1 DEPARTMENT OF THE ARMY CORPS OF ENGINEERS

* / BEACH EROSION BOARD OFFICE OF THE CHIEF OF ENGINEERS

WIND SET-UP AND WAVES IN SHALLOW WATER

TECHNICAL MEMORANDUM NO. 27

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Bureau of Reclamation Denver, Colorado BUREAU OF RECLAMATION DENVER LIBRARY 92099798

92099798 WIND SET-UP AND WAVES IN SHALLOW WATER

TECHNICAL MEMORANDUM NO. 27 BEACH EROSION BOARD CORPS OF ENGINEERS

JUNE 1952 FOREWORD

This paper presents the results of an analysis of certain wind, wavej and water layel data obtained in Lake Okeechobee, Florida, during the passage of two hurricanes, one in August 1949 and the other in October 1950. The purpose of the analysis was to relate wind set-up and wave, heights ob­ served in Lake Okeechobee to wind velocity, fetch, water depth, and surface shape of the lake.

The data used as the basis of this paper were collected by the Jackson­ ville District, of the Corps of Engineers. The planning of the observational program and the development, modification, and maintenance of field instruments represented a joint effort of the Hydraulic and Hydrology Branch of the Office of Chief of Engineers, the Washington District of the Corps of Engineers, the Beach Erosion Board, the Jacksonville District of the Corps of Engineers, and U. S. Weather Bureau. The basic data collected at Lake Okeechobee and serv­ ing as the basis of the present memorandum are set forth in Project Bulletin No. 2, Waves and Wind Tides in Inland Waters, distributed by the Jacksonville District of the Corps of Engineers in June 1950 and containing the data for the August 194-9 storm, and are, as yet, undistributed project bulletin con­ taining similar data for the storm of October 1950.

The present paper was prepared by Thorndike Saville, Jr., of the Research Division of the Beach Erosion Board, who analyzed the data, and prepared the report. Mr. Saville was assisted in the computations and plotting by George Simmons, now of the U, S. Army.

The opinions and conclusions expressed by the author are not necessarily those of the Board, TABLE OF CONTENTS

Page

Frontispiece - Figure 1......

P art 1 - WINB SET-UP ...... !

Figures 2 thru 9 ...... 10

T ables 1 thru 6 ...... 17

P a rt 2 - WAVES IN SHALLOW WATER ...... 21

F igu res 10a thru 1 8 ...... 26

References ...... 36 FIGURE .1 111© SET-UP AND WAVES IN SHALLOW WATERS

PART ONE - WIND SiT-UP

I* Introduction. A program of observation and study of wind set-up and wav® action in inland waters has been undertaken by the Corps of Engineers as a part ©f a general program of engineering investigations re­ lated to the design,, construction, and operation of flood control and navigation projects involving levees and channel improvements on lakes and major reservoirs* As an initial step in these studies, special ob­ servation stations have been established in Lake Okeechobee, Florida, in Fort Peck Reservoir, Montana, and in Denison Reservoir, Oklahoma* It is possible that this program may be supplemented by observations on other reservoirs, lakes, and rivers at a later date. The findings from these studies will be correlated with certain laboratory studies and special ■investigations, such as the determination of the effect of embankment slopes on height of wave run-up and study of slope protection requirements. This paper deals with the development of the relation between wind velocity and duration and the resulting wind set-up (sometimes known as wind tide)'in shallow inland waters based on data from Lake Okeechobee.

2. Winds, waves, and water surface elevations have been recorded for several years at a number of stations on Lake Okeechobee* The records include the two major hurricanes which occurred on 26-27 August 194-9 and 17-18 October 1950* This report deals .primarily with the analysis of the data f r o m these two storms. locations of gages, and the type ©f data ob­ tained from each, are indicated on the topographic map of the lake (figure 1).

3. Theoretical Considerations. Theoretical developments of a wind set-up equation have been made by lellstrom (1), leulegan (2) and Thijsse, among others« The mechanics of the various determinations have differed somewhat, but the resultant equations have been essentially the same, giving

d h „ t\T s or ' S * nF% d x PgD p g 0 , where h * surface elevationj x « horizontal coordinate! d /4x * slop® w a t e r surface! % *» surface shear stressj g = acceleration due to gravity! P - density of water! D * depth with lake surface horizontal! n - a coefficient defined as n * -jf - - + 1 w h e r e T0 is the shear stress along the bottom. This coefficient0 has a value ©f l.f for laminar flow, and for turbulent flow will depend on the particular theory adopted*

1 Hellstrom, adopting the Boussinesq theory for turbulent flow, finds n to vary between 1*30 and 1*15 for moderate to large depths. leulegan assumes an average value of n * 1.2$ for his laboratory tests;

S * set-up, expressed as the difference in water surface elevations at windward and leeward sides of the lake; F * fetch length, oar distance between points of maximum and minimum water surface elevation.

4« The important assumptions involved in obtaining this equation are that it be for a steady state, equilibrium case with the set-up developed by the action of the wind alone, and that the set-up be small compared to the depth.

5« The surface shear is generally expressed as a function of the wind velocity, % = KPaV* where H * a numerical constant ^ . 0 0 3 * air* density V =» wind velocity KnP* VZF Substituting this in the equation for set-up, them S = PgD 6. Frequently the fetch direction, the line connecting the points of minimum and maximum water surface elevation, does not coincide with the average wind direction. In these eases the use of a resultant shear force in the direction of the fetch is generally deemed best, where the resultant force te)=7s<,os & if 0 denotes the angle between the wind and the fetch* It is thought that, since it is the shear force that is producing the set-up •the set-up is determined by a resultant force in the direction of the fetch rather than by a resultant velocity which might be said to be producing the shear force. Hence the cosine term enters the equation to the first power rather than the second power, and the set-up equation becomes 5 = Cos 0 PgD 7. It may be assumed that the set-up at a particular spot maybe modified by the topographic configuration of the lake as it approaches that point or that a convergent effect would occur where the wind blows from a wider portion of the lake towards a narrower portion causing the actual set-up to be higher than that expected; similarly a divergence might be expected if the wind direction were reversed. This convergence or divergence, generally would be of a very minor magnitude, but might reach an appreciable value in the case of a triangular shaped lake with the wind blowing toward (or away from) an apex. Hellstrcm (1) has mentioned this probability, and personnel of the Jacksonville District of the Corps of Engineers nave devised a method of computing this convergence or divergence based on the ratio of the width at the leeward end of the lake to the average

2 width (3)* However, when the leeward end of the lake narrows to or nearly to a point so that there is little or n© width, the method breaks down« If this convergence factor is denoted by I, then the final set-up equation b e comes 8 Co s & > "where A f * The term A will be approximately constant for all conditions.

8« It is obvious that steady state conditions generally do not exist over a lake, as the wind velocity and direction vary at different points across the lake, and also change with time at each point. As the wind direction changes, the fetch and average depth along the fetch also change« The primary difficulty involved, therefore, in the application of the set- tap equation to open bodies of water, is the choice of suitable average values for the parameters involved so that-the average values selected will adequately represent the cumulative effect of the actual values.

9« Method of Computation« There appear to be two different concepts for determining average values which might be utilized for each parameter at any on® time. 0ne of the se is based on the assumption that the action of each strip of water is influenced primarily by the wind and the effect of the bottom in that strip alone, and is essentially independent of the wind and bottom forces acting, oh other areas of the lakef this would- 3ead ' to average values being obtained from the actual value® observed along the fetch alone« The other is based on the assumption that the set-up at any one point is due to the action of the entire water mass as a whole, and, as such, is not produced by the wind and bottom forces acting along the fetch alone, but is a cumulative effeet of all these forces acting on the entire body of water« In this case, average values would be computed from the actual values observed over the entire lake surface. Both of these averages have been computed.

10. The average depth over the entire lake is obtained by dividing the volume of water in the lake by the surface area, and will be constant for' any one storm if "the inflow and outflow of the 3a ke are assumed to be negligible or compensating. The average depth over.the fetch is obtained by dividing the cross-sectional area along the fetch by the fetch length, and will change with time as .the 'fetch changes. Both methods assume that the actual lake bottom may be replaced by a hypothetical level bottom suck that, in the case of average lake depths, the volume of the lake remains the same, and, in the ease, of average fetch, depths, the cross-sectional a r e a u n d e r the fet c h is the same-» .

11» . To determine the average wind velocities, the hydrometeorological section of the Weather Bureau analysed the anemometer record ©torts for ten-minute average velocities throughout the storaj these were then plotted at each station on a series of lake maps, and lines of equal velocity' (isovels) and vectors showing direetion were drawn on each map. These maps were drawn at one hour intervals during the greater part of the storm, and at half-hour intervals during the more critical portions of the storm. The average velocity over the entire lake surface was obtained by planimeter,

3 and since the velocity appears as a squared term in the set-up equation, the average value was assumed to be given by the equation where Vx is the .velocity appropriate to the planimetered area Ax* The averase velocity over the fetch was obtained from the actuation v - V f , + + where ?x 3-s the velocity appropriate to the short segment, Fx > of the fetch. The average, or resultant, direction of the wind was ob­ tained by visual inspection of the flow pattern, and the angle between this average direction and the fetch obtained by protraetor.

12. It is evident that the set-up produced is the result of the wind forces acting over a certain period of time and that it takes the lake a certain definite time Interval to fully react to the forces applied. Study of the differences in times of definite changes in the wind and the corresponding lake level patterns show the lake to have approximately a one-hour lag behind the wind* TldLs is in agreement with conclusions reached by Haurwitz (4) and the personnel of the Jacksonville District for this same storm period. Hellstrom gives an approximate method for computing the time necessary for equilibrium conditions to be reached if a lake, initially with a level surface, is subjected to a wind of constant velocity* Applying this method to Lake.Okeechobee, for a fetch of 35 miles and a wind speed of 60 miles pe.r hour, it would take about 1,6 hours to approach an equilibrium state (i.e. when 9 5 % of the energy required for steady motion has been developed). This figure also is in general agreement with those given by Proudman and Doodson (5).

13. . In the case of Lake Okeechobee, however, the wind is a gradually, increasing one, rather than one of constant speed, so that for any one value of wind speed the initial lake level would already be partly deformed, and the corresponding time to reach an equilibrium state would be less than that computed. Hence the average effective values producing set-up at any one time have been computed as averages over the preceeding hour.

14. Comparison of Theory and Observations, The factor AN in the set­ up equation, computed by substituting average values from the data on the 1949 storm in the set-up equation, is found to be remarkably constant at a value of approximately 3*4 x 10”^, the extreme limits being 3.14 x 10”° and 3.45 x 10“« (see table 1). Keulegan’s experimental work at the Bureau of Standards has indicated that, at least for models in small closed channels, the value of A should be about 3.3 x 10~6. The agreement between these two values is notable, particularly in view of the greatly different conditions involved in the two cases. If Keulegan's value is accepted as the correct one, then the convergence factor N equals 1.03* This is an extremely small value, particularly in view of the fact that the maximum rise in water surface elevation, occurs essentially in the apex of a triangular embayment, and therefore it might be expected that the convergence, or shape factor, would be large although volume computations which would also take into account the extreme narrowness of the upwind end of the lake tend to show a convergence factor of the order of only 1,01 - 1,02. Small though it is,

4 however, with the maximum set-up reaching almost 19 feet it makes a difference of 0*55 feet in the predicted maximum value* With such a qm«n value, percentagewise at least, it is difficult to say definitely that convergence does, or does not, occur, since errors of several per cent may be introduced in the computation of the appropriate average effective values* These errors have been eliminated as much as possible by having the values obtained independently by two different computers. In most cases these values were almost identical, but in some cases differences of as much as two miles per hour in the computations of wind velocity resulted* In all eases, an average of the two values obtained by the different computers was taken as more closely approximating the true value* 15* . Curves comparing the observed values of set-up (So)in the 1949 storm with those predicted by the formula using values of the parameters developed over the entire iate are shown in figure 2* Shown also are curves showing the set-up predicted if the convergence or cosine © terms are omitted, as well as one showing values obtained if no time average, is introduced and the water surface is assumed to reactcompletely and instantaneously to the forces applied* The values of the parameters used are tabulated in Table 1* The validity of the use of time average values of the parameters is demonstrated quite clearly by the plots on Figure 2. Also the choice of a one-hour period for averaging the para­ meter values was apparently justified, as the observed and predicted curves are in close agreement, at least for that part of the storm be­ fore the wind shift* Actually, observed set-up values obtained after the wind shift (of about 180°) are not strictly comparable with values pre­ dicted by using the set-up equation as steady state conditions are no longer approximated* The set-up at that time is due not only to the wind forces, but also to forces set up by the rotational effect of the wind and the water, and to a seiche action. Haurwitz (4) has shown that, in such a case, the set-up may be grdatly different from the steady state condition, depending on the relation between the seiche period bf the lake and the speed of rotation of the storm. Before the wind shift occurs, however, steady state conditions are approximated, and the agreement of predicted and observed values is much better than might be expected, showing a maximum difference of about 4 per cent, 16* The agreement at the higher values of set-up is especially noteworthy as the basic assumption that the set—up is small compared to the depti is no longer valid* Perhaps the continued agreement in this range may be partly explained by the use of average values of depth over the lake. If h denotes the water surface displacement at any point, then the average value of D * h is very nearly the same as the average value of D used in the equation, provided the surface area of the lake does not change appreciably* Then the substitution of the more correct term (D + h) for D would not change the results significantly* 17* Curves for the 1949 storm showing predictions based on values averaged over the fetch rather than over the lake are given in figure 3. Values of the parameters are tabulated in Table 2« It may be seen that 5 although the agreement between predicted and observed values of set-up is not bad, it is not nearly as good as that obtained when average late values were used. The closest agreement occurs when the cosine © term is neglected, which would tend to indicate that the same set-up will occur regardless of the angle of the wind with the fetch* The same disagreement in values after the wind shift is obtained here as was observed with the late average values.

18. Possible Method for Correcting for Marsh Effect. Similar com­ putations have been made for the 1950 storm and are shown in figures 4 (lake average values) and 5 (fetch average values); the parameters are tabulated in tables 3 and 4. The direction of the wind was such that during most of the first part of the storm the rise in water elevation occurred over a marshy area covered with thick grasses and other vegetation. This vegetation produces a damping effect on the set-up reducing the total set­ up due to the increased friction losses occurring as the water moves through the grasses, and to the reduction in surface shear between the water and the wind for those times when the grasses protrude above the water surface* This reduction in set-^up is clearly shown by the curves. It may be supposed that a correction factor , -Cm, can be applied to the predicted wet-up,' S, to obtain the observed set-up S©, such that S© * SCl-Gm). In general, Gm w i | . l decrease with some function of the depth over the marshy area which can probably be expressed dimensionlessly as a ratio of the depth, or the set-up, and some friction length parameter of the marsh vegetation. This coirection factor Cm. has been determined from the data and plotted against the predicted values ©f set-up (as for practical use it is more convenient to use the predicted values than the observed). These plots are shown in figure 6 (lake average values ) and figure 7 (fetch average values) Since only the one storm and area were involved, no attempt could be made to determine the friction length parameter approximate to these grasses. A sharp break in the curve is observed when a set-up of 9.5 feet' is obtained, as at this point, with increasing set-up the correction factor drops rapidly toward zero. It may be that when the set­ up reaches this figure, there is enough depth of water over the vegetation far the set-up to take place essentially independently of the vegetation, and the full set-up results. This might occur if the bottom return flow occurs above the tops of the vegetation, and the water included within the vegetation is still and not a part of the circulatory system of water flow resulting from the set-up. It is probable, however, that this correction factor is also dependent on some term embodying the ratio of that portion of the marsh area included in the fetch to the entire length of the fetch. Then, when the fetch lies outside the marsh area, after some appropriate lag in time, full set-up "teuld be reached. This postulation may also be an explanation of the sharp ohange in the correction factor at the higher set-up stage.

19. Anotter sharp change in the slope of this curve seems to be in­ dicated at a set-up of about 2.4 feet, though this may be Just scatter of the plotted points. However, a sharp change would be expected somewhere in this range, since at some value of setnxp the tops of the vegetation

6 disappear beneath the water, and the surface shear between the wind and the water changes abruptly to its full value.

20. It is recognized that the data used in obtaining this correction factor (Cm) are entirely from one storm, and that the apparent straight line fit may be completely coincidental due to a rather fortuitous choice of the variables. It is hoped that additional data from future, smaller storms over the same area may strengthen the basis for the curve.

21. Values of set-up dbtained by using correction values (Cm ) obtained from this curve are also shown in figures 4 and 5« Good agreement is shown between the predicted and observed values, with, for fetch average values, the best agreement again being obtained when the cosine © term is omitted. Unfortunately, the effect of the marsh prevents any exact determination of the divergence, but a value for N may be determined for the tine (toward the peak of the storm) when the set-up appears to be taking place essentially independently of aEy effect from the vegetation. Ibis value obtained by the Jacksonville method would be 0 . 9 6 , which at least is of the carder of magnitude to be expected. Curves drawn on the basis of no divergence factor as well as for this value are shown. The use of a different value for the divergence would change the slope and intercept of the marsh correction curve slightly, but a similar straight line curve would result.

22. After the wind shift for the 1950 storm a discrepancy is apparent between predicted and observed set-up that is even greater than that indicated for the 1 9 4 9 storm. This, as in the 1 9 4 9 storm, is undoubtedly due to non-uniform conditions.

23. Possible Method for Computing the Rise in Water level at the Leeward End of the Lake. It is usually the rise in water level at the lee­ ward end of the lake that is of major interest, rather than the total set-up between the two ends of the l a k e -. Creager, Justin and Hinds (6) have studied the reported set-up on numerous lates and arrived at an average value of the ratio of the rise at the leeward end of the lake to the total set-up of about 0*571. Study of the Lake Okeechobee data indicates that this ratio is not constant, but varies somewhat with the nearshore slope and the amount of set-up. If it is assumed that the water surface slope may be sufficiently approximated by a straight line and that the bottom profile of the lake may be idealized as shown in the dietch below, then an expression for the percentage of total set-up occurring above or below mean lake level at either end of a rectangular lake is easily obtained. In the sketeh the triangular area Ax = hx c and Ag » h2 (L-c).

7 24 Since the amount of water in the lake (and hence in the cross- se< is presumed to remain the same, A]_ ■- Az and hg * c h]_ L - e

C+h, aha, c+tj g ^ i + (H-hjc+n <*) A (h, +hx ) o+n Of, But

Making use of the equality L-b = H (ct?n a, + c-fa ocz ), this may be solved for c: h, L - h,hz ( cM ar,+ chi crx) 0 SB 111 — f ■" J?,+hz

Then h S. A f k z M ) where $ =. ( c+n a, + cin ar2 ) h , L-c ~ Mz I L+ h , } )

25, Since the set-up, S * h]_ + hj, a quadratic equation in hx or h£ alone may be obtained

h* + (2L- Sf)h, - Si =o resulting in h, - S&-2L ±-/4Li+S'14 % 2Z$ and h Z $ -(2 L + S)h. + SL*Q L Si+2L±T/^.Z*SZ^> z ~ 2

For h x > o > it must be associated with the plus sign$ then, since S * hx+hg, • h2 must be associated with the minus sign* It may be noted that for the limiting case of vertical sides, these equations reduce to hx =* h.2 = S

The percentage of set-up occurring at the leeward end can be expressed as a function of 2L/S(jf s

A /- + ! 5 2 % This function is shewn as a graph in figure 8* From it, values of hx/S have been computed for the first half of each storm, and are compared in figure 9* with observed values. The values of the necessary parameters are tabulated in Table 5* Also shown on the graphs are the average values obtained from each storm and the average value given by Justin, Greager, and Hinds. As maybe seen from the graphs, the observed points are fitted much better by the derived curves than by any constant average value* This is particularly true for the 1950 storm* It should be noted that consider­ able scatter should be expected at times of small set-up, since the values of set-up were determined only to the nearest tenth of a foot (for a

8 two-foot total set-up, an error of 0.05 feet in the partial set-up results in a change of 0.025 in the observed hi/S ratio). The predicted and ob­ served set-ups at the leeward end of the lake are compared in figure 2j also shown are values computed by use of constant percentage values. As may be seen from this curve, the predicted and observed values agree ex­ tremely well except for the final value at 2130. It is suspected that this may be partly due to the wind shift occurring at that time. This shift may have already produced non-equilibrium conditions, giving a somewhat lower set-up at the leeward end than might otherwise be expected. This is partway substantiated by the continued decrease in proportionate set-up at the leeward end as the shift continuesj at 2200 the rise in water level was only 48.1 per cent of the total set-up.

26. Possible Dependence of Set-up on Foim Besistanoe of the Waves. Keulegan, in his experiments • found that for small closed channels the shear streSs is not independent of the length of the channel if waves are produced on the water surface, but that an additional stress is added due to the form resistance of the waves. This would result in the addition of an extra term to the set-up equation, which, for Keulegan’s experiments became S * 3.3 x 10~*y~ / \ÇF „ Despite the generation of waves

of appreciable size (up to 7 feet) during the more violent part of the storm on Lake Okeechobee, the data do not indicate the necessity of any such additional term. It is thought that either the effect of the form resistance of the waves becomes gradually less important with increased length of fetch, until at some undetermined point it becomes essentially negligible, or possibly that the form resistance term is introduced into the tank tests by the restriction of the area of air flow, which would not be observed in nature.

27. Conclusion. If the dimensions of the set-up equation are con­ sidered as miles per hour for velocity, miles for fetch, and feet for depth and set-up, then S * 1.165 x 10”3 £ff/yCi>s e * It will be noted that this

coefficient (1.165 x 10~3) is almost identical with that (1.25 x 10-3) of the so-called Zuiderzee formula developed some years ago from data for the Zuiderzee. This close agreement between coefficients applicable to such widely varying conditions as those in Keuleganls model experiments, the Zuiderzee, and Lake Okeechobee, would seem to indicate that this is a wind set-up formula of general application. It is hoped that further checks on its applicability may be made on other inland waters, pa.rticular- ly on those of considerably greater depth than that of Late Okeechobee and the Zuiderzee.

>■

9 Lake Average Values (depth a velocity averaged over entire lake)

FIGURE 2 20

18

16

14

12

e 0 1 10 a.U I «n

Wind Shift

legend: observed S 0 __ 3.3 x io - 6 VFF/gDF _____3.3 xio-6 V*F/gD cos 0 3.3XIO“6 V*F/gD N cos 3.3x to-6 V*F/gD N

For best fit, N = i.05 N|s 1.08

1600 1700 1600 1900 2 0 0 0 2100 2200 2300 24 0 0 0100 0200 0300 0 4 0 0 0500 0600 26 August .949 LAKE OKEECHOBEE-1949 HURRICANE ** Au«ust 1949 Fetch average volues (depth 8 velocity averaged over fetch) Figure 3

14

l e g e n d : ------observed *S 0j — ,— 3.3X10-6 v*F/gD — .— 3.3 xio”6 V*F/gD cos 0

1900 2000 2100 2200 2300 2400 0100 0200 0300 0400 0500 0600 0700 Q800 0900 1000 IIÒO 1200 1300 LAKE OKEECHOBEE — 1950 HURRICANE 17 October isso is October 1950 Fetch average values (depth a velocity averaged over fetch alone) ,0

•OSS ° O 2 O .045 a .035 AT 9.95

1950 HURRICANE — LAKE OKEECHOBEE 1950 HURRICANE — LAKE OKEECHOBEE Correction foctor introduced for marsh. Correction factor introduced for morsh

Figure 6 Figure 7 Figur« 8 15 PORTION OF TOTAL SET-UP OCCURRING AT LEEWARD END OF LAKE

h, g _ observed - 1950 T i i i 0 TABUS 1 PREDICTED AND OBSERVED VALUES OF SST-up Lake Okeechobee - 1949 Hurricane - Lake Average Values (D * 3*0 f t )

V near V av t ' ( 8 * ° > | F 0 Sc N necessary S « 1.03 | Time (mi) Qnph) (mph) (deg) 9 av Cos 0 av (ft) (ft) M l for S ^ SU 1$00 25 i |o o 35.5 31.1 28.0 1 1 1 4.05 4.05 4.2 1.058 4.17 3.4U10"6 1700 35.5 34.6 32.8 9 5 .996 5.56 5.54 5.3 .956 5.70 $ 4 * 1800 36.0 40.2 37.4 12 10.5 .983 7.34 7.21 7.4 , 1.027 3*38 1^00 36.5 47.9 44.0 13 12.5 .976 10.31 10.08 10.3 1.022 10.38 3.36 2000 37.5 56.1 52.0 20 16.5 .959 14.79 U .1 9 U.O .98? 14.60 i M 2i00 35.0 63.9 60.0 24 22 .927 18.37 17.02 17.5 1.028 17.53 3.39 2i30 34.5 63.0 62.7 20 22.5 .924 19.79 18.30 19.0 1.038 18.84 3U2 22*00 34.5 56.7 61.6 76 35 .819 19.12 15.66 16.2 1.034 16.10 3*45 2400 67.0 64.7 31.0 oiOo 31.5 55.7 61.4 11.0 21.0 .9336 17.35 16.23 11.5-16.0 o Jo o 31.5 46.3 51.0 16.0 13.5 .9724 11.95 11.61 9.0-10.5 03^0 35.9 37.9 42.1 9.5 12.75 .9753 9.26 9.05 6.4 0460 35.7 35.0 36.5 10.0 9.75 .9856 6.92 6.84 4.0 0560 37.5 31.8 33.4 U.O 12.0 .9782 6.08 5.94 3.8 0660 37.5 34.1 33.0 6.0 8.0 .9903 6.02 5.95 4.6

TABLE 2 PREDICTED AND OBSERVED VALUES OF SET-UP Lake Okeechobee - 1949 Hurricane - Fetch Average Values

necessary F D V now V av S' ) I So for S * So 1.08 1 N necessary Tiife (mi) (ft) (mph) (mph) Cos 6 av »

156o 35.5 29 l6f° 35.5 9.9 32.5 30.8 1 3.97 3.97 4.2 1.059 4.29 1.0 5 9 4.17 170o 35.5 9.9 37.1 34.8 .996 5.06 5.05 5.3 1.050 5>45 1.049 5 * S I860 36.0 9.9 44.4 40.8 •983 7.06 6.95 7.4 1.064 7.51 i-o jS 7 .41 19$0 36.5. 9.9 52.2 48.3 .976 10.03 9.80 10.3 1.051 10.49 1 .0S 10.53 20d0 37.5 9.9 61.0 56.6 .959 U .19 13.59 U.O 1.031 14.67 *9$5 14.96 17.18 18.02' 2ldD 35.0 9.9 67.9 64.5 .927 17.17 15.96 17.5 1.101 H 5 2130 34.5 9.9 63.1 65.9 .924 17.65 16.30 19.0 1.165 17.66 1.078 18.53 22(50 34.5 9.9 44.8 59.5 .819 U .40 11.78 16.2 1.375 12.72 1.124 15.21 246b 74.0 01(3) 31.5 9.2 66.1 70.0 .9336 19.55 18.05 11.5-16.0 026b 31.5 9.2 49.8 58,0 .9724 13.42 13.04 9.0-10.5 03O0 35.9 9.4 43.0 46.4 .9753 9.56 9.33 6.4 040b 35.7 9U 37.5 40.2 .9856 7.16 7.05 4.0 056b 37.5 9.9 37.1 37.3 .9782 6 .U 6,00 3.8 06

TABLE 3 PREDICTED AND OBSERVED VALUES OF SBT-UP Lake Okeechbbee - 19$0 Hurricane - Lai® Average Values (D - 7.9 f t )

F V now V av 9 9 av » < 9 * < S (N « H

ft Cm % (8-0.96) S (l-C n ) Timj (mi) (mph) (mph) (deg) (deg) cos 9 av («) (ft) 1 O (ft) a . o r-1 . 0 )

1806 22.7 4 1906 29 23.1 22.9 3 3.5 .9981 2.24 2.24 2.35 1.3 U 19 •395 1.38 23.3 1.5 2.25 .9993 2.36 2.36 2.27 145 •364 .339 1U9 2006 29.5 23.4 1.66 2106 29 25.O 24.2 2 1.75 .9996 2,51 2.51 2 . a 1.6 .362: .336 27.2 26.1 7.5 4.75 .9966 2.86 2.85 2.74 1.9 .333 .306 1.89 220.- 28.5 2.22 2300 28 29.2 28.2 12.5 10 .9848 3.28 3.24 3.11 242 .315 .292 2336 28 30.2 29.2 13.5 12.1 .9777 3.52 3.44 3.31. 2 «40 2400 28.5 33.5 30.8 21 15.1 .9655 3.99 3 ^ 5 . 3.76 2.8 .273 .243 2.74 33.4 22 19.4 .9432 4.77 4.50 4.32 3*36 0036 29 36.4 .216 4 .0 1 0106 29.5 38.5 36.2 21 21.5 .9304 5.76 5.30 5.10 4.0 .245 40.8 38.5 24 22 .9272 6.45 5*991 5.75 4*60 0130 29.5 5.38 0200* 30.5 41.8 40*5 18 21.75 .9 2 0 7.39 6.85 6.59 5*4 .212 .181 0230 31.5 42.7 41.8 27 21.75 .9283 8.11 7.54 7.24 5*98 0300 31.5 46.8 43.5 25 24.25 .9118 8.80 8.02 7.70 6 .3 .2 U .182; 6U1 0330 32 51.1 46.8 36 28.25 .88Q9 10.32 9.10 8.73 7.3 .198 .164 7.37 0400 32.5 51.2 50.0 40 34.25 •8266 11.99 9.90 9.51 8.3 .162 .127 8.10-9*51 0430 33 51.6 51.3 40 39 .7771 12.80 9.95 9.56 9.5 •045 .006 0500 33. 52.3 51.7 a 40.25 .7633 13.03 9.95 9.55 9.6 .035 0 080$ 36.1 43.1 43.5 34 4.7 0830 36.1 39.3 4 2 .1 38 0900 36.1 39.3 40.2 32 33.0 .8387 8.6 7.21 5.1. 100O 36.1 40.3 39.8 19 25*5 .9026 8.44 7.6 4 .9 110O 35.7 40.1 40.2 25 22.0 .9272 8.5 7.89 5.0 1200 35.0 36.4 38.3 24 24.5 .9100 7.55 6.87 4 .4 1300 35.8 34.3 35.4 24 24.0 .9136 6.63 6.05 3.9 UOO 36.1 29.4 30. .8 34 29.0 .8746 5.4 4.71 2.7 17

i TABUS 4 PREDICTED AND OBSERVED VALUES OF SET-UP Lake Okeechobee -1950 Hurricane - Fetch Average Valued

F D V noir V av Time S' fo M l i i l 1 imph) (mph) Cos 0 av N JLX1 S (N-0.96) &L. C'miN-0.96) S'fl-C'n,) 1800 23.0 1900 29 8.5 25.1 2 4 .0 .9981 2.29 2.29 2.20 2000 29.5 8.5 24.2 2 4 .6 1.3 .409 2100 .9993 2.45 2.45 2.35 29 8.7 25.9 2 5.0 .9996 1.5 .362 2200 2.43 2.43 2.33 1.6 28.5 10 28.8 2 7 .5 .9966 •314 2300 2.69 2.68 2.57 28 10 31.9 30.4 .9848 1.9 .264 1.90 2330 3.02 2.97 2.85 2.2 28 10 33.4 31.9 .9777 .242 2.16 2400 28.5 10 37.1 33.9 0030 .9655 3.82 3.69 3.54 2.8 29 9.9 39.0 36.6 .9432 .237 2.81 0100 29.5 9.9 41.0 39.0 .9304 5.29 4.92 4.72 0130 29.5 9.9 44.6 41.4 .9272 4.0 .211 4.01 0200 30.5 9.9 45.2 43.9 .9288 6.92 6.42 6.17 0230 3L.5 9.8 45.3 45.1 .9288 5.4 .138 5.38 0300 31.5 9.8 50.0 46.4 .9188 8.05 7.40 7.10 0330 32 9.8 51.9 49.3 6.3 •185 6.32 0400 .8809 9.25 8.15 7.82 32.5 9.7 53.0 51.7 .8266 7.3 .169 7.25 0430 33 9.7 10.44 8.65 8.30 8.3 .172 53.1 52.8 .7771 11.04 8.35 0$00 33 9.7 8.60 8.25 9.5 .105 8.87-10.61 55.0 53.5 .7633 11.34 8.66 0800 36.5 9.4 48.3 8.31 9.6 .118 9.23-10.89 0830 36.5 9.4 4.7 0900 36.5 9.4 42.1 45.2 .8387 1000 9.25 7.76 5.1 36.5 9.4 43.7 42.9 .9026 8.32 1100 36.5 7.51 4.9 9.4 43.2 43.4 .9272 8.51 1200 36.5 9.4 7.89 5.0 37.8 40.5 .9100 7.42 6.76 1300 36.5 9.4 4*4 35.9 36.8 .9136 6.13 5.60 1400 36.5 9.4 32.0 34.0 .8746 3.9 5.23 4.57 2.7

1 HDjujCrf 3 COMPUTATION OF PORTION OF SET-UP OCCURRING AT IE EJfARD END OF LAKE Lake Okeechobee-1949 Hurricane

40 Lfr Time hl hip hlo So F ctn a z ctn a , hi P 0.571 S 0.55: — 0 V 2 S0 S (N-l) ft-1.03) ' 1600 4*2 2.2 .524 - — ■ 35.5 3110 4540 7650 11.6 .520 1700 5.3 11.8 .520 2.11 2.17 2.38 2.31 2.9 .547 35.5 3110 4540 7650 1800 7.4 4.0 9.25 .525 8.60 •528 2.92 3.01 .540 36.0 3110 4540 7650 6.72 3.25 3.15 1900 10.3 5.8 .564 .637 6.69 .537 3.88 3.98 4.24 4.10 36.5 3440 4540 7980 4.69 .553 2000 14.0 8.1 .579 37.5 4370 4.65 .553 5.57 5.74 5.93 5.74 4540 8910 3.17 .576 3.04 .580 8.23 2100 17.5 10.1 .576 35.0 4450 3530 8 .46 8.35 8.08 2130 19.0 7980 2 .6 5 .590 2.65 .590 10.03 10.3 .542 34.5 4370 3190 7560 10.34 10.01 9.70 2200 16.2 7.8 .481 2.53 .595 2.56 .594 10.86 11.19 10.76 10.41 34.5 4370 3190 7560 2.98 .581

Lake Okeechobee-1950 Hurricane hlo Time So_ hlo So E. ctn ar2 h’ip ¿L (Cm * 0) hlp 1900 1.3 0.7 .539 29 1910 12670 14580 16.2 2000 1.5 0.8 .518 9.8 .525 1.13 .72 .533 29.5 1270 13000 14270 14.5 .520 2100 1.6 0.9 .564 9.6 .525 1.19 .78 29 1270 19000 20270 9.4 .525 2200 1.9 1.0 .526 28.5 1820 6.24 .540 1.31 .86 2300 23000 24820 6.37 .540 4.41 .557 2.2 1.1 .500 28 2540 15000 1.53 1.05 2400 2.8 17540 7.65 .533 5.41 .546 1.5 .535 28.5 2450 5600 1.70 1.21 0100 4.0 8050 13.3 .521 10.1 •524 2.2 .550 29.5 2910 5600 8510 1.94 1.44 0200 5.4 3.1 9.15 .526 7.18 .535 2.73 .574 30.5 3000 5600 8600 6.92 .536 2.14 0300 6.3 3.6 .571 5.67 .544 3.58 2.92 31.5 2910 5600 8510 6.19 .540 5.06 0330 7.3 4.1 .562 32 5120 .550 4.24 3.53 5600 10720 4.32 .557 3.59 0400 8.3 5.1 .614 32.5 6450 .569 4.96 4.19 0430 6400 12850 3.22 .575 2.82 9.5 5.9 .621 33 10000 6400 .585 5.56 4.74-5.56 0500 9.6 16400 2.24 .605 2.22 .605 6.2 .645 33 15000 6400 21400 5.78 5.78 1.69 .636 1.69 .636 6.07 6.07

18 TABLE 6 HURRICANE WAVE SATA FOR LAKE OKEECHOBEE, FLORIDA

H • significant wave height F - f e t c h T • significant irave period t * duration, based on group velocity dg ■ d ep th at gage V f* average wind velocity over fetch d f ■ a v e r a g e depth o v e r f e t c h for this duration

Labs H T F t VF fiatf. Ties (f t .) (sec.) ¿f t .) (f t .) (ad.) (hr.) Lw>h * l ± _ JL 1949 26 Aug 1900 u 3.3 4.9 ioa 11.0 23.9 2.8 41.3 2.59 1240 0.029 2000 12 1.8 4.0 6a 4.5 8.3 1.0 41.0 2.U 390 0.016 2100 12 1#8 3.5 5.0 4.0 aa 1.2 48.0 1.60 286 0.017 2130 12 1.4 3.3 4.5 3.6 6.0 1.3 54.5 1.33 222 0.009 27 Aug 0200 12 4.0 5.6 12.5 10.8 25.5 2.6 62.1 1.93 576 0.015 0300 12 2.7 5.4 11.3 11.6 22.5 2.7 50.0 2.47 870 0.016 0400 12 2.4 5.0 10.7 12.7 20.0 2.7 41.3 2.66 1180 0.021 0500 12 2.3 4 a 10.7 12.5 18.3 2.6 36.1 2.68 1370 0.026 0600 12 2.3 4 a 10.7 12.4 20.0 2.5 33.5 2.75 1480 0.031 1950 17 Oct 1900 10 0.5 2.0 3.3 3a 2.4 0.6 21.5 2.04 410 0.0161 U 1.2 3.0 7.7 ua 6.3 1.0 22.5 2.93 980 0.0353 2000 10 0.5 2.0 3 X 3.6 2.4 0.6 19.5 2.25 500 0 .GU5 12 1.4 3.0 8.6 8.3 7.8 ia 20.0 3.30 1540 0.0412 H 1.2 3.0 7.6 10.8 9.0 1.5 20.3 3.25 1730 0.0374 15 0.3 2.1 8.7 7.7 0.5 0.1 U .0 3.30 201 0.0229 2100 10 0.8 2.0 3.4 3.9 2 .5 0.6 22.7 1.94 384 0.0232 12 1.8 3.1 8.6 8.2 7.8 1.2 22.5 3.02 920 0.053 U 1.4 3.1 7.6 iia 11.5 1.8 21.9 3.U 1890 0.0436 15 0.4 2.0 8.7 7.7 0.8 0.2 16.0 2.75 248 0.0234 2200 10 1.0 2.0 3.3 4.0 2.7 0.7 25.7 1.71 323 0.0227 12 1.5 3a 8.6 sa 7.8 1.2 23.7 2.98 uoo 0.040 U 1.7 3.4 7.6 11.0 12.8 1.8 25.9 2.90 1510 0.038 15 Oa 2.2 8.4 7.4 0.9 0.2 17.0 2a4 245 0.0206 1* 1.2 3a 8.5 10.3 21.8 3.1 26.3 2.68 2480 0.026 2300 10 0.9 2.1 3.5 4a 2.7 0.6 27.0 1.70 294 0.0185 1950 12 1.8 3a 8.5 8.1 7.8 1.1 26.7 2.80 863 0.0378 2300 H 2.0 3.7 7a 11.0 13.4 1.7 27a 3.00 U 30 0.0404 15 0.5 2.5 8.5 7.5 0.9 0.2 17.0 3.23 246 0.0284 16 la 3a 8.6 10.3 21.8 3.2 28.9 2.44 2060 0.02U 2400 10 1.0 2.4 3.6 4.3 2.8 0.6 33.7 1.57 194 0.0131 12 2.1 3.5 8.4 8.0 6.8 0.9 27.7 2.84 700 0.0408 U 2a 4 a 7.9 11.1 12.8 1.5 30.0 3.08 1120 0.0465 15 1.1 3.3 8.3 7.3 1.0 0.2 26.0 2.78 116 0.0263 16 1.3 3.2 8.7 ioa 21.8 3.3 30.3 2.33 1870 0.0163 17 1.9 2.8 8.0 ua 20.5 3.6 30.3 2.03 1760 0.0310 18 Oct 0100 10 1.5 3a 3.8 4a 2.9 0.5 39.5 1.78 153 0 .0U 3 12 2w4 3.6 sa 7a 6.8 0.9 31.7 2.50 530 0.0353 14 3.4 4.5 8.1 10.9 13.0 ia 30.2 3.28 1130 0.0560 15 l a 3.3 8.1 7.1 0.9 0.1 26.0 3.06 105 0.0221 16 1.9 3.5 8.9 9a 21.7 3.0 33.6 2.29 1510 0.0252 17 2.0 2.8 8.5 11.5 19.9 3.5 34a 1.79 1320 0.0254 0200 io la 3.8 3.9 6.8 4.0 0.5 43.5 1.92 166 0.011 12 2.7 3.7 7.9 7.2 6a 0.9 37.2 2.18 389 0.0292 14 3.6 4.5 8.1 11.0 13.2 1.1 39.2 2.50 675 0.0332 15 1.1 3.7 7.9 6.9 1.6 oa 29.0 2.80 150 0.0195 16 2.3 4.5 9.2 9.5 22.0 2.5 38.9 2.54 1150 0.0226 17 • 2.0 2.6 8.9 ua 20.0 3.9 36.6 1.56 1210 0.0210 0300 10 1.9 3.2 4.1 8.9 9.0 ia 41.4 1.70 4U 0.0166 12 2.9 4.0 7.6 7.1 6.8 0.9 37.8 2.34 375 0.0303 U 3.6 4.8 8.7 10.9 15.1 1.6 42.5 2.50 660 0.0209 15 la 3.5 8.1 7.0 1.9 0.6 29.0 2.65 179 0.0195 16 2.5 4.7 9.3 9a 22.1 2.5 a.7 2 X 2 965 0.0205 17 3.3 2.7 9.5 10.6 20.0 3.8 40.6 1 U 6 959 0.0298 0330 10 2 a 3.1 4.5 6.6 12.0 2.6 U .0 1.66 565 0.0178 12 3.1 4.0 7.5 7a 6.8 0.9 45.8 1.92 256 0.022 14 3.6 5a 8.9 11.1 16.8 1.8 43.8 2.56 694 0.028 15 1.1 3.5 7.5 6.5 1.9 0.6 30.0 2.57 167 0.0183 16 2.6 4.6 9.6 9a 22.0 2.5 45.6 2.20 837 0.0187 17 3a 2.9 9.9 10.7 20.0 3.5 43.4 1 U 7 840 0.0302 0400 12 2.8 4.0 7 X 6.8 6.7 0.8 47.0 ia? 240 0.0189 14 3.5 5.1 8.9 10.6 17.0 1.7 48.7 2.10 565 0.0222 15 0.7* 3.2 7.3 6a 0.6 0.1 30.0 2.34 52.8 0.0116 16 2.7 4.4 9.9 8.9 22.0 2.6 47.6 2.04 77.0 0.0178 17 4.0 3.1 10.5 U .5 20.0 3.2 45.7 1.49 75 a 0.0288 0436 10 IX 2.7 3.5 3.8 2.8 0.5 60.0 0.99 61.0 0.0058 12 2.4 3.9 7.0 6.7 6.8 0.9 46.1 1.86 252 o a i 69 14 3a 5.0 8.6 U .5 11.5 ia 49.0 2.25 378 0.0193 15 0.6 2.7 6.8 5.8 0.5 0.1 30.0 1.98 44a 0.010 16 2.7 4a 9.8 9.0 22.2 2.6 50.7 1.83 690 0.0157 17 3.0 3a 10.9 U .5 20.0 3.0 49.6 1 U2 643 0.0182 0500 10 1.2 2.3 3a 3.3 2.8 0.6 60.0 0.84 62.0 0.005 12 2 2 3.9 7.5 6.8 6.8 0.9 45.0 1.91 265 0.0163 16 2.7 4.1 9.8 9.1 22.0 2.6 51.9 1.73 651 0.0150 0900 10 0.5 2.3 2.5 1.0 6.0 1.8 31.6 1.60 475 0.0078 14 0.6 2.8 6.8 5a 7.4 1.6 34.9 1.76 480 0.010 15 1.9 3.8 8.9 7.9 0.8 0.1 27a 3.09 87 0.0389 17 0.9 2.4 5.6 2.7 3.8 0.9 27.7 1.91 390 0.0176 1000 10 0.7 2.5 2.2 1.0 6.0 2.3 44.2 1.24 243 0.0054 14 0.9 2.9 6.7 5.0 7.9 1.3 35.0 1.83 510 oau 15 i a 3.8 8.7 7.7 1.7 0.2 21.5 3.88 293 0.061 17 1.2 2.5 5.4 2.2 3.8 1.0 30.7 1.79 320 0.0191 1100 14 0.9 2.8 7.0 5.1 6.3 1.1 45.0 1.37 246 0.0066 15 la 3a 9 a sa 0.9 0.1 26.0 3ao 105 0.0310 17 0.8 2.7 5a 1.9 3.8 1.0 31.0 1.77 304 0.0125 1200 14 0.9 2.6 7a sa 7.3 1.0 34a 1.68 500 0.0116 15 1.2 3.7 9.3 8.5 0.8 0.1 19.0 4 a 5 175 0.0498 17 1.1 2.7 5.7 2.0 3.8 la 23.0 2.58 570 0.0318 1300 U 0.8 2.7 7.3 5.5 7.7 ia 23.5 2.53 1U 0 0.0218 15 1.0 3.5 9.2 sa 0.8 oa 13.0 5.90 375 0.087 17 la 2.6 6.1 2.4 3.8 0.9 22.7 2.52 585 0.0349 1400 14 0.6 2.6 7.5 5.7 7.6 ia 22.5 2.54 U 90 0.0178 15 0.9 3.3 9.3 8.3 1.5 oa 17.0 4.26 408 0.0465 17 0.9 2.6 6.6 2.8 3a 0.9 22.8 2.51 577 0.0259 19

PART TWO - WAVES IB SHALLOW WATER

1« Introduction. In addition to the wind and water stage recorders discussed in Part One of this report, step-resistance wave gages were installed at seven lake stations, and wave records were taken for one minute out of every ten for the duration ©f the storms* lake stations 12, and 14 were In operation during the 1949 storm, and stations 10, 11, 12, 14, 1§, 16 and 17 during the 1950 storm* 0age locations are shown on the topographic map (Figure 1)« The records obtained were, analysed by the * Jacksonville District (3) for maxiimm height, significant height (the' average of the higher third), and significant period for each one minute strip.

2« Oorrelatton Between Maximum and Significant Wave Heights* The excellent correlation between, the height of maximum waves and the average height of the higher third was noteworthy* The average value of this rat i o far all gages was 1*37 (see figure 1 0 ) . Almost all the individual points were plus or minus 20 per cent of the average, which is surprisingly good considering the many factors which conceivably could build up a ma-idmum wave* This value should be compared with that of 1*87 ♦ 20# found by Wiegel for ocean waves along the Pacific ©oast (7) and that of 1 * 8 1 for the Pearson Type III frequency function model determined by Puts (8) to most nearly fit the statistical frequency distribution of observed ocean wave heights* The difference between these values is probably attribut­ able to the depth, difference between the shallow water encountered in lake Okeechobee and the deep water' of the ocean, but may be due, at least in part, to the fetch limitations imposed on the maturing o f . the wave spectra by the short lengths of Lake Okeechobee» Experimental results of wind generated waves obtained in a snail wav® channel by the University of Oalifernla (9) tend to indicate a value for this ratio of the order 1.2 to 1*45« Wnfortunately there was not enough variance in the depths at the points'where wav® data were obtained in Lake Okeechobee to indicate re­ liably any dependence of this ratio on depth* This excellent correlation between the maximum and significant wave heights seems to imply that a statistical grouping of waves is generated by a storm, and that this group­ ing 'is somewhat similar, though of a different absolute value, whether the storm occurs over a large deep area, as the ocean, or a small shallow area, as Late Okeechobee *

3. Method of Wave Analysis* In the further analysis of the Late Okeechobee data it was thought That since any attempt at prediction of wav© characteristics must, of necessity, consider average wind and wave conditions, some type of time average wave should be used in the analysis rather than the actual wave occurring at any particular time* This would in effect, eliminate the effect of gustiness on the generation of the wave. This was don© by plotting wave height and period against against time for each station, which resulted in a very Jagged curve; a smooth, average eurv© was then drawn through this set of points, a© shown in figure 11 for lake static» 16 during the 195© storm. The wave characteristic® applicable at any given time were then considered to be given by the smooth, average curve rather than by the actual observed data.

21 4# As with the data for wind set-up, the wave data during and immediately following the rapid wind shift has been discarded » Since there was no method for determining the direction from which the recorded waves were coming* there was no certainty as to what was the area of generation» The wave might have been traveling in the direction of the wind and hence generated in the short fetch indicated at that time* but it might equally well have been generated in the rear portion of some pre­ vious fetch* and be just reaching the point at that time» Hence the fetch and wind velocities applicable to the generation of these waves could not be determined« Likewise omitted were those waves which had an appreciable part of their fetch in the marsh area* as it was thought that the waves generated there might be considerably influenced by the marsh grasses* and hence not comparable to the other waves»

3 . Depth-Length Halation of the Okeechobee Bata» Sverdrup and Hunk (10) have previously considered the question of the generation of waves in deep water« Their theoretical curves have recently been slightly revised by Bretschneider (11) in the light of observed data ©n the generation of waves in deep water gathered over the past three years by the University of Calif­ ornia» However, as the maximum depth in Lake Okeechobee is about 14 feet with an average depth of about S feet for these two storms* most of the observed Okeechobee wave data fall into the shallow water classification. The frequency distribution for values of the depth—length ratio (d/Lo) for the 1950 storm is shown in figures 12a and b. In figure 12a the rati© is based on the depth at the gage, while in figure 12b it is based on the average depth over the entire fetch distance» As may be seen* this does not vary the distribution a groat deal» Per the vast majority of the ob­ servations this rati© falls within the range 0.04 to 0,24 with the maximum number of observations being in the 0«08 to 0.12 range for both oases. Values for the 1 9 4 9 storm are not shown* but the distribution is similar»

6« Comparison with Deep-Water Belationships. It is not to be expected that the Sverdrup—Hunk equations for wave generation in deep water should be exactly applicable to Lake Okeechobee» As a first step* however* it is worth considering the relationship which these data bear to the curves determined for deep water conditions by Sverdrup and Hunk« The relation­ ship between wind and wave characteristics is best shown by dimensionless representation. For those conditions where the wav© characteristics are controlled by fetch rather than by wind duration* the data for deep water conditions reduces to two diagrams* one representing wave height and the other representing wave periodo These may be expressed by representing the dimensionless variables gH/V^ and gT/V as functions of gF/V^, where H is the significant height* T is the wave period* F is the fetch* V is the wind velocity* and g is the acceleration of gravity. These parameters have been calculated in Table 6 for the Okeechobee data* and are shewn graphically in figures 13 and 14*

7. There appear to be several ways in which the wind velocity applicable to wave growth can be chosen» It can be taken as the velocity at the gage* or an average velocity over the fetch] also it can be taken

22' as the value at a particular time, or as an average value over some time interval* It seems logical to suspect that a value averaged over both fetch and some time interval would be more applicable to an analysis of the generation of the observed waves, particularly when it is remembered that the observed values of the save are, in themselves, averages* For the graphs shown in figures 13 and 14-, the time interval over which the wind velocity should be averaged has been determined as the time that it tabes the wave group to travel over the fetch, and it can be computed as the quotient ©f the fetch distance and the group velocity of the wave train* The group velocity can be determined from the observed period of the waves at the gage. Since the energy is propagated forward at the group velocity of the wave train, the time interval determined from the group velocity is thought to be more applicable than one determined from the wave velocity. However, plots similar to those in figures 13 and 14 have also been made for wind velocities averaged over the time ihtarval determined from the wave velocity! also for velocities averaged over an arbitrary one-hour interval, and for velocities at the time of the wave observations. Plots were also made for values averaged over the full fetch and over arbitrary § a n d 10-mile fetches. No significant differences were observed between any of the series of different plots, and the only ones presented here are those shown in figures 13 e n d 14 , which, it is believed have a more physically sound basis.

8 . Theoretical curves for wave generation in deep water given by Sverdrup and Munk and a revised period curve given by Bretschneider are also shown in figures 13 and 14 . It may be noted that the observed points on the period graph (figure 13) group rather closely about the Sverdrup- fen if curve, but fall somewhat below the Bretschneider curve! those on the height graph (figure 14 ) consistently fall below the deep water curve. This decrease In wave height and period below the values that would be computed for deep water with the same wind conditions is to be expected, and is due both to tbs effect of depth on the wave velocities, and to the loss of energy due to friction along the bottom. It may be seen that the effect of shallow water conditions is somewhat greater for wave height than for wave period.

9 . It is thought that the plotted points lying above the Bretschneider curves are not too reliable as all of these points were obtained from station 15 where the fetches were less than one mile. Due to the variability in direction of wave travel, the fetch lengths applicable to the observed waves at station 15 probably lie along a direction at some­ what of an angle to the wind, and are considerably longer than if measured in the direction of the wind. The use of a longer fetch obtained from considerations of this variability in wave direction would serve to move these points horizontally to the right, and would probably result in the points falling below the deep water curves.

10, The points have been segregated according to the station at which they were observed, and it may be seen that, for any single station, tbs points fall rather closely about a curve parallel to the Sverdrup-itunk

23 curveo These curves for each station are much more clearly defined in the period graph than in the height graph, as might he expected from the large number of factors (particularly refraction) affecting wave height, and the small time interval (1 minute) over which the record is obtained* On both graphs the order of the stations is about the same, the only significant difference being that, while the curve for station 17b lies below that of station lib on the period graph, their relative positions are reversed on the height graph* One possible explanation Is the refraction conditions at station 17, which, would tend to produce higher waves at this station than might be expected from the wind conditions alone* These curves then, would tend to indicate that the complete shallow water solution would result in a family of curves parallel to the deep water curve and which would depend on some additional factor involving a depth parameter* These curves would approach that for the deep water conditions as a limit*

11* Consideration of Possible Depth Parameters* Xt was thought that the depth parameter differentiating the different curves in a family would be one of the obvious dimensionless numbers, d A , d/L, d/Sf, or gd/ty2 where d is the depth (whether the depth at the gage, or the average depth over the fetch), L is the wave length, H is the wave height,..and g, F, and V are as before. An investigation of these parameters, however, showed that none of these appears to adequately define the curves. Thijsse (12) has shown that the results of studies of the generation of waves in shallow water carried out by the hydrotechnical laboratory at Delft mav be expressed dinensionlessly aw a family of curves of the parameters g D A 2 » This family of curves is shore in figure 15, with the observed values of gD/v2 given beside each plotted point of the Okeechobee data. Two values of gD/v2 are given, one representing the depth observed at the gage, and the other representing an average depth observed at the gage, and the other representing an average depth along the fetch* As may be readily seen, the the general trend of values is correct, but the scatter of the points about the curves is extremely large* For the Okeechobee data the best definition appears to occur with the parameter dffF, where df is the average depth over the fetch*

12» The deep water curves can be represented very closely by straight lines in this region (40

13* Relation of Maximum Possible Significant Wave to Depth* The Jacksonville District has found that the significant wave height does

24 not exceed 0.59 times the water depth (3). The averaged heights used in this analysis do not exceed about 0.5 times the water depth, with average wind velocities approaching 80 miles per hour. It is thought, therefore, that for design purposes, at least for depths up to about 20 feet, this ratio of 0*6 may be used to obtain the maximum significant wave height which should be protected against. H . Relation Between Wave Age and Ihve Steepness. Also of some in terest i s the relationship between "wave age" and "wave steepness". The term wave age, as previously used by Sverdrup and Munk, Is the ratio of wave velocity to wind speed (o/V)j wave steepness is the wave height- length ratio (H/L). A plot of these two terms is shown in Figure 18 with the curve as derived by Sverdrup and Munk shewn for comparative pur­ poses. The fact that the Lake Okeechobee data lies consistently below the Sverdrup-lunk curve is due, to a large extent, to the fact that the wave height from the Okeechobee data is consistently considerably smaller than that which would be obtained from the Sverdrup-Munk curves for a given value of the term gF/V^j hence the computed values of wave steep­ ness are smaller than the Sverdrup-Munk data. This is compensated for to some extent because, due to the shallow water, the values of wave length and velocity are somewhat smaller than the deep-water values, which tends to make thecomputed values of wave steepness larger and wave age smaller than the deep-water values associated with the Sverdrup- Munk wave. The la tte r e ffect i s , however, minor compared to that of the height difference* 15. Future Work. A further theoretical analysis along the lines of the basic Sverdrup-Muuk theory for the generation of waves in deep water is contemplated. This analysis will, introduce the actual shallow water v elo cities with their dependency on depth into the equation rather than the sim plified deep water v e lo cities which have no dependency on depth. It Is hoped that acomplete solution to the equations then obtained may be possible, and that by using the data contained herein to obtain the necessary numerical constants, the determination of an accurate method of forecasting waves in shallow water areas can be accomplished.

25 RELATIONSHIP OF MAXIMUM WAVE TO SIGNIFICANT WAVE HEIGHT

26 MAXIMUM WAVE HEIGHT (FEET) 2 7 Wove Height (Feet) Elevation (Feet MSL) AIU WV HIH, INFCN WV HIH AD INFCN PERIOD SIGNIFICANT AND HEIGHT WAVE SIGNIFICANT HEIGHT, WAVE MAXIMUM HURRICANE AA O LK SAIN 1, AE OKEECHOBEE LAKE *16, STATION LAKE FOR DATA ID IETO AD VELOCITY AND DIRECTION WIND COE 17-18,1950 OCTOBER i. II Fig. 28

Significant Period (Secgnds ) NO. OF OBSERVATIONS NO. OF OBSERVATIONS ( , 2 0 .8 1 .6 2 .4 8 3 .6 4 .4 4 .52 .48 .44 .40 .36 .32 28 .24 .20 .16 .12 .08 .04 02 L DISTRIBUTION OF DEPTH-LENGTH R A T lO (l) (l) lO T A R DEPTH-LENGTH OF DISTRIBUTION AKE OKEECHOBEE, OKEECHOBEE, AKE 19 bevtos L° Observations) (109 29 I960 90 hu 40 atr id shift) wind after ( 1400 thru 0900 gags at depth on Based 80 hu 50 bfr wn shift) wind before ( 0500 thru 1800 HURRICANE ______r. 2 b 12 frg. 1 __

F¡G. ¡4

o I o 1.0

0 .9

0.8

Bretschneider

• ? .V Ol ©o_ a . x _ _ x S>df □ ___— O i V.

x + +

1950 1949 BEFORE (wind shift) AFTER BEFORE AFTER • STA. * 10 o a STA. •■12 ■ STA. * 14 □ ♦ STA. 15

4 FIG. 16 40 ------1 ------1------1------1 1 1 1950 1949 BEFORE (wind shift) AFTER BEFORE AFTER • CTA * IO O a STA. * 12 * © o m STA. * 14 □ 0 1 ♦ STA. * 15 o X a> STA. * 16 O + STA. *17 X o X Center of group for each statior

♦ Sverdr u p - Munk 1 * 1 — 0

g h X ♦ f

♦ ♦ ♦

♦ ------■ A * - »I0 4 A AA 1 — ------X ♦

OJ ■ -t» X > — *• ■ A & < — • -■ X ■ X X

< X ■ + + ■> > Y # — - ■ « * X t ■ X ■ ■ • * • ------— ■ • n X • © % * ®

— 4 □ r © + 0 D a — 3) □ X 1 cP

0 © 01______—1------uO 7.0 8.0 9X) 10. 7 8 .9 10 20 10 X) 5.0 1IO4 FIG. 17 .12

O W,co I oo

O O 0) w

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Keulegan, G. H - Wind Tides in Small Closed Channels. Jour. Res. Sat. Bureau of Standards, IP 2207, V. 46, N. 5, 1951.

District Engineer, Jacksonville, Fla. - Preliminary Report, Freeboard Determinations, Lake Okeechobee, Fla. 1951.

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