WOODS HOLE OCEANOGRAPHIC INSTITUTION Ch CUL SO 1aaB a! hie WOODS HOLE, Mass, | BEACH EROSION BOARD OFFICE, CHIEF OF ENGINEERS i ee ae ese WASHINGTON, D.C. — JULY 1, iss . . : i DEPARTMENT OF THE ARMY CORPS OF ENGINEERS THE BULLETIN OF THE BEACH EROSION BOARD OSCILLATORY WAVES Diagrams and Tables of Relationships Commonly Used in Investigations of Surface Waves WOU UU 0 0301 O0O448b49 e SPECIAL ISSUE NO.| JULY 1, 1948 Sema coaies FOREWORD The study of surface wave phenomena, particularly progressive oscillatory waves in water, has engaged the attention of many investi- gators in recent years and has resulted in significant advances in our knowledge. With increased knowledge the importance of wave action as a factor in the solution of engineering problems of coastal and shore areas has become more and more apparent. In furtherance of its statutory obligation to publish technical information useful to the public concerned with the study of shore lines the Beach Erosion Board has arranged with the University of California the publication and dissemination of this paper for the benefit of those engaged in protecting our shores from the ravages of the seas. Contribution of the DEPARTMENT OF ENGINEERING UNIVERSITY OF CALIFORNIA BERK ELEY This compilation of information was completed under contract NObs 2490 for the Bureau of Ships, U. S. Navy. The major portion of the work was completed by Robert L. Wiegel with occasional suggestions being made by John D. Isaacs and J. W. Johnson. Assistance in making computations and preparing the diagrams was given by M.E. Haet, H. M. Gallaher and Mrs. R. Steele. TABLE OF CONTENTS Page Introduction al Diagrams of Wave Functions ....cccccecscsccccccccs S00000006 ehevsiclcles elton 2 Relationship Between Depth and the Height, Period, Length andy WielloleisbymOteiWavieSieieie clelerelelelc\ elec olclicieleictevelole/e 1 ols) ekellel «loleitoilel’« ore 2 Determination of Wave Height and hae of Water at Point of Breaking..... So0o0oH0000 50000000 9500000000 GO0O000 ofelelshetel sielcls 5 Effect of Capillarity on Wave Velocity....... §6.6:5.656.0 giogdodGce 5 Bifect of Refraction on Wave Birection...c.cccocs.cooccocceccvcsce 5 Surface Wave Heights from Underwater Pressure Measurements ... 6 Forecasting af Wind Waves and Swell......ccccccceccccvcvcecces 6 TAlesmonymuncieions) of G/lnand Gd /Rarics scree oils lceieie els s:cio sic +a elciere 6 ciel 7 Values of tanh 2m d/L5"be/ag, L/L, and C/C, .0.-.+-+-ses+ecene U) Pressunceuhesponsclala CivOnmyerso/ctctciatolorcororo ovalioiololeloloyevc! ee efayolorofoloiele:o/elotolell. 8 Fraction of Energy Advancing with the Velocity of the Wave Crest...... | @haile\.s) 00) 6) 01,01 0) c4(0(01 6) 6, €10\0) 6 010/016 0).0101 01010016 oc oleic cloele oe 9 Ratio of Group Velocity to Deep-water Wave Velocity......0. 606 9 Energy Coefficient..........00% S06 oo0 sdooobOOONMOOOOOGOOoGOO CUD 9 Effect of Shoaling.......0. 500600000 seooeDGe000OC ceoccocccooce 10 Hyp erbolaictsbuncitdonSlorcyera sreverolorioreie ites oloccroiettel ele SO00COOCDOOUNOOHOO 10 Accuracy of Computations......0...0. EictateMoxetcboloionelomnetatcvoreicnel sueroeer cycle 10 TealOMelSfavore cree lovere rovers ec oveesoonrcvace ceo cscosse ooo e De Oscooocvscveccceg tl to 50 Flare Sleverevelcvovelsievelovere SO0660 Eicken ceororonioneioicvciote ciere s00600 dosoooccoocdudude 6 Wo) Zl a ee ee ae 9 ve oe oo) ed OL epee OR of Lh... ee so ee ee see's wo wks . pro-ors otul ar and dnet % » wouehes Ke eon i Aik evita gies void SW Smee gare. eileen Fad somogees emminerT add to Ythoo IV edt adiw giteneves yyrend to sothaent PEE ha ne Bis eins aA mer mn SRL Bie acamineaneesiec tsi Sek ae we wee & TLE re Tee in ke an ok, 8 Ge We Ae oe a ae ee ee Le ey a eee ee ee ee eS Diagrams and Tables of Relationships Commonly Used in Investigations of Surface Waves Intro duc tion The purpose of this compilation of data is to assemble for easy acces-— sibility various functions that are used most frequently in investigations involving various surface wave phenomena. For convenience this material has been arranged in two sections. The first section consists of diagrams which are most useful in instances where a graphical solution gives sufficient accuracy for the particular problem. For those problems where a relatively high degree of accuracy is required a section of tabulated wave functions is presented. Where necessary, a brief summary of the theory and explana- tory notes are given for each diagram and table. The symbols used throughout this compilation are as follows: de = Wave period H = Wave height H,' Deep water height of wave which approaches shore without refraction L = Wave length C = Wave velocity Cq = Wave group velocity Dord = Depth beneath still-water level n = Fraction of energy advancing with wave velocity K = Pressure response factor -o = Subscript refers to deep water M = Energy coefficient o< = Angle of wave crest to bottom contour as = Length of semi-major axis of orbit of water particles Dg = Length of semi-minor axis of orbit of water particles Diagrams of Wave Functions Relationship Between Depth and the Height, Period, Length, and Velocity of Waves: Plates 1 - 9, inclusive, present the relationship between wave period, length, velocity, and depth. These curves have been plotted from the following equations; ¢ =/e8b tanh 27d 27 i (1) L = cT (2) Qn Plate 1 velocity has been eliminated, giving the relationship between wave period and length for curves of constant depth. Plate 2 is a large scale plot of the region on Plate 1 where the wave length is less than 70 feet and the wave period is less than 5 seconds. In Plate 3 wave length has been eliminated to give a relationship between period and velocity for constant depths. Plate 4 is a large scale plot of the portion of Plate 3 where the period is less than 1.8 second and the wave velocity is less than 5 feet per second. Plate 5 shows velocity plotted against depth with period as a parameter. Plate 6 is from Breakers and Surf, Hydrographic Office Publication No. 234 and gives the relationship between wave velocity, wave period, and depth. These curves differ from those in Plate 5 because a small correction for deep water steepness has been made. Plate 7 shows depths in fathoms plotted against wave period in minutes with curves of constant velocity and curves of constant d/L. Velocity is in knots. The equations for these curves are derived from equations (1) and (2); thus, substituting CT for L in equation (1): C = gf 21d Sm 2 Omer (3) Solving for ds: d= GR oatagk 2UKC 2TT eT (4) Putting in the proper constants to change d from feet to fathoms, C from feet per second to knots, and T from seconds to minutes: d = 2.69 CT tanh! (0.00548 C/T) or GWE TU SYAE) lea frog, (1 + 0.00548 G/T) - log, (1 - 0.00548 o/1)| (5) For the curves of constant d/L, the substitution of L2/f@ was made for C* in equation (1); thus, Teens g be anh aug col E (6) Multiplying both sides of equation (6) by ad/*: qe 2 2s d/L tanh Dali ite Thus, for a given value of d/, d and T may be calculated. Td (7) Plate 8 has the same axes as Plate 7 and shows curves of constant wave length and constant d/l. The curves of constant d/L are plotted in the same mamer as those on Plate 7. The equation for plotting the vertical asymptotes of the curves for constant wave length is derived from equation (2). To change the wave length from feet per second to knots, and the period from seconds to minutes: T2633 78acr (8) Then from equation (3): 27 C s 2T1da SS tanh gT (9) In deep water 2. dis very large, and the hyperbolic tangent approaches one. Then: or C = T (10) 0.00548 where C is in knots and T is in minutes. Since the vertical asymptote for the curves of constant wave length is in the deep water region of the graph, the value of C in equation (10) may be substituted in equation (8). T = 0.01275 (I> (11) The equation for the diagonal asymptotes is derived by: 77280 (12) The transitional portion of the curves of constant wave length are plotted by multiplying the L parameter by 0.5 d/l, to get the value d; this point is then located by following the .d/L curve to its intersection with the value of the ordinate d. Plate 9 is Plate 1 from Breakers and Surf and shows generalized curves for the change in velocity, height, and length of waves in shallow water from deep water to the point of breaking. These curves refer to waves that approach a shore directly so that there are no effects due to refraction. The horizontal scale of the graph is d/L., the relative depth. The vertical scale gives the values of the different functions indicated on the various curves. (a) H/H" | A theoretical curve for waves of very low steepness showing the change in wave height with decreasing depth prior to breaking. It is used to give the height of the wave at any given depth, when the period and height in deep water are known. (b) BE/te An empirical curve showing the ratio between the height of the breaker and the wave height in deep water, when the period and depth of breaking are known. This curve, in con- junction with curve (e), can be used in forecasting wave conditions. (c) sines A theoretical curve for waves of very low steepness showing sinec, the change in velocity and length of a wave as it enters shallow water. It is used to give the length and velocity C/C, of a wave at any given depth, when the period is known. This same curve gives the change in direction of a wave as b,/a, it approaches a straight shore line, and the ratio of the semi-minor and semi-major axis of the particle orbit. living (a) C,/C, An empirical curve showing the ratios between the velocity and length at breaking and the velocity and length in deep water, respectively, when the depth of breaking and period Lp/Ly are known. This curve represents a refinement that is usually unnecessary in forecasting wave conditions. (e) 100 HvA/ Eo An empirical curve showing the relative depth at which a wave of a given steepness will break. The steepness is expressed in per cent in order to fit the vertical scale. Breaker This curve, in conjunction with curve (b), can be used in Index forecasting surf conditions. Ci ee A theoretical curve showing the change in wave steepness H',/Lo as the wave enters shallow water. (g) 0.001N An empirical curve showing the correction factor for velocity due to steepness. The curve gives 1/1000 of the value of N in order to make it fit the vertical scale. It is not used in forecasting but it could be used in the determination of surf characteristics from aerial photographs. = Gla) na A theoretical curve showing the fraction of energy advanc- ing with the wave at a given relative depth. The value of n is used in computing H/H", « Determination of Wave Height and Depth of Water at Point of Breaking: All the curves on Plate 10 (Plate III from Breakers and Surf) deal with waves that approach a shore line directly, so that there are no changes due to refraction. When waves approach a shore line at an angle, the re- fraction correction first must be applied. Plate 10 is used in forecast— ing and in the interpretation of aerial photographs. Given values of H' and T define a point for which corresponding values of Hy and d, are found by interpolation between the solid and dashed lines, respectively. To find the wave length or velocity at the breaking depth, dp, or at any other depth, enter the inset with this value of d, follow horizontally to the proper value of T, and read off Lon the top scale. The velocity is then found from the ratio C = L/T. Measure the wave length, L, at any depth, d (not necessarily the breaker depth), and find T from the inset. Enter the main graph with T and follow a vertical line to the proper value of dj- Read off Hp from the solid lines and H', from the scale to the left or right of the diagram. Si Effect of Capillarity on Wave Velocity: Plate 11 is a plot of curves showing the effect of capillarity on wave velocity. Wave velocity has been plotted as a function of wave length both with and without surface tension effects. The per cent error or per cent difference between the velocity as determined by the two velocity equations also is plotted as a function of wave length. It is to be noted that for a wave length greater than 0.4 feet, the error in neglecting surface tension effects is less than one per cent. Effect of Refraction on Wave Direction: Plates 12 and 13 show the effect of refraction on wave direction. They give the angle ge between the wave front and the bottom contours in shallow water for given values of the ratio, dyna as a function of the angle ec, in deep water between the wave front and the contours. This relationship is derived from Snell's law; that is: sinee = C_ Siteeg ..e5 (13) where C is the velocity in shallow water and C, is the velocity in deep water. For given values of d/h 5 C/C,can be found from Plate 9, soée« can be plotted for various values of eos, . Plate 12 covers the range of d/Ly from 0.0 to 0.5. Plate 13 is a larger scale drawing of the region d/L, between zero and 0.1. The curves on Plate 14 (Plate II, Breakers and Surf) give the effect of refraction on wave height and direction for waves approaching at an angle toward a straight shore line where the bottom contours are straight and parallel to the shore. The horizontal scale is d/Lg, the relative depth. The vertical scale is oc , the angle between the wave crests in deep water and the bottom contours. The solid curves are lines of equal es , the angle between the wave crest and the depth contour at any relative depth. When the period and angle in deep water are known, these curves are used to obtain the angle of the waves at any given depth. The dashed curves are lines of equal K, the correction factor to be applied to the wave height in deep water to get the wave height in shallow water. Surface Wave Heights from Under-water Pressure Measurements: Plate 15 gives the factor for computing surface wave heights from data taken by under-water pressure measurements. For a pressure unit located ata distance of 3 below the surface in a total depth of water D where waves of length L exist, Plate 15 gives the value of the pressure response K, which is defined as the ratio of the head of the pressure fluctuation at the submerged point, to the surface wave height. Forecasting of Wind Waves and Swell: Plate 16 is used to determine whether fetch or duration is the limiting factor for wave characteristics at the end of the fetch. The graph is», entered with the duration (in hours) and the wind velocity (in knots) and the corresponding minimum fetch length read off from the lines of constant fetch (thus a duration of 36 hours and a wind velocity of 35 knots gives a minimum fetch length of 400 nautical miles). If the actual fetch is less than this minimum fetch, then fetch is the limiting factor, and an equivalent duration is selected; if the actual fetch is greater than this minimum fetch, then duration is the limiting factor. The latter is the usual case. Plate 17 or 18 is used to determine the wave characteristics at the end of the fetch from the duration selected from Plate 16. The graph is entered with the duration (in hours) and the wind velocity (in knots), and the corresponding wave heights and periods read off from the lines of constant height and period. (Thus a duration of 36 hours, and a velocity of 35 knots gives a corresponding wave height of 25 feet, and period of 10 seconds at the end of the fetch.) Plate 19 can be used to determine the wave characteristics at the end of the fetch when fetch is the limiting factor or when an equivalent fetch has been selected. The plate is entered into the fetch length (in nautical miles) and the wind velocity (in knots) and the corresponding wave height and period read off from the lines of constant height and period. (Thus a 100 mile fetch and a 35 knot wind give a wave height of 19 feet and period of 6.7 seconds at the end of the fetch.) Plate 20 is used to determine the wave characteristics at the end of the decay area. The graph is entered with the decay distance (in nautical miles) and the period at the end of the fetch (as determined from Plate 17, 18 or 19)and the corresponding height ratio, period, and travel time read off from the lines of constant value. (Thus a decay distance of 1000 miles and a 12 second period give a height ratio of 0.47, a period of 16 seconds, and a travel time of 41 hours.) The period at the end of | the decay area is thus given by the graph, the height obtained by multiply- ing the height at the end of the fetch by the height ratio (in the above case the decay height would be 0.37 x 25 = 9.25 feet), and the arrival time by adding the travel time to the time of the map used. Plates 16 - 20, inclusive were developed at the Scripps Institution of Oceanography. These plates are revisions* of curves which originally appeared in the Navy Hydrographic @ffice publication,"Wind Waves and Swell, Principles of Forecasting, H.Q. Miscellaneous Publication 11-275. Tables of Functions of d/L and d/L, In many of the basic equations describing gravity waves various functions of d/L and ais occur. Some of these wave equations were dis-— cussed above and summarized in graphical form in Plates 1 to 20. In evaluating these equations in certain instances it is often just as con— venient and certainly more accurate to utilize tabulated values of various functions of d/h and d/ La Those functions that are presented below in tabular form are summarized in Plate 21. The theory involved in calculating the various terms in the tables is discussed as follows: Values of tanh 27 d/L, b b, /a 5! L/h, : and C oi The basic equation for is wave velocity velocity (where the wave S steepness is small Suet S—— tanh 277 a/L In deep water, that is, where d= 1.01, tanh 27d/L approaches unity and since L = CT and ie = C,T (Note that deep water ordinarily is defined as d20.5 Lk, However in these tables it is noted that the values of tanh 2 ue ce sighs appreciably from unity for the range d/h. = (055) 120) a/7i, = Con ere Sale _— Lo » Co oT: ak thus ce 2 Teetanhe2 ime Rs Peele an =- b_ tanh 27 df Co ae ae Lo 27 and Cc. L/f L =+ _ tanh 277d G/-G=vi=etanheeiima iL Coe L/T T B /b, C/Cy / * Wave Report No. 73, Scripps Institution of Oceanography, March 1948 SO o7/c,* = c/c, . tanh 277 df, = LAL, tanh 27rd/L, C/G, = E/ig therefore, we havex ii) C/C, = L/L, = tanh 27 d/l, (14) The wave length changes with depth, and so it is inconvenient to use a/L as a parameter. The most convenient term to measure is the period, since it is a constant. Thus, L, may be computed easily because ee g Te; therefore, it is most convenient to use the parameter d/L, 277 d/L x Lig = d/,, and LAL, = tanh 27 a/L At any value of d/L, L/L, can be had from L/L, = tanh 27 d/L, and by multiplying d/L x L/L, d/L, can be obtained. In order to build up a table of d/Lg vs L/Le and C/Co a series of expanded accurate graphs were made by plotting d/L, vs d/h and then for the interval of d/L, chosen, the corresponding values of d/L were read off. Then the values of L/By and C/c, were recomputed. In addition, in shallow water, the orbital motion is elliptical and the ratio of the semi-minor (b,) and the semi-major (a,) of the surface orbit is equal to the tanh 27 d/h, i.e., bs /a, = tanh 27 d/L. In the tables to follow values of tanh 2 7 d/L (which is equal to b,/a,, C/O), and L/L, ) are given in column 4 as a function of d/L, or d/L. Pressure Response Factor: In order to make use of under-water pressure instruments, it is necessary to know what height of wave give a particular pressure response at some depth below the still water level. It has been founds that a K = H'/H = P/P, = cosh 27 d/L (1-4) cosh 27 d : (15) where P is the pressure fluctuation at a depth Z below still water, P, is the surface pressure fluctuatien, d is the depth of water (from still water level to the ocean bottom), L is the wave length in any particular depth of water, H is the height of wave at the surface, and H'’ is the corresponding variation of head at a depth Z. % Breakers and Surf, Principles of Forecasting, Hydrographic Office Publication No. 234. #* Sub-surface Pressures Due to Oscillatory Waves, by R. G. Folsom. Trans. American Geophysical Union, Vol. 28, No. 6, December 1947, pp 875-881. The solution of this is, of course, a family of curves, with a para- meter (Z/d), with K plotted against d/i. However, for the purpose of this table, only one value of #/d will be used because usually the instrument is placed on the bottom (/d = 1). In this case K = alt cosh 277 d/h (16) Values of K are shown in column 7 of the tables. Fraction of Ener Advancing With the Velocity of the Wave Crest: According to the irrotational wave theory only a fraction of the total wave energy travels forward with the wave form (that is, with the wave velocity C rather than the group velocity Cys. The equation for this fraction, n, is n=3|/1+47 df, Sinh 471d (qty) Values of this term are shown in column 11 of the tables. Ratio of Group Velocity to Deep-Water Wave Velocity: Cg/C, = CQ/e . C/O, =n tanh 27 d/E (18) See column 12 of the tables for this term. Energy Coefficient: This term is defined by the expressionxx M = we 2 tanh? 27d — L (19) in the equation for the energy of waves E=WLH 2 a Moir EE) (20) L See column 14 of the tables for values of M. * A Summary of the Theory of Oscillatory Waves, Technical Report No. 2, Beach Erosion Board, Washington, D. C., 1942. p 32. x* Wave Action in Relation to Engineering Structures, by D. De Gaillard, The Engineer School, Fort Belvoir, Virginia. Reprinted 1935. 10 Effect of Shoaling: An additional item included in the table is the ratio of wave height in shallow water to its deep water wave height when unaffected by refraction.* This is equal to H/HA) =e ee Bin GIs (21) Hyperbolic Functions of d/L: In addition to the above functions, values of sinh 2 7 d/L, cosh 27d/, sinh 47rd/f, and cosh 4 77 d/t have been put in the tables as well as 2 7 d/L and 4 7d/L. These have been put in as functions of d/L,, that is, for a given d/l, the values of sinh 27 af, and etc. are given. Accuracy of Computations: The values were arrived at using five place figures and then after all of the work was completed, the last place was dropped with the corresponding change of one figure up or dow in the fourth place. Because the basic interpolation was done with graphs, errors of one in the fourth significant figure may exist. Because of this, and because the square of the hyperbolic tangent is used to find the energy coefficient, only three places were reported for a certain range as it was felt that the additional figure could not be justified. Actually, in practice, it has been found that usually three figures are the greatest accuracy to which measurements can be made. However, when dealing with differences, this last place is needed to give the desired results. It must be remembered that occasionally an accumulation must be added to one of the values of d/L. Thus in 477 d/L the accumulation is in- creased by the factor of 47fand so the first differences between values of the sinh and coshof 4 7 d/fZ will jump. However, the corresponding values of n were plotted on an extended series of graphs and a curve drawn through them and then the correct values read off and placed in the table. Thus the true values of n vs a/L, are in the tables. It is to be noted that for the convenience of the user the summary of the various data is presented in two tables. Table 1 shows the various terms for even values of d/l, , and Table II shows the same terms for even values of df over the range°where interpolation in Table I is inconvenient. % Breakers and Surf, Principles of Forecasting, Hydrographic Office Publication No. 234. 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PLATE 3 C—--F-l.-/-SEGC: RELATION BETWEEN WAVE VELOCITY, PERIOD, AND DEPTH PLATE 4 WAVE VELOCITY - FEET/SEC. ‘ RELATIONSHIP BETWEEN WAVE PERIOD, VELOCITY, AND DEPTH 50 Ze » 14 ys “ee 13 ' i 12 s | " 45 SO = (Sk amd j {0 C= (Eran h == 2 | E=cr p { ( 9 I 40 35 30 50 DEPTH- FEET PLATE 5 WAVE PERIOD 9 31V1d SS3Nd331S 3AVM YOS ee SSILIZOTSA JAVM WONS NOILVNINYSL3O Hid3Gd - NOILOSYNOD HLIM ‘ 300 1350 t} 400 a ND\ [Di IP BETWEEN WAV LENGTH, : | ELOC | RELATIO “ \ 1000 PERIOD 110 z000 Al Se £000 = WAVE PERIOD — MINUTES cry beecaee Or 008 oso poP INCSNS Ze + _~ i if FS SNS | 2 ED WIZSNE =r a : aE : Sais S aS S <@ilins i. ss — tla oh | a Jle - ISIE < toPtooX 7 SX c 2 Ly AE Se Fae | Ss | 3 ° 8 3 PLATE -6 10 WAVE PERIOD —MINUTES 0s Ol 0.05 Ol 10 OT 08 13}eM deaq 0} stayjey yd}1osqng----- = JON SAARM 0} SIATaY Jd~1osaredng----- 7 °° z Suyyeaig ye soaeM—— — Supyeaig a10jaq saavq’——— SABM BuPYeeIg 0} stayay 3dyrosqng----- q- 6 31V1d ONINV3SYS JO LNIOd O1 Y3SLVM d330 WOYS HLONS1 ONV LHSISH NI SONVHD “Y3SLVM MOTIVHS NI SSAVM uopjovsjay Aq payoarry (1% uopyenba) 1030e,4 ssaudae3g-- AQPOOTAA aAeM WA Sujoueapy AS1euq jo uoporsry T®A9T 1972M ITHS Ueeueg yydeq WIOWO YA 4SarD aaem Jo o[suy TA °7/4p pue °T/p (0) “€00° 200° T00'0 vO 2'0 s‘0 anojuo0g === 2 A}fo0TS A BAeM----- 9) yysueT aaeM----- nf JUSTOH eAeM----- H CNaDAT sites 53: Bs i : pa = E EEE EERE ltt i fr 14 Ef f ig = o ; ; i tt 1 : f 4, (ae v 10° AY yy oe 100° S00" £00" 200° OL ol al el Ol 3aLV1d SNINVSYS JO LNIOd 1V YSLVM JO H1d30 ONV LHSISZH 3SAVM JO NOILVNINYSL3G SQNOO3S NI‘ 1‘ GOINad JAVM 61 61 8 Zl 9l Gl vl €l él " ol 6 8 Z 9 S bv € 7 7 4 BS qt dnagenge i i fae eaag ea Le 4) lapat f 4 THAT i sell Bi ra 5 ue iid + £ i ba ig is Bc at Hllineas ie { : Hf r . i i Sa sdaee sooas +H id + sides H aap Gog cea] PELE: t a : i a saga pal ie E tt ELT ERS i H ath : FES t 1 +t bgeed bed E HE i + + ig ieee BH ag + Tr} T + r deae H QR one ase +f + ee = ag aa EEPRRS aaa f f sag bdegg goghp-Rageuas 1334 NI‘°H‘USLVM d3gagq NI LHOSISH 3AVM WAVE VELOCITY— FT. PER SEC. EFFECT OF CAPILLARITY ON WAVE VELOCITY- FRESH WATER AT 70° F7 ao PERCENT ERROR eae 0.4 0.6 WAVE LENGTH — Lo Fale PLATE {8 iN PERCENT ERROR | SSS ae EFFECT OF REFRACTION WAVE DIRECTION 80° SHORE LINE > pesBaha = ee vis 70° 60° 50° 40° 30° 20° D/Lo PLATE 12 y : ON WAVE DIRECTION Zz o fe O <— o re LJ or Le O j— O rt Le Lo Li PLATE 13 Plate II bl 3LlV 1d SYNOLNOD H1d30 T31IVYVd “LHSIVYLS HLIM S3HOV3E NO NOILOVYSSY OL 3NG LHOISH GNV NOILOSYIG SAVM NI SONVHD °77/P ‘1 ys Ss ¢ a 20 SO €O <2O 10 200 SOO ¢0O0 WIND VELOCITY PLATE 17 | i OE ee an ie Sella gi | iaeia Hi. ane ee er Eakees 1 ,H,in feet T, in seconds [=aeoer Rsies Lines of equal wove height Lines of equal wave period, ” PPAF ATIVE AE Ea TT A | Sa nen uratio n hours WAVE HEIGHT AND WAVE PERIOD AS FUNCTIONS OF SHORT DURATION OF WIND AND WIND VELOCITY PLATE (8 Lines of equal wave height, H, in feet ——- = = Lines of equal wave period, T,in seconds ¢ ” a SjJouy Ul ‘A ‘AJIDO][AA PUIAA 50 60 70 80 90 100 Fetch,F,in naut. miles WAVE HEIGHT AND WAVE PERIOD AS FUNCTIONS OF SHORT FETCH AND WIND VELOCITY 40 30 10 PLATE 19 oO: O02 31V1d HO133 JO ONS LV GOIYSd JAVM ONV ZONVLSIC AVD3S0 4O SNOILONAS SV HOL34 JO ONS lv ONY JONVISIO AVI30 JO ONS LV LHSISH 3AVM N33M136 O1LVY ONY “SWIL TSAVYUL “SONVLSIG AVDSG JO ONZ LW GOINSd JAVM Saji "ynDu ul ‘g ‘aoduDJsip ADoaq folelo}3 0082 0092 O0ve 0022 s 0002 008! 009! 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