPTEMBER 1953
UNIVERSITY OF CALIFORNIA
r University of California College of Engineering
Submitted under Contraot Nonr 222(17) with the Office of Naval Research
Institute of Engineering Research Wave Research Laboratory
Series 74 :*su© 6
IABORA.TORY STUDIES OP DEFTS DETERMINATION
BY THE WAVE ^LOCITT METHOD
By
Osvald Sibul
Berkeley^ California September, 1953.
TABLE OP CONSENTS
Page
I Abstract « • 1
II Introduction • 1
III Theory 2
IV Laboratory equipment and prooedure 4
V Evaluation of the data . 5
a. Uniform wave« • 5 b. Non-uniform waves 6 o. Method 1 7 d. Method 2 7 e. Method 3 7 f. Method 4 7
VI Results and discussion • 8
a. Uniform waves « 8 b. Non-uniform waves in water of constant depth . 9
1. Effeot of methods of computation 9 2. Effect of relative water depth d/L .... 9 3. Effeot of non-uniformity of the waves ... 10 4. Effect of wave steepness 10
c. Non-uniform waves on a uniformly sloping beach 11
VII Conclusions 11
VIII References 12
University of California Institute of Engineering Research
Series 74 Issue 6.
LABORATORY STUDIES OF DEPTH DETERMINATION BY THE WAVE VELOCITY METHOD
by Osvald Sibul
I. ABSTRACT
The wave velocity method of depth determination has been studied in a laboratory wave channel» The data were reoorded on movie film and were evalua- ted for wave travel diagrams (distance-time diagrams) whioh were then used to make depth predictions. Three sets of conditions were oonsideredt 1) uniform waves in water of constant depth (not used for depth determination), 2) non- uniform waves in water of constant depth and 3) non-uniform waves on a uniformly sloping beaoh. The computed results were tabulated and the predicted depth ob- tained by averaging a large number of individual values» The resulting data are presented in a series of graphs«
II. INTRODUCTION
For military operations on enomy-held beaches, it is very important to know the water depths and characteristics of the coast line» Large scale hydrographic surveys of beaoh areas are rare and the areas covered usually /c\* are small. Furthermore, changes in the beach form can take place quite rapidly . It is obvious that direct measurements of enemy-held beaches are -very difficult. Some measurements have been made by sending boats or swimmers ashore at night to make the soundings,, but this is a very dangerous task, and furthermore, there are no means of knowing exactly where the measurements were made. Although im- provements have been made recently, during World War II considerable attention was given to the problem of depth determination, and different methods were de- veloped to make indireot measurements of beaoh profiles. This report deals with the wave velocity method of depth determination. Several publications have de- scribed the application of this method ^X~*J. Photographs have been taken of a large variety of types of beaohes and evaluated for depths- The results have been satisfactory Insofar as the beach slopes were concerned; however, it was found that relatively largo errors have occurred in determining depths at in- dividual points*. The difficulties in comparing the computed depths with ac- tually measured depths may sometimes be traced back to the fact that the surf conditions often prevented soundiug3 from being taken on the same day as the aerial sorties, and the actual profile may ha-va changed considerably between the time of taking the pictures and the time that the soundings were completed.
Considering, these difficulties, and also to obtain a larger variety of beach and wave conditions, it was decided to conduct some laboratory investi- gations on the wave velocity method of depth determination. Four sets of conditions were considered»
li The velocit-i«** <">f uniform waves in constant depth of water were measured at different distances from the wave generator in order to determine the wave tank characteristics.
2. Non-uniform waves in water of constant depth, for several depths.
""* Wuxitsrs in parentheses refer to References at end of paper.
2.
3. Non-uniform waves over a beach of a constant slope of Ii40. This slope. . was chosen so as to be similar to field conditions previously analyzed.* '
4. Non-uniform waves over beaches with slopes of lill, li20 and lt40 . Each of these beaohes was used under two conditions, a) uniformly sloping without an offshore bar, and b) uniformly sloping with an offshore bar.
Evaluations of the data for the first three sets of conditions have been com- pleted and are presented in this report. The results and evaluation of the fourth oondition, however, will be presented in a separate report.
III. THEORY
If wnves are of small amplitude compared to their length (height less than 1/200 of length) anr4 dept.h of water, and are of constant length and height, the wave velocity C, in water cf oonstant depth may be written asi
C2 = (g L/ 2 7T)tanh ( Zn d/L)
or, considering also the definition for periodic waves
L = C T
Equation (l) may be rewritten as
C = (g T/ 2 IT ) tanh ( 2 IT d/L )
L - (g T2/ . 77") tanh ( 2 w d/L) or
where
(1)
(2)
(la)
(lb)
C = wave velocity in feet per seoond, L = wave length in feet (the distance from crest to crest), T = wave period in seconds (the time interval between the appear-
anoe of successive crests at the same point), d = water depth in feet, and g = acceleration of gravity in feet/second .
Equation (1) indicates that the wave velocity and length depend upon the depth of water and the wave period«, Considering also Equation (2), it is seen that the depth can be found if:
1. thö velocity C and length L are known (Equation 1), 2. the velocity C and period T are known (Equation la), and 3. the length L ana period T are known (Equation lb\
If the depth is very shallow as compared with the wave length, tanh ( 27rd/ L) approaches the value of (27rd/L) and Equation (l) becomes
. <rA (3)
As we see in Figure 1, the two functions(2 TT d/L) and tanh ( 2 7rd/L) are nearly equal for d/L z 0.04, or less.
.
At d/L = 0.05, the difference is 1% and at d/L " 0.04 the difference is only Z%* In other words, for d/L s 0.04 the wave length (and period) praotioally oease to influence the wave velooity and the depth may be determined by mea- suring only the wave velocities (Equation 3). For T = 10 seoonds, the simpli- fied Equation (3) may be used for depths approximately 5 foet or less, and T - 15 seconds when depths are less than 12 feet.
For water deeper than one-half the wave length (d/L > 0.5) tanh 3ITd/L is almost equal to 1 and Equation (l) reduoes to
C02 ' (g W z^
Co = (g T/ 2TT)
(4)
(4a)
Subscript -0 refers here to deep water conditions. In Equation (4) the wave velocity, length and period are not functions of the water depth, and so can not be used for depth determination.
For d/^>0.025 the change in wav« velooity seems to be rather insensitive to the ohange of depth, as can be seen in Figure 1, so it would be reasonable to limit the method in the laboratory studies to the region d/L< 0.25 (or the corresponding d/L0 < 0»23)o To compare the laboratory studies with prototype conditions,we know that in many localities, such as the North Sea and the Baltio, and also wherever the locality is near a storm area, wave periods of 4 seconds may be enoounterad. This means that the method may be used for depths of less than 19 feet, which seems to be sufficient for most landing operations.
For T • 3 seoonds, the method seems to be rather limited to a depth of approximately 10 feet. Considering also a relatively large error for higher d/t, values, it seems to be reasonable to limit the method for wave periods T > 4 seconds. The above named limits are set considering only the results obtained under laboratory conditions. The actual application of the method has shown, however, that no satisfactory results may be obtained for suoh high values of d/L. For nractioal purposes d/L r 0.1 may be considered as the upper limit. On many oooasions it was found that for satisfactory results the wave period T should be longer than 12 seconds. Wave crests of waves of shorter periods are scarcely distinguishable in aerial photographs.
As was previously mentioned, the wave velocity method has been applied on numerous occasions. The slopes of the beaches have been predicted satis- factorily in most cases? however, relatively large errors have been noted in the depths predicted at individual points. Thase errors can not all be traced to the accuracy of measurements; it seems rather that the theory breaks down upon ocoasions. To understand and minimize this difficulty it seems to bo necessary to list and keep firmly in mind the assumptions and definitions that have led to Stokes' theory and Equation (1). Tne assumptions are as follows»
1. the water depth is constant, 2. ti.s ?,7iV6 steepness is small, and 3. the waves are periodic and of uniform permanent form.
At first glance it appears that none of the.se conditions is fulfilled in the ocean. The first assumption (that the water depth should be constant) is almost never fulfilled when the theory is used to determine «the depths
r 4.
offshore from beaches, because the bottom is almost always sloping. Some of wave energy is reflected by the sloping bottom, but all the experiments indicate that reflections are very small and negligible when the slope is flatter than li10. The faot that the gradients of beaches were predicted very closely for flat beaches indicates also that the average wave velocities as predicted by the Stokes theory are not affeoted very much by sloping beaches.
The second assumption (that the wave steepness is small) is usually fulfilled in the ocean. However, it is preferable to use steeper waves, since the definition of the orest is always much better for steeper waves than for flat ones, and the error due to error in measurement will be reduced. Equation (1), known as Stokes' first approximation, neglects the effect of wave height. Stokes' third approximation takes this into account, that is,
(5) C - (g L/2 n) tanhU *d/L){ —-g—-?7^ -g- }
where L, C and d are as defined before, and H = wave height.
This formula, however, can not be used for depth determination, since ii is practically impossible to measure wave heights from aerial photographs taken under operational conditions. Hov/ever, one can estimate the relative wave steepness for a given photograph, and it has been the practice (particularly in Great Britain) to reduce the depths computed by 10 percent if the photographs indioate very steep waves, or by 5 percent for flatter waves'*).
Violation of the assumption that the waves are uniform seems to be the most critical. Ooean waves are always non-uniform, with the waves having different periods, lengths and height£< Thus wave forms are constantly ohanging due to their dispersive qualities* however, the changes become much less rapid in the snallow water naar the beach, as this quality depends upon the relative depth, i.e., d/L.
IV. LABORATORY EQUIPMENT AND PROCEDURE
Experiments were perfomred in a wave channel 1 foot wide, 60 feet long and S feet deep, located in the Fluid Mechanics Laboratory of the University of California, Berkeley« One side of the channel consisted of plate-glass, framed in 3 ft. x 3 ft. steel frames, through which moving pictures of the wave motion were taken. The wave generator, of the flapper type, was loca- ted at one end of the channel- Both the amplitude and period of the flapper movement were adjustable. The period of the flapper movement could be varied between approximately 0.4 second and 2 seconds. At the opposite end of the channel from the wave generator an aluminum beach was installed. The elope of the beach could bo varied from the horizontal to approximately ljlO.
Tc measure wave velocities at a particular point, two double-wire elements were mounted on a single point gage, as shown in the sketch. The distance between the two elements was approximately 1 foot. The elements were connected to a Brush Recorder and the surface time history recorded simultaneously for both elementsv'). The time required for a wave crsst to pass from element 1 to element 2 can be read very accurately from the records, and knowing the exact distance between the elements one is able
5.
H K>NT GAGE
H
DOUBLE WIRE RESISTANCE ELEMENTS DIRECTION OF
TRAVEL
to compute the wave velocity. A sample of the record is given in Pig» 3a.
The first experiments were made to determine the influence of the particular channel upon the wave motion, and to de- termine the minimum distance between the wave-generator and' the point of measurement that is require^ for steady conditions. For this purpose constant depths and uniform waves were used. The wave velocities were measured &t different distances from the wave generator (see Figure 2).
For the remaining experiments, non=uniform waves were generated by moving the flapper manually in order to vary the periods and amplitudes» To obtain a con-
tinuous record of the waves a 35 mm Bell and Howeli "Eymo" camera was used. The section covered by the camera was between 7 and 9 feet in length.
Experimentation with photography showed that clear water gave a surface line in the photographs which was difficult to read. Therefore, it was decided to cover the glass windows in the channel with tracing oaper, stretched tightly against the surface of the glass. A grid was drawn on x;he paper to obtain a scale for She evaluation of the data. When a strong light was directed to the water surface a very clearly distinguished shßdow line was obtained on the tracing paper} hence, the shadow of the water-surface profile was actually photographed. The results were satisfactory. Speoial care was taken to keep the tracing paper as tight as possible to the glass, otherwise the shadow image would not be clear and would result in erroneous readings. In order to obtain a time-scale for the measurements an electrically operated clock, graduated in 0.01 second increments, was installed in the field of view. The arrangement of the set-up is shown in Figure 4.
EVALUATION OF THE DATA
Uniform waves
The wave velocities for the uniform waves were obtained from the Brush records by measuring the time necessary for a wave crest to travel the known distance (approximately 1 foot) between the two resistance elements. The accuracy of the time measurements may be considered to be ± 2/1000 of a second, provided that the definition of the wave crest was good, as is the case,for example, in Figure 3a.
Tn many cases it is very difficult- to detarmine the exact location of a wave crest. Tne maximum elevation is not necessarily always in the miaaie of the wave, ana in addition to that it seems to shift back and forth. The change in wave shape might be considerable even at suoh a short distance as 1 foot, as demonstrated clearly in Figure ^b. At element 1 the maximum ele- vation was at the front of the wave, but upon reaching element 2 it had already shifted considerably backward. It was obvious that the maximums in
6.
Figure 3b oould not be used to determine the wave velocity and that such data should be disregarded. The wave velocities were obtained for a oarefully se- lected series of -caves and averages were oomputed for at least ten waves. The measured wave velocities were compared with theoretical velocities as oomputed by Equation (1). Results are shown in Figure 2 as the ratio of CJQ/OO against the distanoe from the wave generator, measured in wave length D/L . Here C^ is the measured wave velocity, Cc the theoretical velooity, D the dis- tance of measurement from the wave generator in feet and L the wave length.
Non-uniform waves
For the non-uniform waves it was necessary to obtain a continuous record of the wave travel, so the waves were photographed on 35 mm movie film as descri- bed above. The movies were analyzed frame by frame, each wave in the photo- graphs being assigned a number, and the time and location of those waves were measured as they advanced down the ohannel. The data obtained were plotted as wave travel diagrams (see Figures 5, 10, 15, 20, 25 and 30.)
To compute the water depth, the wave velocities, periods and lengths were measured from wave travel diagrams. The wave lengths and periods were the distanoes between the successive wave orests in distanoe and time scale, re- spectively. The velocities were obtained by measuring the slopes of the time- distance curves. For the case of a oonstant depth of water, it was relatively easy to obtain the slope of the ourvs, since the ourves could be represented by straight lines through the experimental points. The lines intercepted sometimes only through the caps, and shifts in lines (for example, in Figure 5, wave No. 18j and Figure 20, waves number 10, 12 and 16) were oaused by the transformation of the waves and shifts in the position of wave orests. Much of the difficulty was experienced in obtaining the wave velocities for a sloping beach. The lines representing the travel of wave crests Tere ourved, but the change in curvature was very small as the wave approaohed the beach. The ve- looity measurements had to be made as accurately as possible, since a rela- tively small error in velooity might cause a large error in depth. For the case of shallow water, where Equation (3) can be used, the error in depth would be the square of the error in velooity, as can easily be seen from this equation. It was found that the most reliable measurements of the ourved lines representing the travel of wave ores is were obtained by approaching the curve with a transparent ruler or triangle from the concave side until the straight edge established the best estimated tangent to this point; thus the velooity was obtained from as large as possible right triangle formed by the tangent and the length and time ordinates. The final value for the velocity was ob- tained as an average of at least five independent measurements at the piven point. The measured values are tabulated in the first part of Tables 1, 2 eto.
Before starting th6 computations, it was necessary to decide whioh combination of C, T and L values should be used in Equation (1} for depth determine.tion? lV.«re wnr« many pcjssibilities. such as using: (a) the proceeding period and length to the crest where the velocity was measured, (b) the following period and length and (c) some combination of the preoeeding and following wava periods and lengths. To demonstrate the different pos- sibilities for computation and to determine which of the methods was most
r 7.
reliable and simple to handle, it was decided to use three different metnoda to evaluate Equation (l) for the depth. For comparison the simplified Equation (3) also was used for depth determination. Here it was necessary to measure only the crest velocity, and the method was introduced as Method 4. The other three methods used were as follows.
Method 1:
'(n-iV-n
TIME, SECONDS
Loco j ion of evaluation
The depths wore oomputed using the following combinations» (a) wave period T(n-1)— n » wave length L/;i_i) —n , wave velocity C/n_]\ of the proceeding crest; (b) wave period T(n_]_} _»n and wave length L/n_j\ __n as under (a), but the velocity of the following crest 6 n . Thus, two computations were jsa.do for each wave, as shown in Table 1, 3 , etc., labeled Method 1. ^e final depth was obtained by averaging all the single re- sults.
Method 2i Kör each crest velocity CQ an average wave period Tn and Ln were computed as follows«
,„ . vi)-° ;*"-(-+!) (T)
L„ = L(°-H—i + ^ -(- -n' (e)
The values Cn > Tn .and Ln were used in Equation (l); the odisputations were com- pleted in Tables 2, 4 etc. under Method 2.
Method 3i
Method 2 is satisfactory for more or less uniform wave trains. To consider also the location of crest n as compared with ( n-1) and (n-4-1) (non-uniformity of waves) Method 3 was introduced. In this method a balanoed wave period Tn and length LQv
(linearly interpolated values and effective at the location where the velocity Cn was
measured) was computed as follows:
, - 2 T(n-lWn* ?n-(n-tl) *b * T,n_-n-n + Tn__(n + 1)
- 2L(n-l)-n xLn — (n-H) T-rrR- l»(n-l)-*a + La K"
•1 \ 1J
(9)
(10)
n> Tn^j and Ln^ were used in Equation (1). The computations The values C were completed in Tables 2, 4 etc., under Method 3.
Method 4t
The only meai3urement necessary he /a was wave velocity, and the depth was
F^ 8.
oom^uted in Tables 2, 4 etc., under Method 4, using the simplified Equation ('3^t
«. • T The computation procedure was the same for Methods 1 to 5 inclusive.
Rewriting Equation (la) we have
tanh (2 7rd/D= CQ/ 5.1Ü T = Cn / C0 (JI)
C n and T were measured, so the value for tanh(2rRd/L) could be computed at once, and referring this value to existing tables^ ' the value for d/L was obtained. With L known, the depth was easily computed.
As an example, a wavo given in Table 2 under Crest 3 will be considered. Under Method 2 it was found that the average period Tnx 0.98 second, average length Ln = 5.45 feet, and the velocity Ca • 4.43 feet/second. Therefore
CQ/ 5.12 Ta = tanh(2 77-d/L)wns found to be 5.12 4jö.96 " 0,9°3
Referring; this value to Wiegel's tables^8) page 45, we obtained d/L • 0.237; knowing Ln we have dc » 0.237 x 5.45 • 1.29 feet. Knowing also the actual depth of water, d^ « 2.00 feet, the absolute error and the percentage of error are computed as shown in Table 2. The negative sign irdicates that the depth was underestimatedj the positive sign designates overestimated depths. The distribution of errors was plotted for different methods in Figures 6-9, 11-14, 16-19, 21-24, and 26~29„
The computed wave lengths LQ » Cn Tn (CQ and TQ were measured in wave travel diagrams) and the measured wave lengths LQ also were compared in Table 1 etc. under general data, but no attempt was made to use them for the depth determination.
At first it seemed reasonable to use only that part of the data where the values of L 0 and LQ were approximately the same, but later it was dis- covered that there was no definite relationship between the error in measured wave length, Ln, as compared with the computed value, L0; hence it was decided to use only the measured wave lengths, Ln, for tho computations«
IV. RESULTS AMD DISCDSSIOH
Uniform waves
Tho first set of experiments were made using uniform waves in order to determine the characteristics of the wave channel* The results are given in Figure 2 whioh shows a comparatively wide scatter of the data a lose to the wave generator. In this region the measured wave velocities, Cm, seem to be higher than the theoretical velocities, indicating that the waves need a certain time to beoorae stabilized after they are generated by the flapper* At a distanoe of approximately three wave lengths from the generator, the wave velocity seemed to develop a constant value, with measured values approximately 5 percent less than the theoretical. There were not enough data available to maks this statement more conclusive % however, considering other observations, it seemed reasonable to maintain the points of measurement as far as possible from the generator.
)
Son-uniform w*ves in water of constant dapth
In the wave theory used, an assumption of a finite and constant depth of rmter is made and then extended to shoaling water by assuming the waves to have the same oharaeteristios in water of any depth a3 they would have in water of the same, but oonstant depth« The seoond set of experiments were made to test the validity of theory in water of constant depth* Tho experiments were made in five different depths of waten 2 feet; 1.5 feet; 1 foot; 0.633 foot and 0.253 foot with corresponding average d/Loav. values oft 0.41; 0.34; 0.33; 0.104 and 0.067. The wave travel diagrams are given in Figures 5, 10, 15, 20, and 26, and the computations completed in Tables 1 to 8.
•
Effect of methods of computations As already mentioned, the computa- tions were completed using four different methods (see page 7). Method 1 was tedious as the computations had to be repeated twice for each crest; also, there seemed to be considerably more scatter in the results than in other methods (see Figures 6, not recommended.
11, 16, 21, and 26). Because of this, the method is
f
Methods 2 and 3 seemed to be of equal value and the computations in- dicated the least scatter in results, as oan be seen in Figures 7, 8, 12, 13, 17, 18, 22, 23, 27 and 28. Method 2, which was slightly simpler to handle, is recommended for the case where the wave travel diagrams indicate a more or less uniform train of waves. Method 3 should be used in oase of highly non- uniform wave8.
Method 4. as mentioned before, should be used only for the oase whers the d/^j value is less than 0.04. To demonstrate this, all the computations were oompleted for the fourth method also, and the results oan be compared with other methods in Figures 9, 14, 19, 24 and 29. The results present little scatter, but the depths are considerably underestimated. The greater the • relative depth of water, the larger the error. At d/^,0av • 0.41 (Figure 9) the average error is - 63 percent; at d/L0 » 0.10 (Figure 24) the error is - 34 peroentj and at d/L0aT a 0.067 the error has been reduced to - 14 peroent. There are no experimental results available or smaller &/L0 values, but all the available data'9' indicate a very good agreement for small d/L0 values. Consequently, thi3 method is recommended for the region of d/L < 0.04. The advantages of this method are the simplicity of application and the fact that only one measurement - the wave velocity - is necessary.
Bffeot of relative water depth d/L: The lower the value of d/L, the less soatter in results, as can be seen in the error distribution plots. This is to be expected beoause the shallower the water, the more depth affects the wave velocity, and hence, the less the importance of phase shift and period of the non uniform waves. On the other hand, for the higher d/L values, the velocity is rather insensitive to the depth, and the curve tanh(2TTd/L)develops a flatter and flatter shape (see Figure 1), so that it is very difficult to select the proper value for d/L for experimental values of tanh(2 IT d/L) =
Cß /5.12 Tu« 'lh";8 results in au uncertain determination of depth* Thus, it- is recommended that those data be used which have the longest possible wave lengthe.
-•
• . ;>jK^M»anr: \r*r z*trjm -v-^v ./--»iai.uj.wtw r* H . ,;
ir\
Bffeet of non-uniformity of the wave»» in addition to the variation of lengths and periods, there are the phenomena ofs
a) the disappearance and reappearanoe of wave orests (see Figure 5, Crests 18-19; Figure 50, Crest 50} «to.):
b) the instability of the wave orests, so that it is not possible to de- soribe the travel of the crest by a single, well-defined ourve (Figure 20, Crest 10);
o) the shift in the crest (Figure 20, Crests 12, 16 eto.);
d) the orossing of different wave orests (Figure 30, Cresxs 21 and 22)j and
e) the breaking of waves»
All the above-named irregularities in wave shape and appearanoe seem to affeot considerably the wave characteristics as desoribed by Equation (l). To demonstrate this fact, the error in depth determination was plotted in the upper portion of the wave travel diagrams (Figures 6, 10, IS, 20 and 25) at the location of the wave where the computations were made. As the case of tanh {Zirä/l.)* CQ/ 5» 12 Tn > 1 is indeterminent, no values oould be de- termined; the blank spots in the error graphs usually are because of this factor* Large errors in computations usually can be traoed to some kind of irregularity in wave shape, as desoribed above under a) to e). The error is large, not only for the computations where the velocities of the unstable orests were used, but also for the neighboring orests, even when the travel diagram indioated a perfect crest for this wave. This oondition is very clearly demonstrated in Figure 5 at Crests 18-19, Figure 10 at crests 6-3, and Figure 20 at Orerts 10 and 12» The errors usually started a couple of orests before the unstable orest, increased to a wia-Hm«» at the looation of the unstable orest and decreased again after that«
Thus, it is recommended that ample data be obtained to assure a definite and unique interpretation of orest lines in wave travel diagrams before starting the prooedure of depth determination. All orests which in- dicate Instability in plots should be excluded from the computations. Furthermore, the crests preoeeding and following the unstable orests (even when they seem perfect) should be excluded.
Bffeot of wave steepness» There were not many data available from this phase of the oxpirimenxs to show the affect cf wave steepness on the results of depth determination. Only Run 4 was evaluated for wave h ights, and the corresponding steepnesses were plotted for eaoh wave in the upper portion of the wave travel diagram (Figure 20). However, additional experi- ments will be evaluated for this affect and the results will be given in a separate report. Comparing the errors in computation and the corresponding wave steepnesses for Crests 5 to 8 (Figare 20) (these orests being reasonably well defined and not surrounded by unstable crests), it oan be seen that the smaller steepnesses have resulted in a higher percentage of «rror* 7h* irror being always negative for the given oase, a decrease in error indioated a higher velocity of travel for steeper waves, as mentioned and demonstrated previously by Eque.tion (5). Definite conclusions oannot be stated without the support of more data.
•
I
11*
Kou-unlform waves on a uniformly eloping beaoh
The third »«it of experiments wa3 done with non-uniform waves over a uniformly sloping baach with a slope of 1*40. The wave travel diagram is shown in Figure JO, and the results as averaged from the data of 26 waves are plotted in Figure 31. The computations were completed for seven stations according to Methods 2 and 3, and were given in Tables 8 to 15. The distribution of error is given in Figures 32 to 45. In this experiment it was found that all the statements made for the non-uniform waves in water of constant depth wer? also valid for the non-uniform waves on a sloping beach.
Bffeot of methods of computation» Method 2 and 3 «rjre fcrund to be of equal va\ue, with Method 2 resulting in slightly smaller depths. But this might be of only local importance and might not be true for different oases*
gffect of relative water depth« It appears that less scatter in predicted depths occurs for the shallower depths, as can be seen in Figures 32 to 39. Figures 40 to 44, which are for relatively shallow w»**•»•, indicate an in- creasing degree of scatter again. This latter can be traced to the steadily.increas- ing number of breakers as the depth decreases. Breaking waves can not be used very well for depth determination»
Kffwot of non-upiformity of wave3t It is recommended here again to use data only where the wave "travel diagrams indicate well defined unique crest lines. Comparing the results ia Tables 8 to 14 with the wave travel diagram in Figure 30, one can see clearly that the high percentage of errors can often be traced to unstable or breaking waves. As an example. Crests 33 and 59 yield the highest error at Station 32. At Station 55 the highest error of +100 percent is encountered at Crest 40 which breaks at this station and crosses with Crest 59. Naturally a great number of large errors are due to the high &/L value (short wave lengths) where depth determination results in uncertain values.
Bffeot of wave steepness * The wave heights wore not evaluated for this run and so nothing definite can be stated. It seems, however, that the depths were overestimated for the shallower portions of the water, which indicates higher wave velocities than predicted by Equation (1). We know that the wave steepness increases as the wave moves over a sloping beach, and so the steadily increasing wave steepness might be the reason for higher wave velocities, hence, overestimation of the depth. Thus, the remark on Pa^e 4 regarding the reduotion of computed depth« for steep waves seems to be war- ranted.
VII. CÜÜCLÜ5I0KB
Depth determination by the wave velocity method gives satisfactory results for relatively shallow water in the laboratory» provided many measure- ments are made to rive a good average. The computations are relatively easy to complete and do not require highly trained personnel. However, it is ne- cessary for the operation* to be supervised by a person h»vi^g; a good under- standing of the mechanism of wave mo-.ion; it is his task to make the choice as to which data to use. The remainder of the analysis can be completed using tabular computation forms by almost any intelligent person with a mln1mum of training. The computations for this report were made by different persons without any special training; and it was found that there was no appreciable difference in results obtained by different individuals.
•...i: * ..,• •• ' ' .'. -
\\
i
.
iz.
The moot important part of the procedure is the proper ohoioe of data. It is suggested the following points should \,s kspt in siaä In msiing the ohoioei
1* The definition of the wave orests in th* wave travel diagra» should be unique and without any shift or break in line.
2. lhe neighboring ways orests to unstable orests (even when perfect in shape) should be ezeluded from the oamputation».
S. Xxolude all orests at or In the immediate neighborhood of the breaking point.
4. laves of the longest wave lengths (or periods) should be used.
5. as much date as possible should be amassed and the final result obtained by averaging ail of the single results at a given looation. The more ir- regular the waves the more data that is necessary to obtain a satisfactory average prediction of depth*
Concerning the different methods of computation, it might be said that*
1. Tor more or less uniform waves,, Method 2 may be used.
2. For highly non-uniform waves Method S seems to be the most reasonable.
5. Method 4, utilising the equation d - C2/g may b» used only where there is enough proof available that d/L < 0.04.
Till, RBFERBHCE3
1. Johnson, J.W. Wave velocity method of depth determination by aerial photo- graphs} university of California, IKE, Tech. Report 155-10, October 7, 1949.
2. Crooke, R.C. and Wiegel, R.L. Beaoh profile determination from timed vertioal aerial photographs; university of California, IBR Tech. Report 29-48, June 1961.
5. Lundahl, A.C. Underwater depth determination by aerial photographs; Fhoto- gramaetrio Engineering, vol. 14, no. 4, December 1948.
4. Williams, W.W. The determination of gradients on enemy-held beachesj The 3eographio Journal, pp. 76-93, July 1945.
6. Beaoh Erosion Board. Longshore-bar«; and longshore-troughs; BEB Teoh. Memo. Io. 15, January 195C.
5. FuehX; R.A. On the theory of short-crested oscillatory waves; Rat. Bu. Standards, Cir. 5212 (Gravity waves), Rov. 28, 1S52.
7. Morison, J.R. Measurements of heights by resistance elements; lhe Bull. Beaoh Erosion Board, vol. 3, no. 3, pp. 16-22, 1949.
„ *.-. M*«*ailfc.rf .*B! - : •. , • t i ;---. • - . .
iö.
8. Hegel, R.L« Tables of the functions of dA, and d/LQj University of California, IBR, Teoh. Report 116-265, ,*n. 30, 1948.
9.
10.
Ifoffitt, F.H. imp lysis of controlled terrestrial photographs taken at Davenport, California, for wave velooity method of depth determination; Unirersity of California, IBR, Teoh. Report 74-3, 1953.
Chien, Hing. Ripple-tank studies of depth determination by wavs velocity method from timed vertical photographs; University of California, IBR, Teoh. Report 74-5, Sepbember 1953.
DEPTH DETERMINATION TABLE I
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I 1 o» OM 3M» 4»» t M 1174 1 II »0» it I 104 0 ••• 0 114» 0 4SO o>oi 1 i* t 'ovnjiM 4 M 0 «2» | MM O-.u... ..« ._ ??••> * » OM Ml 1 04» 1 7» 4 4| il»1 • 01 1*7 IS « » 1» 0 S»7 0 1011 OIM -0 it« •SO.» I> 0>»jl42 ft *' 0 Ml | ÜA1 OAtA-OiM -31 7
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1
DEPTH DETERMINATION TABLE I
LOCATION
DAff. WAVC CHANNtX
WAV« N04-UNlFOflM
MtASUfllU w»'fH 4* 0 293 *t
9tO>£ CONSTANT DCPTM
IIHOll DATA M £ T H U D 2 METHOD 3 MFTMOO 4 G INI 1
Ml
DOT TO
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tit
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»>2 ra ft 4* «• -«• s ^4» M»V 51i\ •«• «•»-4M
m 4 *C«-*M
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S OTS QTT I» r« 1» tM •9 2/ s 3*4 0*00 OKI« 02*0 0007 27* o n 2J5 394 0«O9 Ml« ü?Ui 0 00/ 7 7« 919* 0 0*/7 302 - J
4 on DM» i.n t» 10» S 105 KM !• s 422 or» 0(900 030« 004* • 4 4 DM* > J9 4 1» 0 74 9 net U3CC 0049 • 9 4 3?*» 004* i»2 - 4
• aw • •• 10»»
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2 929
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K> a« i or )*e» 1 4» 7 3* 1 »1 2M (K CM 4 424 0*64 LI 1249 0 23* -O0)S •ft ft* o*t* .r»5 4 ir 0«M •it« 024» O007 -2 7* L1294 OLX4 2 t* - tO
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M 0» OM )»0 If» r to ICO t •* IMS 2on 4«J o«i; 3 114» 0 2.12 •0 021 «31 0*M to« 4 «O 0*1* 04* 07-9» 0 02* • • 70 0 29) -ooo* 079 - »4
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DEPTH DETERMINATION TABLfc 8
»V4I »O -^
tTtnoN . :;' LOUTIO» «AVI CHANNEL OAT» tO-K-tt
•««• MM-UtMrOtllt •US» -'-• J. C «BO II. «.Ott I 40
UNtrOftM »to«
«I H C « A I OATA MC T H00 2 METHOD 3 MKIt*Oti 1 i.l .1 HAI
m T.
UH»1M
i*4T-4 tu
•" mm
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vtLoan tm
W4B)
Lt-
VII «12 W £ «« •9-ftJTT
1 •** <•» L% 9H I», £» £ ««• «a,-*» s
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tf or» 07» 110 t M c»»o 3 97 2*2 309 37* 09*0 Ot«* 0 74C 0090 • 3 94 0 75» 254 3 7* 0*M r*« 074O OON I3M
t» as» OM on» »TO »04 »•70 39* 2 97 • 7ft 407 0*97 Cilt 093* • 0 III •7 K) 0 7»3 289 40* 0 82» 0*31 43 III • 17 1
t» OM IM i ot t M 4tO »«7» 3 ST 3 7» I 91 4 45 O •** 9 1297 0 4*7 0M33 -2ft 10 1 01« ft.« 9 19 OM« 134*1 04M •0.1*4 -1*1
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M 1 to OM 9M 4U t II • a» 149 999 4 74 9 0» OM* 0 iMI 0 433 •OUT 33 4 D979 1 9« 44* 0 770 mm 0431 0 OM • 197
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DEPTH DETERMlNATIO TABLE 9
ftON NO STMiON 55 LOCATION WAVE CHANNEL
OATE 10- 23-92
w«MS N\.'N- UNIFORM M[*9UM0 OtfTH •• 06?» II *LOM 1:40 »norm UNIFORM 510«
6(NH», DAI* METHOD 2 H£ T H 00 3 A\ III . .
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DEPTH OETEHMINATION TABLE 10
STATION 33 LOCtTiOK *».vt CHANNEL 04TI 10 23 • 32
«•VIS NON-UNIFORM MCASVMCO OCPTM *m 0 900 It. 34.0»« I 40 f>or«.C UNIFORM SLOft
SINtMl DAT* ME T H 0 0 2 METHOD Ö .•I If . • CRC3T
NO
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SO 1 ii 0 ftl 1 OS 4 17 3 04 360Ö 3« 3 »4 «0« S 42 OOftS 0I2«4 0 4«3 -0 037 -7 4 |i 03ft!s ftl 3», 0M2 [ISM 0467 0 033 • t
Si 0 31 l 24 1 073 S 23 4 23 373 3 2» 330 IM ft SO 0 392 0 lOftft 0 40S -0 093 19 0 |i049j3M 9 37 0 !"- it»7 lo40ft •0 092 .... 31 1 34 0 7ft 0 3 91 2 43 »If. 3 35 3 33 4 51 8 :l 0 ftSr 0 124ft 0 S9ft -0 104 20ft p94S'300
2 « fO ftft» 2 31
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94
30
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3 SO 2 ft»
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i 4,8 0 79< h'2< i04«8 0 033 70
i \ 1 II I 14 . irS 3 ft» 4 7ft 3 T3 Ml 4 V *C: Ofttl 0 IIftft 0 4ij 0 üi 7 • 1 4 1 170 4 lO 399 1 1
OC23 |iift2,|C4« •OLV« -4 •
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33
40
0 32 I Oft J9M 3» 2 ftft S045 3 ft« 3 3i 12/ S04 0 723 0 14*1 0 4M 0032 •1 04 3990 303 8 02 0 TiO Ui3 0432 OOM ,8
\ 1.0* I 37 1 21 3 7| 4 73 4800 3 4ft 4 1« •Oftft • 19 0 Sftft 0 lOUS 0 422 -0 07ft •13 6 1 190 4 23 • <0 1
OftftT KS9 04J4 OOftft 130
41 1.97 OTft 1 Oftft 4ftO 1 ftft ss;s 3 4ft 3 33 •0 71 5 14 Oftfti OlZftft 0 47S 0 023 so I04I 319 C 3» OftftO 1234 04O6 009S • 9 0 ; \ 42 079 Oft«
0 74
DT2
370
290
2 11
1 H
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238
t 875
3 29
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2 37
213
0 42
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Sftft
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0 09*
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.2301
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44 0 74 Oftft 3 798 lift 3 03 I89C 3 4ft 2 74 3 10 4 07 C 949 0 i»92 O&lft 0 01ft ft 9 3 7fti 290 403 0 ftftl 20- 0325 0O23 4ft \ 49 03) OftO 9 «28 1 •• 1 34 1310 3 Si 290 »«4 4 21 0 »32 0 iftOO 0 497 0 003 -Oft OMft 2 94 4 13 0 930 u» 0490 OOtO 2 0 \ 43 090 0«7 0 73ft • Tft s 00 IB 2 94 2 1« 44 T 3 r< 0 792 0 ie'i 0 32 3 0011 4« 10 730 3CJ 3 '4 0 Tft« 44» 0 3» OOSO 4 0
47 0*7 07» 373 13ft 2 M I 78 3 34 244 •IS 1 3 7J 0 993 0 23O2J 6«M 0 Oft *}T f JO 723 ? 7ft 3 71 0 900 i±* e«*4c 0l4ft 1ft 1 \ 48 0 79 0ft7 311 83« 3 ift 2 «0 3 «ft i I? '0 2 4 30 Oftl 4 0 Ifti2 [ 0 324 0 021 1 - ft ,0*4» 2*4 4 4] Oftftft •m ',)•>« 0 03ft 7 • : \
»V* 3ft9J 5 07 3 40 • 0 490 f-0 O'O •2 0 [0171 2 08 |0 4*3 O007 . 1 40
• •' • " • '-- ' ; • -• ü . .. . 1 . • ,.
OFPTH DETEIIMINUMUN T«pi r
»UN MO STATION 41' LiXATiON WAVE CHANNEL DAT! 10-23-62
»«vf» NOW-UH»''*»'
äLl**t 140 MK>r«.C UNIFORM KOPC
GENERAL C * T A ME T H 00 2 METHOD 3 M* IIIIJO •• .. 1 N 1 Nil
1 COT
*C1
moo v.-.
KPMOD V4JH>
(«Ml
MR
r«
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1.1
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. SB 1 ' °
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ML A 141 1
• tlll*A .. rr, * *i_BHV WAJ l« L M i 791
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19 0 91 2 40 334 ! 334 I 1 . 1 1
24 0« 0 43 3Tr 3M 2 04 P 9/ 3 •« tt 43 21 2 »94 1 0 403 jo 17« 0 924 0099 23» 0 749 279 3 II 04SO 4» •099O O049 tt* \ IS 0 AS 0 M 37« 1.44 3 13 ? V« 3 IS 2 40 •2 2 4 07 0 7 7ft 0 IA44 0 4K> 00O4 • 1 ib 0 7AI 2 4) 140 0404 1144 0434 00*0 134 \ 2« ON 0.41 3 7« •M 2 04 244 3 IA pr 91 I 99 404 0 7M 0 l«9A 0 417 •0004 •1 «4 0 794 2 *» 3 -4 Ott4 44» 944A 0011 4 M \ 27 0 41 l 0« UN > 49 »04 290 324 fc04 4 72 4 17 0 799 OITOO O 429 0 0 D 7A9 7 34 394 0 494 49T7 045S 0041 1 30 \ 29 ( ut OM 07« 3 V4 t A3 JW 2 97 L « II A 3 99 0 749 0 1930 0 396 • oca* »83 0 713 2 »A 4M OA44 Ml 0 4M 0004 0 71 \ 2» 0» 1 tT 0 64 i AO 4 S'3 2»r» )H 2A9 •1 2> 4 40 0 783 0 199» 0 447 0 042 9 89 D749 '?» 3A3 0 414 8W 0-Ä" OOAI Ul% \ 30 : n ,0 DM
1 II
»A4 3 TO 392 3 49 ft 7»! 7-3 Q 9ft 0 630 ü 'iAO 0 419 G0»0 | 2 39 i 078 34A »M i »4M -0 Oti •f 49 \
SI I 0 :» 137 4 00 3453 3 t9 I9QI -9 a 9 M j 0999 '"0996 0 \f T 0 099 13 A IOM 3bo 4 A3 0940 vs 039» •OOM -13 t \ )f lit 0 4? 1 01 171 7*2 1 n^ * i* 9 «>l 2 66 9 2? ' 0*02 lottot a 34b ' 0 079) •• A [>9AO 3 H 402 0 4tt H7O0SA4 •0 041 .14 1 \ ss 0» a 40 P'l 30« It ' 19 J 02 • I4 '6 4 3 45 0 832 10 i900 0 473 ! 004A| II 3 0*43 ? V 3 94 0 891 •CM 0473 0044 II .3 \ M
» 36
060 I 0 ueo 1 7» ... »7» 3 i/ t 94 7 3 4 09
', rti
0 711 O «64-» 0 448 ' 007» | 9 *i
l_ 1 , o roolf »A 3 44 0 82* %K 10448 OOtl 4 94 \
\ 1 0 1 27 * >« 3 K> 4 «4 3 87 1
5 6? [4M 9*4 0 433 JO Mft? 0.440 I t)«r*9 . 9HA 1 n9 Sit 4 IS 04SS <49 0*4* 00*4 S 7T
1? 1 2f 0 «I l 09 4M 3 12 3931 s 97 am i ifi ft 9» 0A40 0 1207 OA» O09O II 7A • OAOJ» A4 ,41 OAM 1247 04431 Q 042 .34 \ SB
3»
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0 »I 1 10 .ooft 3 ib 324 -.4M 3 90 191 1 9/ 9 14 0 682 |0 1329 O 498 | 0 033
t
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0 999J3 43
j IM 044)4 iS43|CkA4i OOS4 • 443 V
1 10 1 37 1 734 3 7» 4 44 4 199 3 36 4 >3 I OS ft 1? 0932 009« I 2T0 4 19 .» 043A 38B7 OJ8V -oon • 440
41 1 17 0 «i iO» 4 it 2 AS >M ) 14 ^ 44 ' A9 9 5H 0»9 0 HOI 0 )'>4 j 0 01' ,w i OI7 !» 34 420 0 443 1219 0401 4)0*4 -4 47 \
4* 0 AI 0 TO g njt i to 2 24 ? 4.- 3 20 J *.• 0 3 »ft 0 990 0 >8tt> 0 448 0033 i '" 079O 2 41 344 0 834 171» 0443 OOtl •*.m \ 43 0 TO OM EM* -f 2 F>4 r is 3 12 ? 12 9 i» 3 48 OMT 0 25f9 0 917 0 092 \"' OMO ,73 3 48 oni IH 09T7 004t 2I4 \ 44 OSS c /:- 202 r ic i " IM f?9« 3 94 3 86 0 B'I 0 2114 cm n mr \n. 0 f44 12 37 3 «1 0 -T- —i 0--4 0094 ISO V 4» ote OM • «TA 2 7» * 24 ? tt. 3 39 })07 • 0 1 4 A) 0 73S 0 KM O 411 0 0*4 in r>«oi 2H 4 Si 0 239 •WO«! •0 01« 144 \ 4« 0 M
OM
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Ott
1 07
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2 90
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u r+2
0 i V* 0 S7? 0 trt3 l? 4f 0 700
0 613
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229
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4« 4 -»O 0 I7i3 0 479 0 094 L m 04V 0071 IA4
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Ru-i HO ._ STATION 4 4'
• OCftllON WAVE CHANNEL 04 TC 10 23 U
Ut I ERMINATION
«».i5 tumuiurvftu »ÄAIAJ4C0 DCfTM *« 0 SSO ft
».OM I «O mi» HE UNjrOCtM store
TABLE 12
GENERAL OATA METHOD Z M E T M 0 0 3 Ul ItltJU 4 i I •» t h A i
NO
! w.s»oo V4-« ItNl
y"* Avcn uNbTM :U»CIH ' *?"! «AVt
WlOCITY
c»
It '
»fa
ml It 1\ A, 4. 4. :S LS 312 T^ Of ft % •M-4.
**,-*- i •.'•'»« *•*' **
9 | . ~
M-A lUftlf) M( 4 * K K •
f. It It hi ., r^ 4.
|M
JL *** ] 'mi mi ] |M
un*
73 096 j 2 99 3 IO
24 OM O 46 p t" 1 1 74 .94 '34 b
; 409
3 04 *«6 4 68 4 14 0 7)9 0 149' O 39i ooo» 0 29 0 7«) 2 2« 40I 0 7M aw osa OOO IM
79 06« 0 91 p 79S 1 94 2 97 ,„ i IÜ iO ) 4 02 !0 699 0 1 5C» i) \?« 1 40 0 743
0 t\T
2 52 3 91 0 714 •429 o\»i OOA 9 44
24 091 0 39 b 79 2 37 ,94 [ 194
: 974
.,. 2 7.
8 t i m • C 8)8 0 1632 om 0 049 itw ; i4 3 47 0M9 703U 0444 cow 140
77 099 1 12 U89A I 49 33G 3 TO ,m 4 54 '0 730 0 '478 O 5AI OOSi 9 86.0 775 2 37 394 0 A09 ITA« Ü47* 0 074 21 2
24 I t| 0 49 Jo 789 3M 1 39 t*r ,n 227] -8 B> 4 02 i O 720 0 1443 0 »97 O0O7 £0 Oft43 * 00 3 29 O 879 243 0437 0 047 144
"J *.W 1 l* i« .29 ! »9i I 9« 3 23 794 0 4 tt 1 0 *90 O I704 0 440 0O9O 29 7 B444 ) 90 3 92 OM7 0« 0 248 74 4
lo 1 19 • 10 i29 ,90 3 39 3479 3 17 147 4 0' % 76 '. 0 990 0U940 0 357 0 013 3 72 1 >29 5429 9 74 09AO 0337 oots -3 72
31 I 10 .0. i 099 3 3 j » 68 392 3 il WB • 0 7» 9 40 0 976 0 I049 0 »48 0 018 9 )9 ( 09* 392 9 39 0 977 «344 05A9 OO» 4 44
3? 1 04 1 04 • 023 its 306 3 i45 3 07 )»3 * 9 24 0 9A7 OK>77 O 537 OOlS 5 72 jl 121 )M 4 77 0 337 OM 0297 0093 19 19
33 t 04 094 D79 308 i 87 1 3*3 2 8* '26 >A 44 404 0 109 0 14O9 0 335 ooir 4M 0 7iO 2 16 549 0 7tt 0 346 OOA) 494)
34
39
0 94 046 Ü70 1 92 3 27 2 39*. 309 I 14 il 9 3 58 0 853 0 20'7 0 483 0 153 MO B4»3 ?06 » 39 0900 7J43 04S7 0 157 59 t
V •J «* i i' ' ^ü imü - " ; :~* ; «•; :-;-. zzt = :; O!*» 0*4, nntr vr j • HI« U »i 9 32 0M3 i«9 0440 0090 297
32 I., 0 94 i :« 4 39 5 19 3 tt 3 31 »75 107 376 O ft79 0 I042 0 393 0 043 12 3 i 094 3M 9 80 JO 991 U* U3M 0OA9 >5 7 \ 38
34
40
094 '* • 039 2 M 5 /• 1 Ü 5 54 144 5 76 9 50 0 650 n MAO 0 393 004» • 2 » i 02l> »27 9I-» 0 637 ma OJ92 0 0-11 12 0
\ i 13 1 37 i WO 3 n» 4 48 • i2 »27 4 0« 98 6 40 0 917 009OO 0 3TO 0O7O 9 U • tit 408 6 S3 0 317 394 C-r» 0022 6 SO
41 1 \7 0*3 i 10 4 M 185 »909 3 20 3 9; 04» 9 63 U964 0 lOTft 0 360 U UiO •" • 034 »79 9 30 0AO9 M9 0347 Q0I7 4M \ 41 0 IS 0 7» J r% 2 9A 2 24 Mi 3 09 f 44 I 2» < 04 0 766 O I609 1 O 3M 0O3A IOM 3 r« I 40 4 03 0 747 •612 ftttt O037 10 M \ 4S 0 76 040 0«>9 2 «4 1 94 tot 5 OO I 02 3 46 5 4ft 0 870 0 2i2i 0 444 0U94 749 0647 tot 3 41 Q>?*0 2190 0446 009& 214 \ 44 o »a 083 :i no 1 73 2 69 III »to It-i I 4» 3 66 0 879 0 2.59 1 0 477 0 iZ7 36 H D4M t II 5 47 OA97 tm 04C9 OI39 397 \ 45 0 93 1 10 <969 2C9 3 '0 =•879 3 17 iOi, f b8 4 »4 .'•64 3 01713 OM» 0 0O7 -2 0 0944 7 60 *" 0 694 G?4ft 0349 •OOO. 0 29 \ 4A
47
I 10
0 49
0 49
0 99
D 798
3 935
3 29
1 38
i 43
• 94
( 34
> 66
2 62
; 92
t M
i 56
446
6 4
4 07
? '4
0 644
1 (<49
0 1347 0 3i9
L _. ow • H
J ft37
1 991 3 47 0 r? «M 0339 C009 * 47 \ \ 1 1
4A 0 94 i 22 U9C 1 69 »t>7 ^66 3 24 : »? • 90 4 t» 0 70 5 0 1389 | 0 »TO j 0020 9 72 0 rm if 26i 4 03 |0 909 lift 0401 0094 • 9 4
\
L. .^
A*M
- 1
1
JBÜ1
1
I 7? 3 12
i
i
1 ;
| i I
0 365 | 0035 9 44 0 649|/ l>2 1) «..lit O 093 13 I
DEPTH DETERMINATION lAÖLt 10
«UN HO STAIiON 4 7' •-O'.&TiQN WAVt CHANNCL OATI 10- 33-32
*m'. NON-UNIFORM WCASUfftO DCPTH 4. 0275 fl |L(Vfl 1*0 PfloriL* UNIFORM SLOPE
GENERA L DATA V. E T M 0 0 2 MCTHOD 3 •/• .
earn NO
URIAI V4-« tMAl |M| *
wort UNOTM
IHI
WAVt LtNOTM
l„-t>«il L" c I0" • Uli :(!,]
m/in! 1
L|
•a '''' 1 - «< ». •*»
IU
T% ^ 911T«, rr^ •«. •»!'«* . - . tl OH 1 2 <• 7 9/
14 0 1» 0 60 0171 1 •• t M 2 079 2 74 it 40 199 4 46 JU61Z 01133 0 239 0 044) -14 9 0**» 20' 4 46 om MS OUT •JOSS •HI V 19 0*0 o n D79 t IS t tl ?if t 49 tl M -IS 7 3 69 O^-O 0 IttQ 0 196 0 019 •S» 079* 2 17 3 M Olli IM 0197 OOi* * I \ M on 0 6* ON 1 M I 10 1 73 3 0« 1 *9 >3i 3 33 0 920 10 792» 0 4 37 0 i*t 39 0 0*42 1 7J 3 tl 0 »94 'J44B l> l*V «* • \ II Ott 1 tl 0*0 ,» 3 90 tl» 3 13 **T en 4 61 f 0*76 ;0 ||I4 U 349 0 070 29 4 0 7IS 2 33 4 01 0 7 60 MM] OSSI 0 lit 41 l
ti it« 0 5« OK) ! 70 1 01 1 St t 13 2 t*4 »79 4 0» {0 995 10 1319 0 1*3 0 090 lit G&*0 . |} t 97 0*54 t*»t 09QE 0 2*7 It * \ '* Ott MO D 77 1 1 04 3 »9 IMS 2 It 2?* -3» 3 94 |o 734 .0 1492 0 343 0 0*1 144 0 673 1 61 ttl 0*17 4003 0*4« IM 4
\ i " ... I 06 III j Sift 3 33 »ts* t 14 3 19 •" 9 66 1 0 900 10 0074 0 264 coo» 33 1 '04 3 29 9 44 0 900 (»741 02»» 000* s 1
ii I 0« i 0» t 079; 2 It 3 t< ,3 019 2 9» 9re ist 9 90 1 0 944 10 0971 0 293 0011 69 1 07» 3 0 950 0 644 097 07» OOi* »1 \ i ss ' o» i t| t «09' 3 3< 3 09 13 •« 7 7., 303; 4 99 t> 6* i 046» 0Gft43 0 766 •0 007 2 9 1 109 1 1» 96« 0 94* -021 02TS 0060 6 2 \ M i II 0 4« 0119. ? M t SO p 74 7 9« ' 01 >6 •• 4 Ü2 0 636 !o itOl 0 379 ) 0 094 » »? St* 0 771 Mt 0462 0*77 «4 4 \ M
u M
041 0 74 0 9*9 i It 2 »9 1 7| 2*4 3 04 0 9*0 '0 3330 0 602 0 3t7 11» 0 J M> • 53 7 M \
or. 1 JO |
1 07 1 t tO 4 4a fl *••> ' 27 1
3 49' 3b2 9 43 0 961 0 1074 0 397 0061 2*1 0 9*9 2*3 4*9 0*11 mi 03*2 00*0 it 7
\ \
»7 1 40 0 I» i 1491 4 I« 7 ii aus 2 69 »£&! 7 67 9 M 0467 '0O6l7 0 79* 0023 I 4 1 ON 1*4 B 9* 0 »13 OBO oso; OGftt 11 4 \ M 01« 1 14 " 019 i 1 S3 3 69 L* 01 3 it jilt! 909 9 19 0 602 lo 11O6 e 394 00*»
... J 114 1 000 2*4 9 ü 0*IO wf» 032J 0044) 17 4 \ >• 40 4 41
i -. .. - -1 i
.r: .,1 \ ,,. .41 1 II 3 f 9 '3 »3 IIS 6 39 0 46i 0 0639 0 MB | 0 093 1» 3 1 2*4 3 67 * * 0417 0*41 0*9 0 063 1*3
41 4, 0 16 • '33 4 tl 2 99 js4i, ? 9; [3 33 2^6 3 ft lo9«9 0 0*63 0 302 ' 0 027 »1 1 0*3 3 20 5 46 0 63« 096? ouo 0065 irOO \ » ait Oil »•3 2 37 7 23 |? JO 2 17 2 4k 49 4 29 ( 0 700 -•<••(!•• 0 3<6 ; 0 043 19» DI90 230 4 29 0 TOO IMJ OS*« 0 049 19* \ 41 0*1 0 33 0«? 8 33 i 96 i 9*9 ? TR 1 99 * .3 3 43 |O «09 0 1'71 0 344 I 0 049 29 1 D640 1 »7 3 17 C649 «7i 0349 &094 94 t \ 44 0 93 0 P5 Oil | i 43 t 60 1' 949 ? M . 99 1 26 3 93 0 6>9 0 18'6 0 337 0 04c 29 6 0633 1 67 3 34 0 4%* •C64 OVl 0 104 »7* \ 41 0 «ft 1 1 7 i Oi 2 SO 3 t« J7 71 1» 2 at- 26« 9 it 0 94| 00960 0 272 0 003 [)**9 2 70 904 0 6*2 IM OtT» 0001 -0 7 \ 4« 1 17 0 4» D It 3 10 1 2» W 22 t 7) ,2?2 0 4 -9 0 643 0 ittt 0 F73 0002 0 7 0*7i 1 »3 3 43 0 '90 !K* 03« ocsr 13 4 \ • 4T
4«
0 47
0 SO
0 50
I It
D*MJ6
DU
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1 *.'
1 49 ! 39
3 ?t »99
7 •• it 3« 1
3 00 it 67
0 74
49
t 4«
4 »9
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0790
1 3«
tCM
193
IM
i i
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0 660 OlMt 0 32* '0047 If 1 om «ca 03*7 0.01t »4
AM« 1 0890 i bn 1,., r~ 1
0 326 0 03; i» 3 Ott» 2 41
j
036* a oat 30t
DEPTH DETERMINATION TABLE 14
RUN W ST4T.0N 50' uOC*'*N WAVt CHANNEL OAK 10 - 23 • 02
«*vt> NON-UNIFORM -1!U50 OfFTH .. 0 tOO ft H.OK 140 noFitl UNIFONM «LOFC
GENERAL DATA METHOD 2 ME T N CO 3 MiTtiub •* i. 1 \ 1 i. .\ 1
'JOI ML
w&oo V4-« (Ukl
mLaJ 1 •»« «AV€ 1
1 - no mi
WAVE UfiOClTT
c. Ift*ml
I r*T„i U 6 <2 T„ TS »-•» 6i2 I%
Ct t «*.*•
4t. *• j I
1. , • ** * 4H •V A lll'l
•l! M a
i m S.S\
IS 0«, jittj 09* R94A 1 31 2iOi 706
1 96 \! ' 14 0 74 2 41 209 • 94 4 49 0 644 0 0*70 0 169 ' 0039. -ITS 0 94T 1II 4 S3 0697 «X»!U» -OOi» •19 9 Y IS 09* 0 36
0 91
0670; t 31 j 1 03 i i»
0 44A1 0*4 ! 1 71 IllTS
2 09
2 »7
1 39
1 36
•2> 6
14 9
S 43 0 607 0 MIO 0 >89 . 5oil j -» 5 0 5<6
0 437
1 43
0 93
7 68
2 Si
0 7*6 •MVtlOMll 0 041 90S
IT 0 61 I 33 OBMl I «5 ; 3 66 2 66 2 94 5? 4 II 0 603 Oilli JO tV4 ; 0094 j 470 )«C9 ttt 4 .4 0 710 »413-0 391 • 0 ill 10 6 \ ta » 0.40 CMSj 34» 1 1 02 It336 2 69 2 33| -0 25 4 43 0 607 0 "20 iOt6i OOU 30 6 091« 1 60 3 16 0 999 tosaastbi oit5 • 1 6 \ 11 0 40 • to 090 1 1 02 1 3 17 [tON 2 49 1 99 -929 4 09 0 609 0 i>35 [0 736 ÜÜ« 110 O-IOol 1 64 3 07 0H3 i909J0t79' 0 079 S9XS \ M I K OLM 1 0* 2 M 1 Z 73 Is 79 t 37 ? ^( II 2 * 4£ 0 4 39 0 0748 0 206 ' 0 009 4 0 I04flt 79 & »4 0441 OT«!otit! oo«i • 0 \ 11 0 »t 1 S3 1 076 2 OS ' S »» ft *9 1 9« 306 !!99 9 30 0 320 0 0919 | 0 243 j 0 043 tl 6 1 09< 11 51 6 31 0 611 isM, 0234I OOI* 190 \ Jl > u 1 It lt95 3 19 1 t M |3I*9
__ 1 ..I _. 2 47 3 03 .31 * 6 27 0 394 0 0M3 0 20- 0 00 f 3 9
0 M66 0 204 0 0O4 t 0
1 MB 1» tt
3 3M 1 0*
0 36O|O.M
.17
1 M
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0 394
1 i*S
1 *I0
09» OSO) 0 007 36 \ M 1 St \ »4 0Ü
1 \ V \
37
a» I.M
•
I 59
0 71
.... 1 16
| II 4 II
3 47 1 2 09
3 ii
3 03
2 84
t t»
1 SO
t*i
4 29
-16 1
I 94 0 4 76
3 86 |0 316
CCS30 0 262 0O62 ' |l 0
0064» 0 196 ,0 004 -2 0
1 000.141
0 993'2 74
3 <t
6 03
0 303
0 441 or» otn U OlS
4ft 0
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0 n 1 14 0995 1 4* 3 4| K 499 2 68 2 6' 694 4 73 0 610 OM27 0 290 , 00*0 , 40 0
I 0 67T j t 09
i 4 4* 0 643 itisjots* 0O94 IT 0
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1 a*#.o«»4> 0O94 4f 0
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FIGURES I - 4
METHOD 1
gO (.
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-60 -«5 PERCENTAGE ERROR
COMPUTED 8 ß I DEPTH, ° ° ° <? ,
m FEET £ S 8 8 S 3 8 ö o ö - - - -
882§
S3
28
s a <V> CM
FIGURE 6
30
w £ 25
3 20
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PERCENTAGE- ERROR
COMPUTED DEPTH,
IN FEET
I -60
METHOD 2
15 15 JO
g 8 8 s a a o o - - - -
8 8 982? o o -' -
? 8 8 CS. CM CM
8 8 £ <\i c\i cv
FIGURE 7
& is
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PEHLtN!Äi_ ERROR
COMPUTED DEPTH,
<». FEET
ll 1
METHOD 3
1 8 ?
8 * COMPUT D DCI'Th,
IN FEFT
<S '5- ,-•
METHOD 4
FIGURE 8
c> <2 Q P o o o o ,» lA QJ f- a) oi o ..
,S 2 *- ® Oi G - P4 (JOOOo- - -
FIGURE 9
RUN I Non-uniform waves in constant depth
Depth of water d - 2 00 feet
d/L0ove> 0.414
FIGURES 5 - 9
TIME IN SECONOS FIGURE IÜ
ERNCR
COMPUTED DEPTH,
IN «ET
6 §§£§? E d 3 o u -r " "
b 3o--- t 8
FlGURt II
IS '»
I 10
POS8S?0E
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DEPTH, m FEtT
um
METHOD 3
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ft s K e Ö Ö Ö
in I § ? §
PlRCENTftGE EBROR
COMPUTED DEPTH, i« FEET
» —rTrrvm • ö METHOD 2
!" ~JrIT"_ gM ~r
L'::::ta:i.::::::: -60 -45 -30 -'5 IS 30
§ I § I § i i 1 o o o
FIGURE I?
g UJ
$ 11
10 8 s UJ a. t,
PERCENTAGE LP.P.OR
4- ii i in
METHOD 4
--
i
DEPTH, IN FEET
FIGURE 13
-60 _ Sm O 1"
W Q fn o «-» O ^
fl 8 SO s M Q h- y> H 2 ü ^ c o o o
45 -30 -15
FIGURE 14
RUN 2 Non-uniform waves in constant depth
Depth of water a • I 50 feet
d/Loove • 034
FIGURES 10 - 14
ts a 10
TIME, IN SECONDS
14 16
FIGURE 15 M -T—1
] K 60
S-o l
3 >s I
r-
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§ z0 h P i
h is m
1. — - •' f
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:| i 2 *° s io i •
' 1
COMPUTED DEPTH,
IN FEET
-4ft -30 -IS 0 IS » 45 PERCENTAGE ERROR
COMPUTEO DEPTH,
IN FEET
-45 -30 -IS 0 IS 30 46
i. 88 S ü 3 O O O O "V -
i $i 8 § *5 O ö o —
FIGURE 16
1 88 ? 1 ii §
FIGURE 17
to 40
36
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8 29
ft 20 UJ & < is z
s io
1 IETHOD 3
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6
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rj£pyu
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COMPUTED
IN FEET
-SO -45 -50 -IS 0 15 30
8 8? ?
1 - - - FIGURE IE
L> t> Ö O O
# if <n iq a o o ö ö o
FIGURE 19
RUN 3 Non-uniform wo <es ;n constant depth Depth of wafer d « I 00 foot
d/L0ovt» 0 335
FIGURES 15 - 19
TIME. IN SECONDS FIGURE 20
I" * 5
METHOD I
1
r
ji[r__r -rrlll li — — rttlll ilnr r r r
-30 •ia PERCENT** ERROR
«««•>ä!8!SS!!l I DEPTH. 09OOO9O&O O
IN FEET liiilfl«! 8 oooöoooöd b
1 I
FIGURE Pj 55
W 30
I e& 8 20 6
a '« Is 10
«• 5
ocnrruTuv
1 METHOD 3 |
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35
8 » UJ I 25
5 20 b ui 15 3
B 5
PEPTtNTftOE ERROR
COMPUTED DEPTH, IN PCCT
I I
L
II
1 1
r Iffl
METHOD 2
8 5 8JJJ s 00 0000 Ö
ii iiii i 00 u o o ö b
FIGURE 22 35r
JC
2D-
20-
PERCENTAGE
M METHOD 4
I I I f 1 L 1 !" 1 111" JJLJLI t
COMPUTEO DEPTH, IN FEET
ISSSIII o ö o o o o o
Splfiis o o o o o o o
FIGURE 23
0000 b 09 COMPUTED
OEPTH, OO00 O 09
FIGURE 24
RUN 4 Non-uniform waves in constant depth Depth of wottr d • 0.533 foot , d/L0
0.104
FIGURES 20- 24
10 12 14
TIME, IN SECONDS
»i i i i i i i i i i i i i i • --n 1 METHOD 1
ä!50 !
S
1 r r- I I _T1| | [ Jr. f] 1 _L '
rr nil F brfl I TH m ^ftfcJÜf* -45 -JO 15 0 15 SO 45 60
COMPUTED IS 1 ilSS OCPTM. 9 9 9 9 9 d 9 * FEET S £ E I |j S S)
I ss Ö gg §
5SBS $ ss i do o
FIGURE 26
pEss$ra
COMPUTED DEPTH, IN FEET
J METHOD 2
23° 1 § 25 __ U - - 4- 1 o
1 '• 1-
f ,._ .._ ..! rjr_i ... _I_„_i]i;L. J1 [
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o o 9 d d d d d
<v N w CJ wtS N 5 : N K » ° o d d
FIGURE 27 so
25
20
b IS
9 I10
2 •
ERROH
COMPUTED DEPTH,
IN FEET
rA- METHOD 3 1
• ST
1 ::_:„._ ! JC.__:::j 1! 1 Iltl !
SS S S MS 9 d d o d ° o
|s i £ ss* ° S o o b o ° FIGURE 28
O 15
w J 10 z ui
8 5 If
COMPUTED DEPTH, IN FEET
^P
1 METHOD 4
o o o
5 8 g c 3 -
ft w (^J A o 0 o o
& 9 f; en 5) (Cj Kj (\j O O o
F-"JGor<F. ?9
RUN 5 Non-uniform waves in constant depth
Depth of water d • 0 2 53 foot c'/Lo,,,.1 0.067
FIGU'ttS 25
WAVE TRAVEL DIAGRAM
NON-UNIFORM WAVES ON SLOPING (I 40) BEACH FIGUI<I 30
r
133d Nl *«31VM 30 H±d3Q
HYD-676S
FIGURE 31
50
:: IM
METHOD 2
FIGURE 32
_ SUSIS S ! 51 COMPUTED boooooo o oo
OCPTH, * • * S* fi • * 9 jf 1*4 FttT 9 f I * * * * S i.
OOOOOOO o oo
i 1111 I I II JOr
«SB» Station 32' Depth dm • 0 650 foot d/U-,'0 168
1 | «it i riöü 2
FIGURE 35
j
ll T 1 _j -
! ! f 1 l r i
OOOOOOO o
ooooooO O ES
1 METHOO 3
FIGURE 36
1 1 II
•TJ
-IT L - 4- • -J ] ij ".ifiTtt:"'" .ill—[ JI__.I
-15
stotion 35' Depth dm> 0.873 foot
d/Lc ' 0 148
90 45 «0
1 I i BO II 1
! i i i i ii i i i m -- - - - 1 rl 111 1 rt,MU0 *
!!!! !!!!!!; >••*?
i
r r
._ _f i J:._L .___[ 6393 5 US S o o o o o o d o
OOOOOOO o
|METHOO 3
FIGURE 38
_T 11
j. L Tr i -! Lu. 1 r j rr ^ ±:::.ii .o::::::::: -90 -IS C 19 30 48 (O
Stotion 38'
Depth dm ' 0 500 foot d/Lo«, -0 129
8*8 I l»Sa O Ö Ö o OOOO
^ o ö o o ö c o
COWPUTIO OCPTH, IN
Station 41
Depth dm • 0 425 foot
1 ME THOD 2
J_ FIGURE 40
• r j r
'-l~i±z O OOOOOOO O o
b ooooooo o o
METHOD 3
•15 MM
FIGUPE 41
J 15 30 45 75
Stotion 44
Depth dm • 0 350 loot
I U ' 1-OlM« ' ^ W^W
2 3 S 3 3 ä H * £ I J I bo o oooooooo o ° o o o
U § ssMisM li ^2 Oo o oooooooo ö q ooo
MINI M I
Station 47'
Depth dm' 0 275 foot
j METHOD 2
FIGURE 44
___ ::i:;:::::::j M:_ _ it ._„____. n iffir [it:::::
> 1 1 1II I III I I S SSS S 3£ 3SS % * b böboo6ooo6n ö o
b Ooooooböooo o b
1 MUM II METHOD 3
FIGURE 45
1 1 ' j
i::tL:[iidi::dl 15 SO 45 60
Siotion 50'
Depth dm= 0 200 fool
< IGUKf.S 3.' • 'lb
DISTRTHTTTON LTST
UNCLASSIFIED TECHNICAL REPORTS
Copies.
13
Addresses. Copios
1
Addressee
Director Institute of Engineering Researoh University of California' Berkeley 4, Calif.
Chief of Naval Operations Navy Dept, Washington 25, D.C. Attnr 0p-533D.
Geophyslos Branoh Code 416 Offloe of Naval Researoh Washington 25, D.C.
Director, Naval Research Lab. Attn» Tech. Information Offioer Washington 25, DaC.
Offi oer-in-Char ga, Offioe of Naval Researoh, London Br. Offioe, Navy #100 PPO, New York, N.Y.
Office of Naval Researoh, Br. Offioe 346 Broadway, New York, 13. N.Y.
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Offioe of Naval Researoh, Br. Office 1030 Bast Green St. Pasadena 1, Calif.
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ONR Resident Representative University of California Berkeley, Calif
Offioe of Technical Services T)«p+.„ of Crmimaroe. Washington 25, D.C.
Armed Services Tech. Information Center 1 Documents Service Center Knott Bldg, Dayton 2, Ohio
Asst. Secretary of Defense for Research and Development Attnt Ccmm. on Geophysics &
Geography Pentagon Bldg, rfashington 25,D.C.
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Commander, Naval Ordnance Lab. »Yhite Oak Silver Springs 19, Md.
Commanding General, Res. & Devel. Dept of the Air Force Washington 25, D.C.
Chief of Naval Research Navy Dept. Washington 25, D.C. Code 466
U.S. Navy Hydrographie Office Washington 25, D.C. Attn» Div. of Oceanography
Direotor, U.S. Navy, Electronics Laboratory San Diego 52, Calif AttntCodes 550, 552
Chief, Bureau of Yards and Docks Navy Department Washington 25, D.C.
Commanding jeneral Research and Develop!ient Div. Dept of the Army Washington £5, D.Cc
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5
Copies Addressee
1 National Research Covmoil 2101 Constitution Ave. 'Washington 25 D.C. Attnt Comm. on Undorsea Warfare
1 Project AROWA, U.S. »aval Air Station BldS R-48 .Norfolk, Virginia
1 Dept of Aerology U.S. Naval Post Graduate School Monterey, Calif.
1 Commandant (0A0), U.S. Coast Guard 1300 B. St. N.W. Washington 25,D.C.
1 Direotor, U.S. Coast & Geodetic Survey Dept of Commerce Washington 25, D.C.
1 U.S. Army, Beaoh Erosion Board 5201 Little Falls Rd. N.W. Washington 16, D.C.
1 Dept of the Army, Office of the Chief of Engrs. AttnjLibrary. Washington 25, D.C.
1 Mr. A.L. Coohran Chief, Hydrology and Hydr. Branch Chief of Engineers Gravelly Point Washington, D.C.
1 Commanding Officer U.S. Naval CE Research & Evaluation Lab Construction Battalion Center Port Hueneme, Calif.
1 Dept. of Engineering, University of Calif. Berkeley, Calif.
1 The Ooeanographic Inst. Florida State University Trilahassee, Florida
1 Head, Dept of Oceanography University of Washington Seattle, Washington
Copies Assressee
1 Bingham Ooeanographic foundation Yele University, New Haven, Connecticut
1 Dept - Conservation Cornell University Ithaoa, N.Y. Attnj Dr. J. AyerB
1 Direotor, Lamont Geological Observatory Torrey Cliff, Palisades, N.Y.
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1 Direotor Chesapeake Bay Inst. Box 426A RFD #2 Annapolis, Md0
1 Head, Dept. of Oceanography Texas A it M College College Station, Texas
1 Dr. Willard J. Pierson New York University University Heights New York 53, N.Y.
1 Direotor Hawaii Marine Lab. University of Hawaii Honolulu, T.H.
Director Marine Lab. University of Miami Coral Gables, Fla.
Head, Dept of Ooeanography Brown University Providence, R.I.
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2 Direotor Woods Hole Ooeanographio last« Woods Hole, Mass.
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1 Chairman, Ship to Shore Continuing Bd. U.S. Atlantio Fleet Commander Amphibious Group 2 FPO, N.T., N.Y.
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1 Dr. M. St. Dennis David Taylor Model Basin Navy Dept. Washington 25, D.C.
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Missouri River Division Corps of Engrs. P.O. Box 1216 Omaha 1, Nebraska
Commandant of Marine Corps School Quantioo, Va«
Sir Claude Inglis, CIE Dir. of Hydraulios Research % Office of Naval Res. Br. Offioe Navy No 100, FPO. N.Y., N.Y.
L Gommandant Hq. Marine Corps R-4 Rm 2131 Arlington Annex Washington D.C, Attnj Lt-Col. H.H. Riohe
Chief Andrews Air Fcroe Base Washington 25, D.C. Attn» lar. Stone.
! U.S. Army Transportation Corps Researoh and Development Fort Eustie, Va. Attag Mr. J.R. Cloyd
5 British Joint Servioes Mission Main Navy Bldg Washington 25, D*C.
L California Academy of Soienoes UU.LUOU USW »tun.
San Francisco, Calif. Attnt Dr, R.C. Miller
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