RESEARCH DIVISION COLLEGE OF ENGINEERING NEW YORK UN!VERSITY DEPARTMENT OF METEOROLOGY ON WIND GENERATED OCEAN WAVES WITH SPECIAL REFERENCE TO THE PROBLEM OF WAVE FORECASTING \PERSTARE EY PRICSTARE XE MIDESERRAG Prepared for C . OFFICE OF NAVAL RESEARCH =C Contract No. Nonr-285(05) ot I Washington, D. C. NHS poeek 7 GSSEhOO TOED O MMOD 1OHM/18 ON WIND GENERATED OCEAN WAVES WITH SPECIAL REFERENCE TO THE PROBLEM OF WAVE FORECASTING By Gerhard Neumann Prepared under a contract sponsored by the Office of Naval Research, Washington, D.C. Preliminary Distribution New York University College of Engineering Department of Meteorology May 1, 1952 : 1 ® tend Lh ey | : Tp Tele "i. 7 Ts P me 8 ual - Nami Tes cha 2 AE eT! ik ete ar a ' a> Py io =" Or i yg ; y - wule = of = SN Ste Senna one Sug tenet ssa, start Teeb a - “ay 7 npr “aa? es “gm eae! ts; qa t¢.@p! \ ae 2 Sawer es coh eri an 3 => | ciaenieihedaeee a atom ia ye pe tStereet! | gh Mies A ee eee an d 4 eae wattag 9“ wens _ A om a.) Se — stn san ete A = sea i Table of Contents Not at ion e e e e e e e e e e e e e e e e e e e e e e e ° e e Abstract e e e e e e e e e e e e e e e e ° ° ° e e e e e e ° Introduct ion e e e e e e e e. e e e e ° e e e e e e e e e e Chapter I. The composite nature of wind-generated waves 1) 2) 3) according to observations Comments on observations . .... +. © « « © © « © « « Characteristics of waves in the fully developed state at different wind velocities . . . « « « « © «© « «© « « Composite wave patterns. Interference phenomena. Groups (Of waveS.% < % 3) «96 «). Density of air (1.25 x 1073 g em™>), Horizontal component of displacement of a water particle. Vertical component of displacement of a water particle (elevation of sea surface relative to undisturbed level). Horizontal component of particle velocity at the surface. Vertical component of particle velocity at the surface. Mass transport velocity at the surface. Wind velocity at “anemometer height" (about 8-10 m above the surface). (v_ wind velocity immediately over the surface. ) - Wave height (a = amplitude). Wave length ( x = 27/, = wave number). Velocity of propagation of wave (phase velocity). Group velocity. Mean total energy of wave per unit area. Effective stress of the wind at sea surface. Normal and tangential component of wind stress. Dimensionless factors of proportionality. Wave steepness. Wave age (B, = 6 /¥,)- B(1), Pes Different stages in characteristic wave ages. a2 Yon = 3 76fy Resistance factors as functions of Bp. = o/v ane fe) fe) Y = er of ot Yn? Yt Resistance factors as functions of 6 = &/v. y°(B) Effective friction coefficient. | AL Mean rate of energy transfer to wave due to normal wind pressure. A, Mean rate of energy transfer to wave due to tangential wind stress. A=A_ + A, = p'v3B-C(g): Effective energy transfer to wave due to wind action. C(B) Effective resistance coefficient (dimensionless) as function of 8. s = 0.095 Sheltering coefficient; s* = 2s; s' = s/2. Cea s7T5 = sax n = 0.1075 Te 1.667 Constants used in empirical relationship 6 = f(f). p = 0.062 Du Mean rate of energy dissipation due to viscosity. les Viscosity of sea water. Dans D Mean rate of energy dissipation due to eddy viscosity in turbulent wave motion. M Eddy viscosity coefficient at fully developed sea. M(p) Eddy viscosity coefficient at growing wave motion as function of wave age B. B(p) Effective factor of energy dissipation due to eddy viscosity (dimensionless quantity) as function of 8. ex/v- Dimensionless fetch parameter. gt/v Dimensionless duration parameter. eH/v- Dimensionless wave height parameter. pi ba - a , ye : 3 i= age f mae de Me — | ue ~ pee = asi | Rae ZS Se bre dnhY eal SORANTRROP UE Roh sn git 7 7 ; te De 2 = ,uaves (imi Bes fo - an ont? she oie ss Fae? rer nat? ge iG: alvin 7 " ) - yt apres. br etpe o . gitson!’s é hae? savse ‘ant F eri eee ee gee 0 ahd - _ 7 7 “was at eo? oa eae apis “eres Ww ee Dae AIH Ys rb pseu : byb oy nk ar _ m4; 7 = EF f femens: Sl itl AVEW outs PR ral Po" oe! 3. ay cera hited tens a * ps Pe sea I Tk . 6 ot Sk cane a ew: Oe apaiih vis in PGR, 9 Se br ‘woth! : Shed he: ex! 14, re a i 5 7 dey ve Paar oe! y Pr weil, avr,’ les -¢ Neue * oN ‘bd P59 i AG ay) ye eee ee - a oT « “He ee a =f ar) are 1 7 ve Wes Ce WI tenner >’ di: tela tene.s. — - ec! e ae eee. rn a a : a ; 7 bey 2M a -@€ £5 id @ vat tee apersy ap te AB is Forage afish fi ' A= : 7 - ; ; ' "iG oe 7 eae 9 Niece w oc 9h PPG! in tsedfed “3e0, o cont "Ragas eee In te MeN aie vases a By ih, He zt “eet = . ae - at : fy es TE « oT yan Maar . oe! ah pict eb *, Pgh et es ~ - s {2 WOE, = 7 : ae = % rh 7 7 fits . ve vases faate . ae fe Maes we on “8 vibe AS te ads i) i High oe ie idee 2 ieee ie re oy ee y - eras bee hal ti ; : mine «aa ee aad ! te a) jae 40 eters “| ia ” a = ? CA v es a = 3 a ' 7 ar Pen an Lkgo4, ? a pete i a eg net we = ns. a - eee oy te ‘es + ie i. 2% ul iis © 4 = 4 : 7 fl Prue aa ae ny ea, 3 i yee Tarp ! +) he Ss ' & Jo | DAY 7h hed leey Fr si ie ass wt o Tift i gece = a5, Na = utah ry INAS Las sr = Si ag £9 a equi dvi vit ae! ai Tied % iJ Oe afi 4 / pe ache oy" p Niele wes vi aes, ee oy ? oe ES liagte, ree i re tt His ye, a ‘tle ne Bi Pah as Ho ws abirdasas i= cui tee tite elon re ot. in 7 ooh oy Tea ANN odtta wh: iedeacddamara ee ers! Veet tries 50> pf slrvewae aaibpatiite na inn Teeny, : cies pre want, eo $M, sie haath te’ a si _ ? - - a 6 i i ‘ fi [ : bie : i) al ~ a iy = = 7 : : “ _ : 8 - os * , 7 : re se 7 ce a, : 7 ad aera Din uy Oo — et ae mn | rae ere - Abstract Sea state forecasts are based on a combination of empirical knowledge and theoretical relationships of wind and ocean waves. Because the sea surface wave pattern usually consists of a locally produced sea and superimposed swell, two separate procedures of fore- casting are necessary, both generally based on oceanic weather maps: forecasts of waves generated by the direct action of wind, and sea swell forecasting. After combination of these two independent pheno- mena, one may estimate the actual state of the sea surface pattern at a given locality. Chapter I of this report communicates some results obtained by observations on the composite nature of wind generated waves. The characteristics of dominating waves in the fully developed state at different wind velocities are discussed. A conspicuous feature of the rough sea is the phenomenon of interference. Observed periods, lengths and heights of waves vary through wide ranges, and in this complex wave motion outsize waves occur, aS a consequence of interference patterns. An attempt is made to explain some striking features of composite wave motion by principles of interference. Chapter II deals with the growth of the dominating waves. Re- lationships are derived between the waves, the wind velocity and the area of water over which the wind blows (fetch), or the length of time that the wind has blown (duration). The numerical application of these relationships requires the knowledge of the amount of energy available for the growth of waves under various conditions. In this report a first attempt is made to estimate the difference between the energy supply by wind and the energy dissipation by eddy viscosity at different stages of wave development. These re- lationships have been developed on the basis of theoretical consider- ations and application of empirical laws. The results are presented in graphs and tables for practical use, by means of which the height, length, period and velocity of dominating waves at different wind velocities, fetches or durations can be estimated. The comparison of theoretical results with observations shows satisfactory agree- ment. ON WIND GENERATED OCEAN WAVES WITH SPECIAL REFERENCE TO THE PROBLEM OF WAVE FORECASTING Introduction _ The most familiar cause of sea surface waves is the wind, and it is on these wind generated waves that this report dwells. In re- cent years, questions about the growth of waves under the action of wind, and the dimensions of waves in the fully developed state have been brought -into the foreground of special oceanographic work. It was due to necessity that this recent period of sea wave investiga- tions was begun. Nobody who follows the sea or depends in any way on the state of the sea can ignore the behavior of waves. Not only navigators, sailors and fishermen, but naval personnel, naval engin- eers and seaside dwellers who have to protect their coast against the attacks of the sea, are deeply interested in this problem. The state of the sea, or the sea surface roughness pattern, is a complicated combination of many waves often including swell. The rather steep breaking waves of irregular appearance result from local winds, whereas the more regular undulating swell will in general have been generated in an area far away from the region of observation. For analyzing quantitatively the state of the sea at different wind velocities, it is required to separate these two independent pheno- mena. When describing or forecasting the state of the sea under dif- ferent conditions, independent forecasts of local wind generated waves and of swell are necessary. After combination of the two separated procedures we may get the actual state of the sea surface pattern. Many new ideas from recent progress on this subject have proven to be useful for important practical work. Present methods of fore- 3 casting sea state are based essentially on the earliest American re- port on a method for forecasting sea and swell by H. U. Sverdrup and collaborators in 1942, later revised and published by H. U. Sverdrup and W. He Munk [1]. Its publication stimulated further investiga- tions of wind and sea state relationships, so that more comprehensive empirical data were soon available. Upon surveying the results of investigations up to the present, we find a lack of knowledge with respect to certain aspects, sometimes as it seems, to basic problems. There is the question of the "characteristic" waves, or those waves existing at a certain wind velocity when the sea is fully arisen, and also on the period, wave length and height of waves. First, we must know the characteristic pattern under different conditions at a sea surface which has complex wave motion. The definition of the "significant wave" given by Sver- drup and Munk is not very satisfactory. It is a statistical defin- ition which takes merely a certain average for the highest waves and approximately the wave length of the longest wave present at the sea surface. The steepness of these waves, when fully developed, is about 1:46 (ratio of wave height to wave length), but there are much steeper waves present with a velocity of propagation smaller than the wind velocity. These characteristic "seas" cause the broken appearance of the sea surface and obscure to a large degree the presence of longer and flatter waves. Furthermore, there are some questions about the energy transfer from wind to waves and about the growth of the waves under the action of wind. What type of pattern of sea surface (wave heights and lengths) is to be expected when a wind of a certain mean velocity has blown over the sea surface for one, two, three or more hours? Or, what type of pattern may we expect to find when observing the waves at different distances from the windward shore? This report is concerned with questions, which aim primarily at the practical ends of the problem of wave forecasting. It deals merely with wind driven waves, while the problem of swell forecasting will be considered in a later report. Chapter I. The composite nature of wind generated waves according to observations 1) Comments on observations It seems very difficult to define a distinct wave motion at the sea surface, because of the composite pattern of the "sea," as the undulatory motion of the sea surface in the case of wind-driven waves is called by seamen. Besides the wind-driven sea it is possible that one or more different types of swell are running across or in the same direction as the waves generated by the local wind, which fact further complicates the resultant wave motion and therefore the observations at a given locality. The swell is not causally con- nected with the local wind, and this report does not take into ac- count its behavior when travelling over long distances at sea. The discussion at first pertains only to waves under the direct action of wind, considering the matured state as well as the state of wave development. Therefore when trying to observe wind generated waves we have to eliminate, if necessary, in the best possible manner any swell that appears. Sometimes this separation seems very difficult, especially when taking mechanical wave records, like the recent wave records of H. R. Seiwell [10]. They were obtained by automatic re- cordings of wave pressure variations at the sea bottom. The results of H. R. Seiwell obtained by application of the principles of general- ized harmonic analysis to such oceanic wave observations, and the con- clusions drawn from the mathematical treatments--as far as they have been published to this date--may be summarized after H. R. Seiwell and G. P. Wadsworth [11] as follows: "1, Periodogram analyses performed on oceanic wave records do not appear to give correct geophysical information. The numerous wave periods, and bands of periods, indicated by this type of analy- sis do not necessarily possess physical significance. 2. Application of the hypothesis of generalized harmonic analy- ses to western North Atlantic wave records indicates that ocean wave patterns are not complex interference patterns resulting from combi- nations of many wave frequencies, but frequently consist of a single sinusoidal wave ("cyclical component") on which is superimposed an “oscillatory component." (In the case of the published wave record 53-X [11] the latter component appeared with the same period as the cyclical component.) 3. The cyclical component appears to be that generated under the influence of a dominating oceanic meteorological situation, and the oscillatory component by local winds and other local disturb- ances tending to change the basic ocean wave pattern." The physical meaning of these mathematical results seems some- what obscure. But there is a fact to which we have to pay attention when considering the results of Seiwell's investigations: the re- cords of Seiwell, as far as published, comprise sea bottom pressure observations and the results mentioned above are based on these press- ure-recordings. However the pattern of pressure-fluctuations at the sea bottom does not agree with the actual pattern of complex surface waves. Shorter waves are filtered out, and in his papers of 1948, H. R. Seiwell [10] calls special attention to the fact that surface wave lengths less than 240 feet were not registered by the Bermuda wave recorder at a depth of 20 fathoms. The average bottom period of waves at Bermuda was about three times that at the surface and at the Cuttyhunk observations the average period of bottom waves was about two times that at the surface. Furthermore, the longer waves of the complex wave motion are influenced by shallow water effects when approaching the location of recordings from the deep open sea. The results obtained by pressure-records are therefore not strictly comparable with observations at the surface of the open sea. The pressure-recordings evaluated by H. R. Seiwell indicated that ocean wave patterns are not complex interference patterns re- sulting from combinations of several (two or more) distinct waves, but frequently consist of a single wave. This fact is perhaps the consequence of filter-effects: the superimposed shorter waves dis- appear, because they are filtered out by damping. During his stay at the Woods Hole Oceanographic Institution in the summer of 1951, the author became aware of new unpublished wave recordings made by Seiwell in 1950 at the sea surface. The director of the Woods Hole Oceanographic Institution was kind enough to per- mit the use of these data and to look into the results of mathemati- cal analysis as far as these data were available.* Unfortunately most of the original data and the results of analysis were lent out, but the remaining part of these observations already seems to indicate important differences compared with the former observations and re- sults, which were obtained at small water depths by pressure record- ings. This new material, collected in 1950 in the vicinity of the Bermudas, comprises sea surface measurements taken by photographic *The author is very much obliged to Admiral E. H. Smith and Dr. C. 0. D'Iselin for permission to make use of these unpublished data of H. R. Seiwell. recordings of sea surface wave heights from which discrete values are scaled at equidistant time intervals. The most striking feature of the results of the mathematical analysis seems to be that the wave patterns at the sea surface are dominated by more than one "cyclical component" in most cases, which agrees with the findings of the author. But in his notes on his work, H. R. Seiwell points out that the autocorrelation structure of data containing more than one cyclical component becomes complicated and usually does not per- mit identification of the periods involved. Its use is limited to a means of revealing the presence of more than one period, and further information concerning the identification need be obtained by some form of a Fourier Transform of the autocorrelations into a power spectrum. One may look forward with interest to the results of analysis of Seiwell's surface observations in 1950. Another method for observing the state of the sea under differ- ent conditions, particularly from ships under way, is based on stereo- photogrammetric pictures. This method [2] clearly reveals the com- plicated configuration of the wind affected sea surface, and makes possible exact morphological measurements, though the present possi- bilities of this method are limited too. It is a rather expensive method and there are not many observations available at present. A recent summary of the results obtained and an outlook on future de-=- velopments with respect to stereophotogrammetric wave pictures in rapid sequence by A. Schumacher [3] shows that the improvements of this method may be expected to prove very useful in future work. Besides these more expensive methods, there are several simple methods of measuring waves from ships under way by direct obser- vations. Compared with some highly developed technical methods for modern geophysical observations, these "old fashioned" simple “observations by eye" may perhaps appear too primitive today. Most of them were proposed and have already been used in the past century. But a great deal of our empirical knowledge on waves today is based on such direct observations, and up to date these simple observations by eye have proved very useful when taken carefully and utilized critically. There are descriptions of these methods in several textbooks on waves (V. Cornish [4], 0. Krummel [5], H. Thorade [6]) and more recently they are briefly discussed in H. 0. Pub. No. 602 by H. B. Bigelow and W. T. Edmondson [7]. The time intervals between succeeding crests of waves at a fixed locality* can be estimated from shipboard with a fair degree of ac- curacy by observing the rise and fall of the water, observing for example, foam patches, seaweed or other bodies drifting just beneath the sea surface. This method was shown to be very useful by V. Cornish. But in taking such measurements, it is necessary to observe a large number of single vertical oscillations under the given con- ditions, because the single values of periods and heights of succeed- ing waves differ over a large range. Furthermore, it should be warned against simply averaging these widely scattered observations. There is much confusion in our empirical knowledge of the relation between wind and waves due to the fact that one has considered only some in- dividual observations under certain conditions, or that one has sum- marized the measurements by taking "average values" over a large range of variation. In the fall of 1950 and the winter of 1950-51, while crossing *This quantity will be referred to hereafter as "wave period." 10 the North Atlantic Ocean with a slow-speed freighter, the author took the opportunity to make systematic wave observations. The voyage started October 14, 1950, at Hamburg, aboard the "Heidberg." She docked at ports in the Caribbean Sea, in the West Indies and in the Gulf of Mexico, and returned through the Straits of Florida to Hamburg again, where she arrived February 3, 1951. Crossing the Caribbean, the Gulf of Mexico and round the West Indies there was often a good opportunity to observe the state of wave develovment due to different fetches. Besides systematic observations of wave periods and heights together with exact wind measurements, the author provided for special studies on wave groups and for the mak- ing of moving pictures of sea waves. Altogether, about 27,000 ob- servations were collected under different conditions in the North Atlantic, the Caribbean Sea and the Gulf of Mexico, the values rang- ing up to wind velocities of 22 m/sec. In order to get an idea of the daily North Atlantic weather situation and its history, two weather maps were drawn daily on the basis of ship reports. The methods of observation are described in more detail in "Deutsche Hydrographische Zeitschrift" [9], where the observations themselves are also published. wind velocities The observations aboard the "Heidberg" related mainly to measure- ments of time intervals between succeeding crests ("periods") and heights of succeeding waves in the complex wave motion, simultaneous with wind measurements by cup-anemometers. Between 70 and 150 or more single measurements of periods at a given locality under nearly ue steady wind conditions were collected and related to the measured wind speed. In this way, about 27,000 single observations have been placed together in a set of more than 250 "series," comprising the range of wind velocity between 2 and 22 m/sec. Separated from these series, further sets of wave measurements at limited fetches were gathered which will be used in Chapter II. The single measurements of each series were utilized by count- ing the frequency of the observed periods, taking period intervals of 0.5 seconds, and then representing them in diagrams showing the frequency distribution of periods atacertain locality and at a cer- tain wind speed. Observations under conditions with rapid wind changes are here omitted. This method of counting time intervals between succeeding crests naturally provides no exact mathematical analysis of water level fluctuations or of time series in general. But it seems to be useful just for practical purposes. The result provides a realistic picture of sea surface state, showing the per- iods which are to be expected in the characteristic composite wave pattern, the range of periods under different conditions, and the more or less frequent appearance of periods in certain "bands? In the composite wave pattern the rather steep "breaking sea" was the most conspicuous undulation as long as the sea surface was under the influence of generating winds, whereas the superimposed Smaller waves, as random components, did not disturb this pattern or influence it essentially. But the heights and periods of succeed- ing waves at a fixed locality mostly scattered through a large range, and in the ever-changing wave patterns of the sea surface longer waves with greater periods appeared occasionally, clearly marked by smaller 12 steepness, but running in the same direction as the wind and even as high as the steeper "tumultuous sea." Following these longer waves by eye from a look-out position on the mast, they appeared to change their shape rather soon and disappeared, being replaced by steeper and shorter waves. In figures 1 to 3, three series of observations are represented as examples of frequency distributions for observed periods at nearly steady wind conditions. The characteristic periods cover a more or less broad interval depending on the wind velocity. At lower values we find mostly a rather sharp limitation of the observed period in- terval. This means that waves with periods lower than a certain value have only minor significance, and do not visibly influence the striking pattern of sea surface roughness. Figure 1 shows a scattering of observed periods between about 2.5 and 8.5 seconds, and some distinct peaks in this interval. Be- sides a frequency maximum between 3.0 and 4.5 seconds, maximum peak- ing occurs between 5.0 and 6.5 seconds, and at 7.5 to 8.0 seconds. The single observations of this series were obtained when the ship drifted with stopped engines about 150 nautical miles southwest of the Azores. The wind was steady easterly with a velocity of 9 m/sec. In figure 2 are represented the observations of a series at 13.5 m/sec wind velocity. The periods scatter between about 4 and 13 seconds. Maximum peaking occurs at about 6.6, 8.7, and 11.8 seconds. Similar distributions are to be found in other series. At a wind velocity of 16 m/sec (figure 3) the range of periods is between 6.5 and 15 seconds with maximum frequencies at 8.3 seconds for the short- est characteristic waves, between 10.0 and 11.5 seconds for the next 13 Or ost Ne I@) (cor Oe Ne ae 14T(sec) 6 7 U. 7 = = Lo Gg E— KW — ‘. W\\ _ N “N° eae i s SAReeereenzots : a She: =a. WK = - A Se CMF OFF SOO On oOr =e tc Ne Se ess = ae Se Fig.3 Fig. Fig.!-3 Frequency distribution of observed wave periods T (sec) at different wind velocities. For explanation of To liok and Ty, see text. (Steady wind conditions). longer, and between 12.5 and 14.5 seconds for the longest periods of waves present. To present a simple interpretation it seems that two or three frequency bands could be defined as characterizing the observed data in the case where the sea was generated by a quasistationary and quasi- homogeneous wind field. Hence the usual conclusions drawn from the observations would be that the sea surface roughness pattern con- sists of discrete bands which indicate dominating periods in the com- posite wave motion. It may be pointed out here that a nearly continuous distribution of observed periods is found in series of observations under more complicated wind conditions; for example, where a stronger wind at the windward part of the fetch generates a fully arisen sea, while the wind decreases to the leeward, where the sea is observed (see page 26). The prediction of certain characteristics of period patterns may be based on the prediction of the distribution function for in- dividual wave periods. Statistical data for individual wave periods at given wind velocities indicate regularities in the distribution, namely, the range of characteristic periods, the most frequent per- iods, and the asymmetry of the distribution. Some examples for fully developed sea are presented in figure 4, covering the range of wind velocities between 5.5-6.4 m/sec, 7.5-8.4 m/sec, 10.5-11.4 m/sec, 11.5-12.4 m/sec, and 14.5-15.4 m/sec, respectively. In these distribution graphs all data are collected from observations at fully developed sea within the given wind velocity interval. The graphs show the percentages of observed characteristic periods at 2] T sec T sec 2 s 2 BS S o 7 o T sec Fig4. Frequency distribution of characteristic periods at different wind velocity v(m...) (Fully developed sea). 16 the given range of wind velocity. Ina later report ("Instructions for practical wave forecasting") certain "standard types" of period distributions will be presented for practical use, covering the wind velocities up to 24 m/sec. The results of all measurements are summarized in figure 5, taking the periods of maximum frequencies out of each individual series and plotting them as a function of wind velocity. This dia- gram shows the range of characteristic periods between wind velocities of 2 m/sec and 20 m/sec. At low wind velocities of about 4 m/sec we find a scattering of observed periods at fully developed sea between 1 and 4 seconds, at moderate wind of about 10 m/sec between 4 and 10 seconds, and at strong wind of about 16 m/sec between 7 and 15 seconds. In most cases the lowest characteristic period Ty appeared clearly separated from the highest period To but very often second- ary maxima were indicated in the frequency distribution between two limiting periods, as shown in figures 1 to 3. These different per- iods, indicated by peaks in the diagrams of the different observational series, are marked in figure 5 by dots, circles and crosses. At low wind velocities the intermediate period T3 seems to approach the period To) whereas at higher wind speeds it seems to approach Ty). At winds lower than 4 m/sec the occurrence of higher periods was less striking and at light winds less than 3 m/sec no observations of characteristic periods higher than Ty are made. But in all cases the observed periods scattered over a certain interval, as is generally to be expected in interference patterns (see section 3). The results of observations represented in figure 5 indicate 17 (*stoqufs Aq poexedTpuy o1e suoTqeadesqg) *(oes/W) A A4fFoOoTaA putM Jo suotyouns se (J) SovAeM OFASTTa4yoOeLTeyo Jo SpoTded *C*STWq (22°S/iu) A 02 SL OL S 0) aes (0) Sj e OL Le ; x as 3 6 p a L Sb 18 regularities in the relation between wind and sea state at differ- ent wind speeds, which do not stand out if only single observations under different conditions are taken at random, or when the observed periods are averaged over the whole range of scattering values. This perhaps explains the fact that the older attempts to determine emptrical relationships between wave periods or wave lengths and wind velocity led to less satisfactory results. The most conspicuous feature of the sea surface pattern at all wind velocities is, as already mentioned, the occurrence of the steepest characteristic waves, which are slower than the wind velo- city. They are indicated in Fig. 1-3, 4 and 5 by the period Ty. The curve Ty in Fig. 5 is based on previous calculations of the re= lationship between the dimensions of fully developed wind waves and the wind velocity in the composite wave motion (G. Neumann [8]). The condition that the energy supply by wind to waves equals the energy dissipation by turbulence at fully developed sea leads to a theo- retical relation ar (22 - 1) T, (sec) = a, Sin “Vv (la) z ln 182.5 - s inv where v is the wind velocity in cm/sec, O> the propagation velocity of the fully developed "longer waves" in cm/sec (see [8]} g the ac- 1.667. Bs 137 formula (la) leads to the same periods as evaluated formerly [8], celeration of gravity, and the dimensionless constant r 1n denotes the natural logarithm. With the value 0,/v except at wind velocities lower than 3-4 m/sec, where the periods given by (la) are slightly larger than the periods given in the paper 19 of 1950. But the difference does not seem to be very important. Formula (la) will be derived in Chapter II. The periods of the fully developed “longer waves" are, as it seems, fairly well approximated by a linear relation T,(sec) = ary = = Bev = 0.877v (v given in m/sec) (2a) This relation is based on the assumption 05 = 1.37 v as sug- gested by empirical evidence and already used in the papers of Sverdrup and Munk [1], and of Neumann [8]. At present, it is diffi- cult to say whether this relation is exactly a linear one, or whether there is a slight curvature. But possible deviations from the linear proportionality as given by (2a) seem to be of the second order. The period T3 in Fig. 5, although not always clearly separated from Ty or T,, indicates an intermediate wave with a velocity of phase propagation very near to the wind velocity. The broken line T3 in Fig. 5 is calculated by taking 03 = v and represents the re- lation T3 (sec) = = v = 0.64 v (v given in m/sec) (3a) These three waves which presumably dominate the composite sea surface pattern in fully or nearly fully developed state may be called 1) the "sea" (T)); or f ,-wave ( 0,/v = Ba) 2) the "longer wave" (T,), or Bu-wave ( G./v = a) 3) the "intermediate wave" (T3), or B(1)-wave ( 03/v = 1) The periods and wave-lengths of these three waves are given in Table 1 at different wind velocities up to 24 m/sec. 20 Table 1. Periods T and wave lengths X of dominating waves at different wind velocities. v(m/sec) 2 4 6 8 To Yo" 14" 16 18. -20."* 24 T, (sec) 0.615 1.42 2.36 3.37 4.5 5-7 7-0 8.3. 9.7 12.3 14.5 A; (a) 0.58 3.16 8.7 17.8 31.5 50.8 75.1 107 146 198 327 To (sec) 1.76 3.51 5.25 7.03 8.8 10.5 12.3 14.0 15.8 17.6 21.1 A 2 (mn) 4.8 19.2 43.2 76.8 120 172 236 307 387 481 693 Mi(sac) 1.28 2.456 3.84 5.12) 1654 7.7 920 10.2 11.5 12.8 15.4 3 A3™ 2.56 10.2 23.0 41.0 64 92 126 163 207 256 370 The results of some measurements of wave heights are represented in figure 6. These measurements relate to observations in the open sea, and as much as possible to stable weather conditions. They were taken by estimating the eye-height above the water line when the ship was on an even keel in the trough of the waves, and leveling the wave crest with the horizon. During the time between October 25 and 30, 1950,in the region SW of the Azores more reliable measurements have been made from the drifting ship at low wind velo- cities between 4 and 10 m/sec. The ship drifted slowly athwart to wind and sea, and accurate measurements could be made from the over- hanging bow by means of readings at a sounding line with marks hang- ing down into the water. The opportunity for making such measure- ments from the drifting ship arose several times during the voyage. The variation of heights of succeeding waves is the most striking feature of fully developed sea, and it seems impossible to decide which height is the characteristic one. A single measured height has only little significance. It was attempted to determine al {e) 2 4 6 8 10 12 14 16 18 20 22 24 v m/sec Figure 6. Observed heights of characteristic waves at different wind velocities. the average height and the upper limit of the height of succeeding characteristic waves as accurately as possible. These limits are indicated in figure 6 by vertical lines. A single cross (x) means average heights without estimation of the upper limit. We find at a wind velocity of 15-16 m/sec a variation in the height of character-~ istic seas between about 4 and 8 m, which may be interpreted as an interference phenomenon. But at higher wind velocities it is diffi- cult to determine the upper limit exactly if the long rolling sea interferes suddenly and the steepened high crests fall forward, often forming breakers of several meters in height. In the open sea, successive waves always differ considerably 22 in height and length, and from time to time a wave comes that is considerably higher than the common run, or its time interval from the preceding crest to the following crest observed at a fixed local- ity is much longer than the time intervals of the waves before and after this particular wave. These longer or higher waves disappear on their track after some undulations, changing their pattern con- tinuously. If it were possible to observe these conspicuous waves long enough from a high point it would seem that they are replaced by others not so conspicuous, while waves with uncommon heights and lengths are growing up at another place. Thus waves with a height of 3.0 to 3.5 m were observed on the "Heidberg" when the usual height was about 1.5 to 2.5 m (wind velocity 5-6 Beaufort). During the storm in the eastern North Atlantic on 27 January, 1951, at 20 m/sec wind velocity from time to time the wave height attained 12-13 m while the average of the common run was about 9 m On 13 January, 1951, at noon, still larger variations occurred at 14-15 m/sec wind velocity. On the average, the common run was about 4 m in height, while some individual waves attained a height of 7 m or more. In the evening of this day, the wind increased to about 8 Bft. with squalls to 10 Bft. and the wave height of single waves arose to 10-11 m while the average was about 8 m (see Fig. 6). The curve in figure 6 represents the relation H = 0.215 $- 2_7l- 66758 2 (4a) with T=T,. Formula (4a) is given by the empirical relationship H = 0.215 X:e72- 6678 23 used in Chapter II (see also [8]). In (4a) Bp = S/v, and GO, T and A are the propagation velocity, period, and wave length of the considered wave given by (la), (2a) or (3a) as a function of ve. If v is given in m/sec, we get H in meters by means of (4a). Table 2 shows the mean wave heights at different wind velocities calculated by (4a) with T = f(v) according to (la), (2a) and (3a) for the three dominating waves. At moderate and high wind velocities the difference between the height of these waves is small, and es- pecially H3 nearly equals Ho. Table 2. Mean wave heights (meters) at different wind velocities v(m/sec). v m/sec 2 4 6 8 109s 0 Feb 16. gc liGws 20 24 Hy O,O5°2 OF26970866w 1635 HO SEs Feels 4sas ro oNa7. Ss 9.8 14.6 Hy, O11 0.43 0, 968" 157" %25:7°9366) 5339562 8"e886) 10077 aoe H 05106" O41": 0593" 1.66 °2.6:°3.7 (5.1 6.65854 10.4 , 1580 It must be pointed out that the height of characteristic waves, if taken as a mean and if considered without respect to the actual wave length is less typical for the state of the sea than the steep- ness of the waves, and the everchanging sequence of wave heights and periods at the rough sea surface with its composite wave motion. This feature also seems to be of more practical importance than any average values. Steep and high waves may be dangerous for ships and amphibious aircraft, aircraft carriers, etc., especially when the waves run in groups and form high breakers, whereas flat swell or "dead sea" even as high or perhaps higher than the steep breaking waves is for the most part less dangerous or not dangerous at all 24 in the open sea. But these longer and flatter waves achieve sig- nificance and practical importance when entering shallower water or when approaching the shores (surf). 3) Composite wave patterns. Interference phenomena. Groups of | ER eS waves. Everchanging patterns and single outsize waves are striking features of the sea surface roughness, and we have to consider these facts in practical wave forecasts. Just these single waves which often cause heavy breakers in storm areas have to be taken into ac- count for practical purposes. The source of outsize waves of this sort is the merging of those waves that advance with different velocities, because longer and faster waves are constantly overtaking the slower running ones. This always happens when there are two or more series of character- istic waves present, and the phenomenon is called interference. More complicated patterns of outsize waves may occur when waves are run- ning in different directions, combining a series of characteristic waves produced by the local wind and "dead sea" (very young swell) running from another direction. This happens under certain meteoro- logical conditions, for example when passing cold fronts with pro- nounced "wind-jumps," or in some sectors of hurricanes, especially in the "eye" where "dead sea" of different directions and locally produced sea interfere. The so-called "cross-sea" is a well known phenomenon, where waves are piled up in single interference patterns to considerable height, often forming irregular dangerous breakers in the fierce sea. But it seems that this tumultuous cross sea is usually dissipated in a rather short time. 25 More complicated sea surface patterns also may be encountered when the wind blows in violent squalls. The effect of violent squalls in gales is~-as every seaman or everyone who has observed the sea in the more stormy parts of the oceans knows--a rather rapid increase of the waves. There are more breakers to be observed and the sea surface is covered with more white foam patches taking on the appearance of more fierceness. But it seems that at first mainly the smaller superimposed waves are increased in their size. Over- taken by longer and faster running waves they pile up the crests and fall forward in foaming breakers (see Chapter II, section 6). The sizes of the largest waves produced by squally wind may correspond to a wind velocity somewhat higher than the mean wind velocity or about the average velocity of the wind within the squalls. Furthermore, we have to take into account that the time average of the wind velocity for the most part is not constant over the whole fetch. This leads to further complications and makes the practical wave forecast much more difficult, especially if there is an increase of wind velocity to the windward of the fetch. Consider, for example, a locality of observation on the western part of a low in the North Atlantic Ocean within the region of NW or NNW winds. It occurs that this region westward of the cold front has a large extension and the wind direction is approximately the same in this sector, but the wind velocity may be 2 Bft. degrees or more higher in the north- ern part than at a given point of observation in the southern part. From this northern region "dead peal arrives at the point of obser- vation, perhaps not even as high as in its region of origin, but high enough to disturb the interference patterns of local sea at the southern end of this fetch. It may happen that these more 26 complicated interference patterns produce a higher and longer sea than that which would be the normal state with respect to the wind conditions in the middle and southern parts of the fetch. Perhaps it may give the impression that the (local) waves have been de- veloped to larger size at the utmost end of the fetch as a result of increasing. On the growth of the sea under the action of wind, see Chapter II. In the following, an attempt is made to explain some strik- ing features of complex wave motion by principles of interference, considering at first steady-state conditions at the end of a fetch over which a steady wind is blowing and assuming infinitely wide waves. Let Aas dos r, be the wave lengths of the three character- istic waves mentioned in Chapter II, one ory o3 their velocities of propagation, and Ay» Ao» a, the amplitudes of these waves. As- suming sine wave profiles with first approximation, we have Oo,t-x Qot - x QG3t -x : io ay sinar(—*>_—) + agsinen(—= —) + asinar(— >). This equation contains both of the variables, the time t and the fetch x. We chose x = O and t = 0 so that at this place and time all three waves have the same phase and y = 0. With regard to an observer at sea, who is measuring the waves at a fixed locality as a function of time, we consider the variations of the resulting wave motion at a given place x = const. Let us take as an example a wind velocity of 16 m/sec. There we have to expect the following wave lengths, periods, wave velo- cities and heights, supposing the wave motion is fully developed 27 according to (la), (2a), (3a), (4a) A, = 107 0 A> = 307 m A3 = 163 m Ty = 8.3 sec T, = 14.0 sec T3 = 10.2 sec O41 = 12.95 m/sec Oy = 21.9 m/sec 03 = 16.0 m/see a, = 3.0 m a> = 3.4 m a3 = 3.3m Fig. 7 shows the resultant wave patterns at a wind velocity of 16 m/sec. The upper wave train (a) represents this theoretically constructed “wave record" at a locality x = 0 for a time interval of 260 sec. The lower wave train (b) represents a "wave record" during the same time at a locality x = 550 m, which means about 1/3 nautical ‘miles away from the locality of "wave record" (a) in the direction of wave propagation. These theoretically constructed "wave records" indicate that groups of large waves are to be expected only inter- mittently at_a_ certain locality. At x = 0 in our example they occur at the beginning, at x = 550 m at the end of the "wave record." The groups follow with a time interval of about 45 sec between each group. Within these groups some waves are growing up to considerable height. But in nature these waves presumably do not attain their exact theoretical height. When approaching a certain maximal steep- ness which depends upon the wind velocity, they become unstable and the crests break over. Because of a certain regularity in the occur- rence of these outsize waves in groups it is to be expected that there is a certain regularity in the occurrence of high preakers too. When a "family" of distinct groups has passed the locality of observation, the sea surface patterns gradually change their char- acteristic appearance. By and by the distinct groups disappear and 28 (0) Wo1y Pulmumop WOGG Ppaj090| S! (q),p10de1 aADM,, SOlj1]O90] jUasajjIP OM} 4O SPUODAS QZ 4O |OAJIJU! |WI} D 4Oy Sps0da41 BADM Payndwod das/Aug| JO AjI00jaA Pulm D 4O SaADM Ppadojanap Ajjny yo wsayjod ananm 2°b14 —_—_— (90S); oz! on oo! 06 08 OL 09 os ov o£ oz ol ° ov o¢ x4 or ° ° Hs 2p LL 2'01—>| AAT AR - : wa we 8'6—+s—_ 0°41 O'0I—ae- ¢' 8 0'0| —ste-—._ 0°] —= -— 0'pI yanvaue roi—e yuanvaue 6+1—+ vanvaueY J), 29 the striking contrast between extremely high waves and lower ones is smoothed out. The waves at the sea surface now show a more irregular pattern without distinct groups, until after some time new groups of waves may arise. When discussing the results of observations in Section 2, a large scattering of periods over a certain interval was pointed out. Scattering periods are to be expected in any way when con- sidering the resultant interference patterns of waves,and the ex- ample in Fig. 7 shows these different time intervals between suc- ceeding crests by horizontal arrows. Thus at the place x = 0 after passing the high crest at t = 2.8 sec, we would observe the next crest after 9.2 sec and the following crests after 7.0; 14.9 (double- wave," 7.73 7-2)3 10.23 9.03; 8.03 14.0; 10.1; 8.63 8.4 seconds and so one These periods are to be compared directly with the represent- ation of the observations in Fig. 3, obtained at a wind velocits of 16 m/sec in fully developed wave motion. Fig. 8 represents some profiles of wave patterns, showing the variations of dominant waves with space and time at v = 16 m/sec. The wave profiles are computed for time intervals of 3 seconds and extend over a distance of 1600 m as shown by the scale of distance at the upper horizontal line. By means of these profiles some outstanding features of complex wave motion may be explained. For example, we find the striking wave ay at t = 0 at the place x = 350 m, but, when progressing, this wave decreases in height very rapidly as indicated by the dashed line which is drawn from & downwards to the right hand side. After 15 seconds this wave has moved about 200 m farther and is to be found 30 A B C D E F G H ~ ee 100 200 ie) 400 a 600 {700 800 900 1000 1100 1200 1300 1400 1500 1600m == Tas a x rial L.\ . L t-Osec | eal eee oe ; : - BOLD waa NaS, t-6sec ‘la’ oss t-9sec pa me NZ t= 12 sec oS t=15sec : Fig. 8. Profiles of wave patterns, showing the variations of dominant waves with space and time at a wind velocity of 16 m/sec. at B only as an unimportant elevation, whereas the following crest a5 has risen to a wave of considerable height. We find similar rapid variations at other places too, when examining the changing wave patterns with regard to space and time, for example, between G and H at a distance of 1400 m. This rapid decay on the one end 31 and growing up of single waves on the other end is a fact well known from observation of the sea. But besides this, there are other waves which change their form much more slowly, as indicated by the wave placed at x = 550 m (B) at the time t = 0. When it progresses a distance of 280 m, we find this wave after 21 seconds only slightly changed in the crest b)- If we should observe the wave motion at the point B (x = 550 m on the distance scale in Fig. 8) we should get the "wave record" (b) represented in Fig. 7 (the lower one). After passing the wave crest b, at t = 0 the next crest (b,') follows 8 seconds later, slightly higher than the preceding one. The next wave would be the wave with the crest ai» but this wave almost disappears in the following broad trough and in many cases it will not be possible to recognize this intermittent wave. At the time t = 23 sec the next high wave crest, @>») passes the locality of observation. The "period" between by ' and a> or the time interval between these two striking waves is 14.8 seconds. Waves like these occur rather often in the composite surface pattern of waves and may be called "double-waves" [9]. In all cases the scattering of periods of suc- ceeding waves and the continuous variations of the shape of wave profiles in rather short time intervals are striking features of ocean waves. Similar wave profiles like those in Fig. 8 are represented in Fig. 9 at a wind velocity of 19 m/sec. Within the relatively short time of 21 seconds the "wave group" between points A and B changes its shape considerably when progressing a distance of about 350 m. At t = O we find a group of waves between x = 500 m and 32 B ¢ 500 1000 { 1500m e Sa a ae eee eee Pe \' bs h\ Cy \ = : 2 {i IN a t-Osec NY Sw, \ \ \ \ i ‘ \ \ \ \ \ \ \ \ 4 = f Vaan t-3 \ \ \ \ \ ‘ y \ \ a eee . T 1 7 ~~ t-6sec aw, ae as MH ‘ Mi \ LEN x (die tQced 7 Wo) eee a ‘ . \ \ \ \ \ = LES /\ bo \ Co , a Sra ey, v , t-12sec ‘ \ \ \ y \ t i \ \ X=04: N T X77 t- 15sec \ : \ 1X. a \ (Ain = 3 i — t- \ \ew/ ; t-18 sec ‘i \ a \M! Do 2 a — ZL) 1 \ =, (C3) Te Fig. 9. 1, t= 21 sec Profiles of wave patterns, showing the variations of dominant waves with space and time at a wind velocity of 19 m/sec. x = 1000 m with the crests ai bo) by. After a time interval of 12 seconds the height of the wave by has decreased while the small elevation in the trough of the "“double-wave" between points B and C has grown up to Coy forming a long stretched crest. During the same time another wave @5) grows up at the rear edge of the "group." 33 This wave now plays a role similar to the wave a, at t = O, where- as the wave a, after about 21 seconds is to be considered as the “middle wave" of this group. The former front wave by has almost disappeared after 21 seconds, while it seems as if this wave is re- placed in the group by the former "middle wave" bo. The “ group" as a whole remains behind the waves. This well-known property of wave groups is observed very often at sea and the wave profiles in Figs. 8 and 9 show these variations by superposing the three char- acteristic waves evaluated in Chapter 2. The displacement of the highest (not individual) waves in Fig. 9 would lead to a "group velocity" of about 8 m/sec, whereas the steepest characteristic wave, the "sea" at a wind velocity of 19 m/sec has a velocity of phase propagation. of 16.4 m/sec (Ty = 10.5 sec, according to formula (la), Oi = (g/2r)T,). Theoretical “wave records" or "wave profiles" like those in Figs. 8 and 9, are based on the assumption of three characteristic waves, and are valid in the fully developed state of wave motion. If the wave motion is not fully developed, it is first necessary to calculate the dimensions of the characteristic waves in their dif- ferent stages of generation at given wind velocities. In any case, it seams possible after these computations, to evaluate character- istic wave profiles or “wave records" for practical purposes. These profiles allow us to predict some striking features of composite wave motion as required by practice, for example: (1) the periods of succeeding waves, the heights of succeeding waves and their steepness. The range of these elements and their average and maximum values. 34 (2) The time interval between the steepest or highest waves in the complex pattern of wave motion observed at a fixed locality. (3) The occurrence and the behavior of wave groups and if possible the appearance of “high breakers." Sb] Chapter II The growth of waves under the action of wind 1) Introduction and general remarks Any attempt to calculate the growth of the waves under the action of wind requires the knowledge of both the energy transfer from wind to waves and the rate of dissipation of wave energy in different phases of wave development. The sea may grow only in the case where the supply of energy by wind exceeds the loss of energy by friction and turbulence. At a given wind velocity the waves attain their fully developed state when the energy transfer by wind A equals the energy dissipation by frictional forces D, which are connected with turbulent wave motion. This means that the sea is "mature" when the energy balance A -D=0. As to the question of energy transfer by wind it seems to be necessary to take into account the complex pattern of sea surface roughness, because the stress or the total drag of the wind blow- ing over a rough and wavy air-sea interface depends to a large ex- tent on the "short-wavy" roughness, which is always superimposed on the main profile of larger, striking waves. The energy transfer from wind to waves is due partly to normal pressure components and partly to tangential pressure components (tangential stress, drag), and to do a net amount of work both of these wind force components should be regarded as acting on the particle velocity of water due to the wave motion. Therefore, the actual velocity difference be- tween the particle speed and the wind speed immediately at the sea surface must be considered. But at the present state of our know- 36 ledge it seems very difficult to estimate with the necessary de- gree of accuracy the value of the single components of wind force under different conditions, especially when considering the compli- cated sea surface pattern with its complex wave motion. Further difficulties arise in the calculation of the work done by the wind on the wave motion, when the actual wind speed at the sea surface or the actual velocity difference between the motion of water particles and the wind is sought. Under these circumstances it seems more expedient at present, to make use of empirical relationships between the "effective wind stress" at the rough sea surface and the wind velocity. The re- sultant action of tangential stresses and normal pressures may be estimated by certain observations and related to the average wind speed at "anemometer-height." "Anemometer-height" is defined as a height of about 10 m above the sea surface, where the vertical in- crease of wind velocity with height is relatively small. The second question which becomes quite important in this prob- lem is the question of energy dissipation with turbulent wave motion. The energy dissipation may be due to viscosity or molecular friction, in the absence of turbulence. But in the case of ocean waves, the energy dissipation by eddy viscosity (which Ekman calls turbulent- friction or virtual friction) has to be taken into account. This turbulence in the uppermost layers of the sea seems to be caused mainly by the breaking of larger and smaller waves, associated with whirling and stirring up of the water masses (eddying) due to un- stable wave motion. If there are any other more efficient causes which lead to an additional eddying of the sea surface water, that is, to an additional turbulent motion, the wave motion or wave- o7 formation will be damped out faster. It is well known that a rough sea can be calmed to a certain extent by whirling water which is upwelling on the windward side of a drifting ship. This "natural" kind of wave decay is often used by captains of ships which have to lie to when caught in storms. Lying athwart or nearly athwart to wind and sea, the ship slowly drifts to leeward, causing by its motion a zone of upwelling water to windward. This whirling, up- welling water is an additional source of turbulence and energy dis- sipation with regard to the local wave motion and acts - as exper- lence shows - as a damper on waves, and at first the steep breaking waves are damped. It is this whirling water caused by the drift of the ship which destroys to a certain extent high and heavily breaking seas mostly far away from the ship to windward, or far enough not to become too dangerous for the ship. At lower wind velocities the same effect of upwelling water is to be observed, and during the voyage with the "Heidberg" these additional effects of turbulence could be studied when the ship drifted southwest of the Azores for several days with engines stopped. At a wind velo- city of 8 to 10 m/sec the relatively small but steep waves of the common run were breaking about 50 m to 100 m windward of the ship and the upwelling water in the drifting path near the ship was al- most free of larger breaking waves. Similar effects of additional turbulence are also to be ob- served in the wake of ships under way. This is most striking in the case where a slight breeze causes smaller waves or ripples at the sea surface and the sun shines on the water. One can follow the wake of the ship over rather long distances as a nearly smooth, 38 bright path across the rippled surrounding surface which, with its rough wave motion, appears darker. With a very weak wind, when the first small wavelets grow up in the initial state of wave formation, the energy dissipation ap- parently is determined only by viscosity or molecular friction (if the water is not disturbed by strong currents, for example, strong tidal currents or other turbulent motion). Since surface tension is of decisive influence on small wavelets (besides gravity and viscosity), a minimum wind velocity for the generation of initial waves is found (Neumann [14]). This limit is a wind speed of about 70 cm/sec, and the first ripples that will be generated are those with a wave length of 1.75 cm. If the wind speed is 90 cm/sec the wave length of the ripples has increased to 5.2 cm, provided we dis- regard capillary waves which are generated together with the longer (gravity) waves. (The wave length of these capillary waves at a wind velocity of 90 cm/sec would be about 0.57 cm.) These "initial waves" grow in such a way, that their steepness increases with in- creasing wind speed. At a wind velocity of about 123 cm/sec these primary waves attain the steepest form possible for stable waves. The ratio of wave height to wave length is in this case H/A = 1 which is the maximum steepness computed by Michell based on the theory of Stokes. The wave length of the ripples in this state of development generated by a wind of 123 em/sec velocity is about 1O.5icm..* If the wind becomes a little stronger, the pointed crests of *In shallow water layers, where the bottom friction is an influenc- ing factor, the wind velocity for generating waves of maximum steepness is higher than in the case of "deep" water, and the wave length of the ripples with maximum steepness is shorter than in the case where the waves "do not feel" the bottom [14]. 37 the wavelets take on a "glassy"appearance. The wavelets have ex- ceeded the initial state. They become unstable and "break up" at the crests. This "break up" rather soon changes to a distinct "break over}' when the wind speed increases, and observations show that at a wind velocity of about 150 cm/sec the crests of the fully developed ripples clearly fall forward. The wave length of these small "break- ing" waves is about 20-25 cm. In the further development of wind generated waves into the state of real ocean waves, the turbulence which is now present and connected with the breaking unstable waves in all phases plays an important role. This turbulence (which implies energy dissipation) increases rapidly with increasing wind velocity. The first attempt to calculate the growth of ocean waves due to the action of wind at different stretches of water over which the wind has blown (the fetches) and at different durations of wind action, was made by H. U. Sverdrup and W. H. Munk [1]. It was the first approach to a scientific basis for this special problem of great practical importance. Until this remarkable work, the results of which have partly been used in practice since 1942 and which was published in its final form in 1947, no one had attacked the com- plicated problem of wave forecasting. This first attempt incited further investigations both on theoretical and empirical bases, so that new ideas and more comprehensive observations were soon forth- coming. Even if we know today that some of the assumptions made in this treatise do not hold or are an oversimplification of the mechanism of wave generation, this first attempt has its value as a pioneer work because it stimulated further research and new approaches, like this report, and presumably other scientific work 40 which will follow. H. U. Sverdrup and W. H. Munk compute the work done by wind to waves ("significant waves") by normal pressures and tangential stresses separately. But there are some uncertainties in their as- sumptions and the forthcoming calculations seem inconsistent with the degree of accuracy which is required in the energy balance, when considering the growth of ocean wave motion under the action of wind. For estimating the tangential stresses and the work done by this wind force component on the wave motion, Sverdrup-Munk had to assume a "resistance coefficient," Y; which was constant at all wind velo- cities, although they mentioned that this would not be true in the case where the wind velocity differs too much from the wave velocity. Under these conditions the value of vr would probably be greater. In the present state of our knowledge we have to take into account the fact that there is really substantial evidence that y> varies with the wind velocity ([8], [13]) and with the stage of wave de- velopment. The assumption of a constant y--value is only a rough approximation. A similarly insufficient approximation is represented by the assumption that the tangential stress (drag) over the wave profile is constant at different parts of the wave even if we assume a constant resistance-(drag-) coefficient a over the wave. Besides these important questions of energy transfer from wind to waves, which in the meantime were also discussed by Schaaf and Sauer [15] with respect to the tangential transfer, it must be mention- ed that Sverdrup-Munk do not include the dissipated energy in the energy budget of growing and fully risen waves. After mentioning the idea, they disregard turbulent friction, arguing that turbulent 41 friction or eddy viscosity would give too rapid a décrease of wave height, and that it would be necessary to introduce a smaller coefficient applicable only to wave motion. These conclusions are not quite clear and perhaps not admissible. If in the upper layers of the water where wave motion takes place, a certain state of turbulence is present, this "disordered motion" superimposes itself upon the regular wave motion too, without regard to the causes of this "disordered motion." (See the notes on observations in the wake of ships underway and in the windward upwelling water of drift- ing ships.) In comparison with this "turbulence" all secondary effects of eddy viscosity, which are smaller than this "turbulence" (including molecular viscosity) may be neglected. L. Prandtl [16] states: "Man kann annehmen, dass wenn zwei Ursachen vorhanden sind, die einen Austausch hervorbringen, der wirklich eintretende Aus- tausch ungefahr mit dem erdsseren von beiden Austauschbetr&gen ibereinstimmt." Our present knowledge of turbulence in the surface layers of the ocean at different stages of wave development is very meager, and we have to suppose that this turbulence depends to a certain degree on other oceanographical and meteorological conditions. Oceanographic observations indicate that the coefficients of eddy viscosity are about 1,000 to 100,000 times as large as the ordinary viscosity coefficients. Assuming that dissipation takes place by ordinary viscosity only, the effect of friction is neglected in the energy balance by Sverdrup-Munk. These authors explain the observed decay of waves only as the effect of air resistance against the advancing wave. When wind-generated waves spread out from the 42 fetch area into a region of calm or into a region where the wind velocity is small compared to the wave velocity, the waves naturally meet an air resistance. Due to this air resistance a loss of energy takes place and a decay of waves will be observed. In certain cases, turbulence may play an unimportant part in damping the wave motion, or may be absent altogether, which happens perhaps when the waves travel through a region of smooth sea. But this fact does not ex- clude the significance of eddy viscosity when the waves are growing in the wind area or when they are maintained by wind action. In this case, the turbulent state of real "wind sea" and the energy dissipation by eddy viscosity is not to be neglected in the energy budget. Similarly, eddy viscosity has to be taken into account when considering the decay of waves (swell), if the waves travel through a region of turbulent sea. Wind generated waves present themselves to the observer as a series of more or less "hill-like" irregular crests separated by intervening troughs. The formation of the typical "short-crested wind-waves" may be explained partly by the turbulence of the wind and the different wind pressures on the windward and leeward slope of the wave profile. In any event, it is to be expected that irregu- larities of the air current counteract the formation of long-stretched wave crests. This especially seems to be the case where very "young sea," with rather short wave lengths, but characterized by a great steepness, is generated. If once arisen, these shortcrested waves will themselves disturb the air motion and react on the state of atmospheric turbulence over the sea surface. On the other hand, especially in stormy weather, the fully developed sea with its 43 characteristic waves has more long stretched wave crests, as al- ready pointed out by Cornish (see Thorade, [17]). This fact must be founded in the nature of ocean waves and perhaps it corroborates theoretical results. Taking short crested waves into account, Jeffreys [18] could show mathematically that the waves with long stretched crests require a smaller amount of energy from wind for growing than the short crested waves. The long crested waves there=- fore have the better chance to grow than the other ones. The intensity of turbulence or the irregular fluctuations of the wind perpendicular to its average direction may be assumed to be nearly of the same order of magnitude as the fluctuations in the average wind direction superimposed on the mean wind velocity. For a sufficiently long time interval, the fluctuations perpen- dicular to the average wind direction and their effects on irregular wave motion probably cancel out. A characteristic wave motion and a significant "sea" therefore will be developed only in the average direction of wind, and this fact seems to be proved by experience at sea. The characteristic "seas" propagate in the direction of the average wind, if we consider undisturbed "wind-seas." The irregularities of wave motion, mostly concerning smaller superim- posed waves, may be considered as “perturbation effects" when deal- ing with the growth of the sea under the action of wind and when trying to comprehend the essential features of ocean waves. But nevertheless, we have to keep in mind that the ocean waves with their limited crests are to be considered strictly as a“three dimensional phenomenon," when going into details. This was shown by Jeffreys (see Thorade [17], p. 34). When considering the growth of the sea under the influence of wind action, many possibilities of generation have to be taken into account. It happens very often that the wind encounters an "old sea," has to destroy it (or partly destroy it) and to generate a new wave motion. Therefore, allowances must be made for waves that are present when the wind starts blowing. Or, in another case, the wind may increase with time and the "sea" grows slowly with this in- creasing wind speed. However the simplest case is where the wind in the generating area is constant in time and space and begins to blow over an undisturbed water surface. First we shall assume a wind field of constant velocity and direction in the following con- siderations. The present approach to wind waves forecasting deals with the growth of the complex sea in the area of wave formation depending upon the wind velocity, the stretch of water over which the wind has blown (the fetch), and the length of time the wind has blown over the fetch (the duration). A first attempt was made to take into account the composite nature of wind generated waves as in- dicated in Chapter I of this paper. It seems that the development of composite wind generated wave motion from small steep waves to the case of fully arisen sea is not a continuous process. Discon- tinuities are to be expected in certain states of composite wave formation, because the relatively short "sea" is to be found as a characteristic steep, breaking wave in all further stages of ocean wave patterns up to fully developed sea. Probably we have to assume these rather steep, breaking waves in all higher stages of develop- ment so that longer waves with a higher amount of energy may be 33) generated at the rough sea surface, until the net energy supply by wind equals the energy losses by dissipation in the composite ocean wave pattern. Until now, it has not been possible to observe a continuous growth of certain waves after they have attained a cer- tain height, length and maximum steepness. These waves breke heavily in stormy weather and they did not disappear or increase when the fetch increased or the duration of wind became longer. But when ex=- ceeding a certain fetch longer and flatter waves rather soon emerged at this rough sea surface with its steep wave motion, and these waves probably grew independently of the rough sea as individual waves. The superposition of these longer waves with a phase velocity great- er than the wind velocity, and the characteristic "sea" with a phase velocity smaller than the wind velocity finally led in the fully de- veloped sea to significant fluctuations of wave periods (time inter- vals between succeeding crests at a fixed place), heights, and to the occurrence of outsize waves, groups of waves and other phenomena, as described in Chapter I. This first attempt to consider the growth of complex ocean wave motion under the action of wind, and the ideas involved in theoretical calculations are based to a large extent on observations and on empirical relationships. Both of them are incomplete today, ani it is quite possible that more comprehensive information on complex sea wave motion in the future will lead to better approxi- mations and theoretical treatments. This approach does not claim to have the desirable completeness. At present, it seems that we are still rather far away from a complete understanding of the prob- lems of ocean wave generation and behavior. At any rate, it seems 46 necessary for wave forecasts to consider the complex nature of ocean waves as they present themselves to an observer at sea. Fur- ther theoretical work and observations well suited to support new ideas are needed to compile the knowledge which may lead us step by . step to a satisfactory solution of all the problems about ocean waves and their generation. 2) Energy transfer from wind to waves The energy transfer A from wind to rough ocean waves depends at a given wind velocity upon the pushing and dragging forces which act at the sea surface, and therefore it depends upon the present state of wave development itself. In this way, the actual roughness conditions of the air-sea interface become important with regard to the problem. The effective wind force @ may always be split up into a component Ge acting normally to the "wavy" interface, and into a component oy acting in the tangential direction, as represented in Fig. 10 which shows a schematic profile of a rough wavy surface. At present, it seems very difficult to estimate the real dis- tribution of both of these components Ce, and c,) over the wave profile with the necessary degree of accuracy, especially when the rough irregular forms of ocean waves are considered. Therefore, an attempt has been made to split up the resultant effective resistance of the actual rough sea-surface into a "pressure-resistance" and a "friction-resistance." This has been done for practical reasons by considering the smooth form of the wavy air-sea interface and approximating it by a "general wave profile"("Hauptprofil,” G. Neumann [8]). The resistance of the "rough" superpasitions on this general profile, which would have to be counted as "pressure- 7 Fig. 10. Schematic wave profile with a rough surface (7), and wind force components ( 7, Z,) of the effective wind stress T. (@ = ‘velocity of wave propagation in direction of the wind velocity v. resistances" in the exact meaning of the definition, will be attri- buted to an effective "frictional-resistance" with regard to the general profile (L. Prandtl, [16], p. 160). In this way, the wind force acting on each surface element of the rough sea surface may be divided into an effective pressure force and an effective friction- al force, where the first one may be considered as a normal com- ponent and the second one as an effective tangential component with respect to the general profile. The work done by these single components on the waves at present cannot be estimated with the necessary degree of accuracy, if we consider their actions separ- ately. It only seems possible to examine these effects in a merely formal manner, if we consider a very simple main profile of waves, as is shown, for example, in Fig. 11. We assume a simple sine wave, for which the elevation relative to the undisturbed level may be given by 48 Fig. 11. Streamlines of the air over a wavy surface, and distribution of wind force components Ge and Cy in schematic representation. 7 = 8 sinx(x =- Ot), (1) where a is the wave amplitude, %= 2r/, the wave number and © the velocity of wave propagation (phase velocity). The horizontal com- ponent of the displacement of a water particle at the surface is — =a cosx(x- Ot). (2) With these expressions, we have the vertical (w,) and horizontal (u,) component of partical velocity as given by wy, = 97 /t = - axocose(x - ot), (3) and axosinw(x - ot). (4) ei = 9& /ot a When considering waves of finite amplitude, Stokes' theory con- cerning irrotational waves leads to the important result that upon the completion of each nearly circular orbital motion the water particles have advanced a short distance in the direction of wave propagation. The average velocity of this forward motion at the sea surface during one wave period is uy! = ar 2e-o. Thus Stokes' waves with finite amplitude are accompanied by a horizontal mass transport of water. Taking this into account, the horizontal component of particle velocity at the water surface may be written U. S++ u.' = axrcisine (x = ot)l+ Seria 2 (4a) The average rate at which energy is transmitted to the wave by normal pressure is anes fea dx, (5) and by tangential stress, considering (4a) 1 A beam Ade Besa (6) fe) Whether these forces do a net amount of work on the wave motion or not depends upon the distribution of Ca and ( along the wave profile. In order to do a positive amount of work on the wave, the wind force components have to be in phase with the components of the particle velocity. Because the normal pressure on the windward slope of the wave profile is on the average greater than on the lee- ward slope, these pressure forces in general will do a positive amount of work, as long as the phase velocity of the wave is smaller than the wind velocity. In the case where the phase velocity O exceeds the wind velocity v, the wave form encounters an air resistance, 50 because the wind relative to the wave acts like an opposing wind. When © > v, the vertical particle velocity and the normal wind force component are out of phase by a difference of 7, and energy is given off from the wave to the air. In this case the wave motion is slowed down. There will be no objection to considering the action of this normal pressure component T,, and H. Jeffreys [18] took it into ac- count as the main source of wave energy. He assumed the effective pressure component to be proportional to the product of the density of the air p', the square of wind velocity relative to the wave velocity (v, - ao), and the slope of the wave profile 97/dx. These assumptions are the most plausible which may be made with regard to the action of normal wind force components as considered here, but some uncertainties still exist when going into details. For example, there is the question of the accurate definition of the difference Vo oe where Vo is the wind velocity immediately at the sea sur- face, and questions about the so called "sheltering coefficient" under different conditions (wave form). However, it seems that there are more difficulties encountered, and opinions differ when the action of tangential wind stress compon- ents is considered. If we assume that the wind blows over a general wave profile which may be considered really as an ideal "smooth" sur- face in the hydrodynamic meaning of this word, then only viscosity stresses would act. The work done by these viscosity stresses would probably be small compared with the effect of normal pressure com- ponents. This perhaps may be the reason that Jeffreys did not take into account a transfer of energy by tangential stress and considered oil this process as negligible. This really seems to be the case when the generation of initial waves or wavelets at very low wind velo- cities is considered ([18], [14]). But if we refer to actual ocean waves, we have to regard a certain general profile with all the smaller superimposed waves, including ripples, as a “rough" wavy sur- face, where the "effective frictional forces" are determined not by viscosity but by the "roughness" of the wave profile. Therefore the possible work done by these effective stresses, Tis on the wave motion may be of the same order of magnitude as the work done by normal components acting on the main wave profile. The distribution of the effective stress C4 over the wave pro- file probably depends upon several factors (roughness, wave forn, wind distribution over waves, etc.), all of which may be different under different conditions. If we assume with Sverdrup-Munk [1] that Cy is constant along the whole wave profile, than in the inte- gral (6) the periodic term would vanish. This means that this force would not do a net amount of work at the horizontal component Uy (in (4a)) of particle velocity. But the assumption Tc, = const seems to be an oversimplification which annuls wind effects of the same order of magnitude as the effects considered with the normal pressure components. Even though it seems very difficult at the present state of knowledge to estimate the accurate distribution of T, along the wave profile, we have to assume that Ct, is different at different parts of the wave. The horizontal stress component fay be written 2 Cy = p'f, Vo 3 where p' is the density of the air, Me the wind velocity immediately 52 over the "surface" (the wave), and f, a "resistance coefficient." This dimensionless number depends upon certain surface character- istics and its value has to be determined under different conditions. Here, the same question arises for the actual or “effective” wind velocity Vo° But we may leave this question open at first and re- gard only the relative distribution of Vv, over the rough wavy sur- face (Fig. 11). Then we have to expect that v, will be greater over the crests than over the troughs, and “TC, will be greater over the crests too, even if we assume a constant “resistance coefficient" over the wavy surface. Therefore, energy is also transferred by the tan- gential stress which the wind exerts on the wavy surface. The effect of this drag is to speed up the motion of particles at the wave crests and to slow down the motion of particles at the trough; but the speed- up is greater than the slowdown, so that a net increase in wave energy results not only by "normal pressures" but also by "frictional forces." It is seen that the attempt to consider separately the effects of the single wind force components encounters many difficulties and uncertainties, even if these effects may be written in a merely for- mal waye Let us assume that TC, on the windward slope of the waves is re- latively greater than on the leeward slope and T, ~ 0 fax. Then we may write for the distribution of the normal pressure component over the wave profile Ty = Cp, + Tleosx(x - Tt) (7) where x om represents a constant pressure value over the wave profile. If a relatively higher wind velocity over the wave crests is con- sidered, the distribution of wind velocity over the wave may be 53 written in the form vy = v,[1 + 75 sinx(x - Gt)] where 8 = 2a/, , and v5 means a constant average value of the wind velocity over the wave profile. Dropping terms of higher order in 5, we have wet = pol + 275 sinx(x - ot)], and - 2 oe = p'f, vo" = p'f, ¥oq[l + 278 sinx(x - ot)] (8) Here, p'f, ae means a constant value of tangential stress over the wave. Fig. 11 represents these simple assumptions, where in the upper part the streamlines of the air over the wave are shown in a schematic distribution. This distribution may or may not be symmetrical to the wave profile. But in any case it is to be expected that the streamlines are crowded over each crest, so that over the erests a bundle of streamlines with higher velocity rushes ahead to- wards the lee. An asymmetrical distribution of the air current in this case may perhaps affect the wave in such a way that a flatter windward slope and a steeper leeward slope of the wave profile may result. In each individual case the distribution of air currents may be more or less aSymmetrical and complicated. It would be suffi- cient to consider the effective wind force components to be composed of a series of harmonic terms of wave lengths A, 2A, 3A, ......, but of these terms only the one in phase with Uy and Wo does a net amount of work. For the average rate of work done on a wave of length A and of phase velocity ©, we get together with the formal expressions (7) and (8) according to (3), (4a), (5) and (6) 54 AL = + Tso rox cos x(x - Ot) + Ge cose (x - Ot) ]dx A AL = 4 ea To. | [wS0 sinx(x - Ot) + 20°6°o sin? x(x - ot)+ + 16° ]dx after the term with 6° in the expression for A, is dropped. The terms with sine and cosine do not contribute to an energy transfer from wind to waves, and on the average after integrating over one wave length, the result is ul role ' e AL 16 ia o (9) 2 2 er p'dé : fon i es Pes A 1B (0) t The accuracy to which AL and A. can be separately evaluated is perhaps not sufficient, because the effective wind force components depend upon several unknown factors of the hydrodynamical character of the sea surface (G. Neumann [12]). But one might see that both of the components may contribute to an energy transfer, and only in the case where in (8) T, = const. over the wave, does the average work of this drag at the particle velocity Uy become zero. H. U. Sverdrup and W. H. Munk [1] therefore only consider the work done by the stress Glee = const. at the mass transport velocity Uy» which accompanies Stokes' waves of finite amplitude. Let to ' = 2 ce +p fly, =), ; = = where CeO for OVv,, then A =+2rop't (¥.-0)° .o . n ee Pot, XV ‘ 95 The dimensionless coefficients of proportionality fn and f, fin (9)) prohably depend upon the actual conditions at the air-sea inter- face (stage of wave development, wave form, relative wind velocity, stability of the air above the interface, etc.), and therefore they will not be constants. Let 8, = S/v,, and collect the dimensionless factors by putting Yon = 3 Of, and Yo, = 2°6°r, . We then have A. =+p'y Cie w Grae 36 (10) no = on‘Fo fs) 0° Po A, = p'Yo4 (By) ¥> By - (11) The wnknown "conditions" of the rough surface are now involved in the dimensionless quantities Mian and Toe: We may call these quan- tities "resistance factors" or "frictional factors." They are given as functions of Bp = %/v (or p, = 9%/v, resp.), because the hdyro- dynamic "roughness" of a surface not only depends upon the height of roughness elements (in a geometrical sense), but also upon the steepness 6 of the waves and their form, which highly determines the roughness conditions of the sea surface. Because 6 = f(B), and the roughness involved in fa and f, may be expressed as a function of 8, it is to be expected that the quantities Yen and Yot are functions of B too (G. Neumann [8]). In the formal equations (10) and (11) neither the single resist- ance factors, nor the effective wind velocity Vo are known with the necessary degree of accuracy. But it seems possible to estimate the effective wind force - the "effective stress" = by means of emp- 56 irical relationships, which may lead to a satisfactory approximation. Let 2 A= A, + A, = p'vplyy(B) + yy (8)(2 - 8°], (12) the combined action of both wind force components or A= o 'v3B-C(B) ; (13) The effective factor C(p) with the meaning of a resistance coefficient may be estimated empirically and determined as a function of 6. The hydrodynamical characteristics of the rough wavy sea sur- face are now implied in the dimensionless quantity C(p), and its value is related empirically to the wind velocity at "anemometer height," say, 10 m above the mean sea surface level. This means that C is determined by observations in such a way that it refers to the wind velocity at a certain height by definition. This empirical method is not a very satisfactory one, but steady pursuit in this direction may in the future yield a means of determining better approximations. At present this empirical way seems to be the only approach. For estimating the resistance coefficients, several different methods have been tried. As already mentioned, H. Jeffreys [18] considers only the normal wind force components, and assumes for the effective wind pressure a priori a certain distribution. Let the effective pressure component be p*; then we have according to H. Jeffreys p* = Sp'(v - &)* dy/ax H. Jeffreys calls the coefficient of proportionality, 8, the "shelter- ing coefficient" ("Streamlining coefficient" according to Sverdrup- Munk {1]). The work done by this force per unit surface area and unit time on a wave, will be 57 or with (1) and (3) <= + ptes(v - yoo % (14) The dimensionless factor =e sn? (28) = 306° (15) corresponds to the resistance coefficient. (14) may be written again A= p'v8-CCB), with C(p) = z sr-6-(1 = p)° in analogy to (13). Based on observations of waves in their initial state, Jeffreys evaluated 3 = 0.30 (0.27). Another attempt for estimating the resistance coefficient was made by H. Motzfeld [19]. The distribution of the pressure p over wooden wave profiles of different form was determined by experiments and measurements in a wind tunnel. For the width b = 1 of a given profile, the pressure resistance S 4G [> sina ds, fe) where S is the "length of unrolling" of the wave, ds an element of the wave profile, S, and a the angle between the direction x of flow and ds. With dy = ds-sina, the integral has the value Waa { p(y )dy . The pressure resistance over a wave length A is given as the plane area bounded by the curve p(y). If we put 58 Wa 2% pteg(v - «)+Aa yg sec?) (26a) for the surface area of the width one and the length A , Cq may be evaluated by the values of p determined by experiments. The results of Motzfeld's measurements seem to indicate the proportion- ality eqn (15)3/?, but it seems uncertain whether these results of measurements over smooth, rigid wave profiles are applicable to the conditions at the actual sea surface. Using an empirical relationship [13] between the wind force exerted at the rough sea surface and the wind velocity, Corr = f(v), the author [8], [13] made an attempt to estimate the effective value of the wind force as a function of wave development at differ- ent wind velocities. By means of a relationship between 6 and 8, as suggested by Sverdrup-Munk [1], it was possible to relate Core = F(p(v)) where, according to a previous paper [14] Cy = sax = sO , (16) with s = 0.095 was used as suggested by empirical evidence. Both assumptions (15) and (16) lead to nearly the same numerical values Cas when considering initial waves with the steepness 6 = 1/10. This follows with Jeffreys' assumption from (15) 3r°8* = 2106-1075 and from (16) sw& = 2.98-107°, But when considering the generation and the growth of initial waves with increasing wind speed it seems that the relation (16) holds good. The determination of the Cg-value at different conditions of the sea surface is necessary for an exact evaluation of energy transfer from wind to waves, and further in- vestigation of this point is indicated. Another important empirical relationship deduced from experience, 59 which we shall use to a large extent in the following considera- tions, is that the steepness of the waves, & = H/a , depends on the stage of wave development. Such a relationship has already been suggested by 0. Krimmel [5], but H. U. Sverdrup and W. H. Munk [1] first related the ratio H/, to the "age" of the waves and used this relationship for theoretical discussions. The stage of development or the "wave age" can be conveniently expressed by the ratio of velocity of wave propagation O&O to wind speed v, that is Bp = S/y. The relationship & = f(f) is shown in Fig. 12 where the observed corresponding values of H/A and 9%/y were plotted. While Sverdrup-Munk did not fit an empirical curve & = f(B) to the observed data and chose another way to the solution of the problem, we put, in accordance with a previous paper [8] 6=2ne"P for 1/31/3 a definite relationship be- tween the steepness and the wave age already appeared fairly well established by older observations (see [1] and [8]), this is not the case for the “initial state" where B<1/3. Based on the collect- ion of data by Sverdrup-Munk only four not very reliable measure- ments in the state B<1/3 were available. In Fig. 12 some further 60 *esuTT LTIng Aq umoyus dtTysuoTyeTed peunssy *d = A/-O OTZEI OUR YSUTeZe peqgoTd V/H = 9 sseudee93s eARM “*2T °3TA ool e e e OS © ysu0d= Q Ov O¢€ G2 Od orm | (,,549qp!aH | S'W) uupwnaN o (oom | (€) YUNW- dnsapsaas Aq payoa|j0o e Le 2SUOI}DAIaSGO 6! e@1 21 91) Sil pile el 21 jg ii Of 60 80 £0 9°70 GS'O ~0O. £0 20 1,0 .e) vt rn OD DOr O W X % Ul § 277 61 observations are plotted, obtained during the voyage with M. S. "Heidberg" to the West Indies. The observations were taken near the coast of San Miguel (Azores) and along different coasts of the Caribbean Sea. These observations indicate a rather rapid develop- ment of wave steepness in the stage of very young waves up to a maxi- mum value of 6 = 1/10 to 6 = 1/7. This steepness seems to remain nearly constant until the wave age attainsavValue of about p = 1/3. In this early stage of development a rather intensive breaking of these very young waves is to be observed. But this state of wave de- velopment with B<1/3 passes so rapidly, especially when the wind is stronger, that the exact relationship is of very little consequence to the later development of the sea. At present, all observations of very young waves seem to indicate that it would be a good approxi- mation to assume rather steep waves and 6 = const for the stage of wave development B<1/3, and to neglect the initial stage (perhaps B<0.05) where the first disturbances grow with increasing steepness. In the first stages of wave generation let us consider a gene- ral wave profile with the steepness 6 = 2p. If we assume that nor- mal pressure forces on this steep wave are the effective forces needed to do a net amount of work, we get from (10) A= o's@pv3(1 - g)°8 ’ (19, where the resistance coefficient has been replaced by (16). With the dimensionless quantity c,(p) = smp(1 - 8)? (20) we have A= p'vg.c,(p) for p1/3 to be given by (25) as a function of the stage of wave development or the wave age B, it follows with (17) from equation (12) that A= o'wBLy7(B) +s'rne 7 - g)7] o A/a pS (27) is the rate at which work is being done by the wind on a wave, which is characterized by its wave age 8. With Co(p) = y2(p) + stm ne "(1 - gs)? 5 1/31 A= p'v>p*C,(B) sd awa tec sBie (29) where 66 c(B) = (8) + s'rne 7m (1 - 8.) =g*rn-en.° (l= px)> (30) In the stage of fully developed complex wave motion C3 (B) is given by the expression (25). 3) Energy dissipation due to turbulent wave motion Energy may be dissipated by viscosity (Dy ) or by turbulent motion in the wave (Dj). The viscosity of the water is so slight that this effect can be neglected when dealing with the growth of real ocean waves. Only the process of generation of primary wave- lets, which are not breaking, is apparently influenced by viscosity [14]. But the "sea" evidently is a turbulent wave motion, and when considering its growth under the action of wind the effect of turbul- ence has to be taken into account by introduction of an "eddy vis- cosity" or“virtual friction." The dissipation of wave energy due to a viscosity coefficient les is given (H. Lamb [21]) by Du = 2» (22)° Ga" . (31) When dealing with turbulent ocean waves which are characterized by a phase velocity 6 , or at a given wind velocity v by the ratio a7v = 8, and the steepness 2a/,A = 5, we may write in analogy to (31) D, = 2M(B)r-g6" , (32) where the ordinary viscosity coefficient #* is replaced by a turb- ulence coefficient or coefficient of eddy viscosity ML em71g sec7-}, This coefficient M naturally is not to be regarded as a physical constant characteristic of the fluid like lad (at a given salinity and temperature of sea water), but it will depend upon the state of the "sea" and on the wind velocity. Therefore M is written in (32) 67 as a function of 8. Formula (32) follows from (31) when putting 2a/, = 6, and eliminating the phase velocity of the wave by o2= 8h, (33) In fully developed sea, where A = D, and where the "B, ~wave" (B76) and the "B,-wave" (the “sea") (61,6) (see Chapter I) are fully developed, we have with (29) and (32), considering (25) and (17) 8ur2e nZe72t om p typ y*(B.) + s't n eT Pm(1 - B.)* -s*rn eT Bm" (1 - pce (34) Since Bn = f(v) in the fully developed sea, it is to be expected that Mis only a function of wind velocity, M(v), whereas in the case of not fully arisen sea, M depends also on the stage of wave de- velopment. With reference to Chapter I, 2, we put pit = 1.37. From (34), or with the aid of an empirical formula for the ef- fective stress given in a previous paper [13], ; a72 1 2. = Tere = P'k(v)v"3 (wv) = 1 = we) ot which agrees with equation (24), when putting Bp, f(v) as given by equation (60) in this paper, we get 1? 'k(v)v3B * = ELE eo 8 ie nee not Pm From this expression, and with p' = 1.25+1073, g and 3 =.1,667 980, n = 0.1075 M = 0.1825-107* y/2 Cem72 g sec]. (36) The coefficients of eddy viscosity computed by this formula are of the same order of magnitude as the values of the “Austausch- 68 coefficients" in the upper layers of the ocean, as far as they are known. But the values given by (36) are to be expected only in the case where the complex wave motion is fully developed. Therefore, they depend upon only the wind velocity, and the coefficients M represent maximum values at_a given wind velocity. In the stages where the sea is not fully developed, that is, when the sea is still growing, the M-values will be smaller. We denote these smaller values by M(g), and relate them to the “age" 8 of the longest wind-generated wave present in the composite sea. The information currently avail- able on the state of turbulence and on the eddy viscosity in the sur- face layers of the ocean at different wind velocities and at different Stages of wave development is very meager. With respect to the nat- ure of the phenomenon it seems reasonable to assume that the state of turbulence increases with increasing development of the sea by obeying an exponential law. Evidence of this will be presented later but here it will be assumed that the increase of turbulence, and of eddy viscosity in layers with turbulent wave motion at a given wind velocity obeys the Oa?) M(gp) = Me Pm /tor lep0.1. It has already been mentioned in connection with (18) that these earliest stages of turbulent wave generation are so rapid that the exact form of the relationship is of very little consequence to the later development of the sea. It is to be expected that the change from ordinary viscosity to turbulent eddy viscosity does not occur continuously. This change to turbulent wave motion probably will take place when the maximum steepness of the "ripples" is at- 12 tained with the ratio H/A 1/7, that is at a wind velocity of about 123 em/sec [14]. At a wind velocity of v = 125 em/sec, for- mula (36) would result in a value of M = 3.2 [em7t¢ sec7-] for the “coefficient of turbulence" (eddy viscosity) in the uppermost layers of the sea where wave motion takes place. The wave length of the ripples at this wind velocity is 11-12 cm, and these wavelets are already breaking up at the crests. Below this limit of about v = 123 cm/sec the initial waves have a steepness less than 1/7. They are stable, and the dissipation takes place only by ordinary viscosity. In the case of fully developed (not breaking) initial waves it follows from the condition A = D, according to (21) and (32), that 2 2 pvp S22 (1 = p)° = 2ummgo® , (39) where Jaa is the coefficient of ordinary viscosity, substituted in (32) for M(g), and where in (21) 70 c,(8) = 2 (1 - g)*. With the condition (39) we get v2 = —Atnee ° (40) p's(1 - B)“B If “= 0.018, g = 980, p! = 1.251075, s = 0.095, B = 1/3: and 6 = 0.138, then v 120.3 cm/sec for the wind velocity necessary to generate and maintain this type of initial wave motion with a steepness of 6 = 0.138 and a phase velocity 6 = : v. This result agrees fairly well with the more exact computations on the generation of initial waves given in a previous paper [14]. In this special case a slightly higher wind velocity would result from more accurate considerations, that is, v = 122.7 em/sec. The small discrepancy of 2.4 cm/sec is due to the fact that in (39) or (40) the effect of capillarity is neglected. For practical purposes, capillarity effects only become important if the wave lengths are smaller than, say, 10 em. Therefore only when considering the generation and maintenance of primary wavelets, does surface tension have coane taken into ac- count. In Table 3, some numerical values of the coefficient of eddy viscosity (coefficients of turbulence) are given for different wind velocities and for three distinct stages of sea development. M is the value for fully developed sea as given by (36), M(1) is the value given by (37) for the state B = 1 and M(B) the value for Bin as given by (38), when B = 6,. For very weak winds (1-2 m/sec), the coef- ficients of eddy viscosity approach the value of ordinary viscosity in the first stages of wave formation. But with increasing wind velo- city and increasing sea, the coefficients of eddy viscosity increase very rapidly, and at moderate wind velocities of about 8-10 m/sec 771i their numerical values are between about 50 and 500 emg sec", Table 3. Coefficients of eddy viscosity (turbulence coefficients) [em-l g sec-l] at different wind velocities and 6, = f(v). v(m/sec) 2 4.6 8 10 -12 14 16° 18 20 24 Bn 0.425 0.53 0.60 0.66 0.70 0.74 0.78 0.81 0.85 0.88 0.94 M 10.3 58 161 332 577 912 1350 1860 2520 3260 5120 M(1) 0.57 562 20.551 99 174 274 409 583 800 1382 M(B.) O208 1.3 5.4 1664.36 73 132 216° 355° 536 1134 Figure 13 represents the relationship between M, M(1), M(p,) and v, respectively, where for comparison the "“Austausch coefficients" or "Koeffizienten der Scheinreibung" are plotted, as given by A. Defant in "Dynamische Ozeayographie" (p. 76) or by Sverdrup, John- son and Fleming in "The Oceans," according to the results of H. Thorade and W. Schmidt. These values show a rather good agreement with the curve for M, that means with formula (36). Only for strong winds are the values given by Thorade and Schmidt somewhat lower than the M-values given by (36). Their numerical values are between M and M(1), and this probably indicates that at very strong winds the stage of fully developed sea was not attained in all cases, when the observations for the determination of the "Austausch-Koeffizien- ten" by Schmidt or Thorade were taken. As an example of the in- crease of the coefficient of eddy viscosity with rising sea, Table 4 shows the M(B) values at a wind velocity of v = 10 m/sec as a function of increasing "wave age" 8B. 72 (*stToquAs Aq umoys AgysoosfA Appe jo sqyueToTJyeoo fo SONTeA *(g ese eAeM) QUedOTeAep BES Jo sazeyS YUeJEeTITp qe (7-295 W)A AZTOOTSA puTM 9yg pue (7-088 7 youn A4Tsoosta Apps Jo squefTotyyeood oyuy uUsemgeaq dtysuotyzerley (,.99s 6,.wo)W 0001 001 01 aposoulL’H puDd }P!WUudS'’M O} Bulpsoo00 SJUdIDI¥J9OI- YOSNDSNy oO eT O| G| O02 S2 (99S/W)A 13 Table 4. Coefficients of eddy viscosity (turbulence coefficients) M(g) [em-1 g sec~1] in different stages of wave development at a wind velocity of “vy = 10 m/sec. . B O.25.0525 0.337 °0.367 0.450555 0.65 O27. 2.0 21.15 1.25 sh. 3y7, a DS RS eS SS LT M(B) 5685°8515° MO.1 12.0 15.8 22.1 3059 36.5°> 99 202); 326 577 4) Energy equations A steady supply of energy by wind is necessary to cover the losses of energy by virtual friction in turbulent wave motion. Only in the case where the energy supply exceeds the energy dissipation may the sea grow. Let E be the mean energy of the wave motion per unit area of the sea surface, A the supvlied energy and D the dis- sipated energy per unit surface area. Then, for the total energy E-A per unit crest width of a wave with wave lengthaA , a ms at (EA) = (A - D) >». (41) This equation states that the individual change of wave energy with time equals the difference between the supplied energy and the energy lost by friction and turbulence. Associated with the wave motion is a flow of energy in the direction of wave propagation. This energy flow per unit time across a vertical “control-section" of unit width and a depth below which the wave motion is negligible,is cE = + pga-o , (42) where G6 is the velocity of wave propagation (phase velocity) if we consider deep water waves. The average energy per unit area of a wave with the amplitude a = H/2 equals E = é gpae ; (43) 74 Equation (42) can be interpreted either to mean that half the energy (E/2) is propagated with the phase velocity, or that the rate of transmission of total energy E is equal to the group velocity ¢ =.0'/2. From (41) aE dhe = _ A + E AP a (A D)A (44) or age +c g8) +E ah. +c ga) =(A-D)JA . (45) Let us consider two cases: Case A: If a constant wind with mean velocity v blows over an un- limited sea room (fetch), and if the energy added is the same every- where so that the waves grow at all localities at the same rate with time t(duration), then E A sel erage and equation (45) reduces to a ae (46) With (43) and the substitution (33) we have 2 a pga $2 + P82) QT _ 4g _p, (47) Case B: If, on the other hand, the duration,t, of wind action is unlimited or in practice long enough to produce a steady state, but the fetch x is limited, then for local steady state conditions In this case, E a A = or Ge and a has to be determined. From equation (45) we have (3s +E gay =A =D (48) Again replacing A by @ and considering 1 ga. 2 Qo A x G ox it follows with (43) that 2 a gpa cau = zs c(gpa $3 + F x Sh Dy (48a) or, putting c = 6 /2, we get 4 gpa- =. + — gpa ga = A-D. (49) Since Bp = @ /v is given by the empirical relationships (17) and (18) as a function of 6 = 2a/f , we get by this substitution for Case A: doce -~o3 $F -a-d; p<1/3, 2 ee a ee p into te %3 - r™ )9o = a-D 5 1/3 C4(B) - Benn M(B) e2TB or 77 ap = 2 = 2.-erp 2 e = ; Fp eg Sen (Ry = ES dp 3; 1/3<8<8,. (56) Here the dimensionless quantity Bo(B), which takes into account the effect of dissipation, is given by Greene -2rp By(p) =F MS) eg (57) Case Bs: The growth of waves at limited fetches as a function of the fetch x. Similarly as in Case A equations (53) lead to the following expressions = 6r° 2 3 dx = os Vv SCY = By BY dp 3 B<1/3 (58a) and 3672reB ap. ane ey? Be te) : < ax mil go ad Co(B) = BACB dp 3; 1/3 B1/3 and the assumptions at the earliest stages of wave development, a discontinuous change results at B=1/3. In nature, we have to expect a certain continuity in slope of the steepness 6 when 8 increases above the value 1/3. It would have been possible to establish another empirical relationship for 8<1/3 to approach a more continuous change between § = 0.2 and B = 0.4, perhaps as indicated by the broken line in figure 12, but such at- temps would not contribute to a better understanding of the mechan- ism of wave generation. Therefore it seems to be more expedient to wait for more complete empirical knowledge. For the practical aim in question, it seems adequate to smooth the numerical values by assuming a continuous change over 8 = 1/3. The smoothing is 79 ‘ase aAeM JO suoTgouny se (d)q pue (d)o setatqguenb sseTuotsueutq | ome) “pL =°s ta Oonoaoro w 02 (g)a‘(g)o 80 indicated in Fig. 14 by a broken line. But if the original values of C and B (full lines) were used, the results would not be changed essentially for larger "fetch parameters" or "duration parameters" (see section 7). The reason for this is that the earliest stages of sea development are so rapid in the state B<1/3, that the rela- tively short fetches (or durations) are of very little consequence to the later development of the sea. Table 5 gives the values of C,(B), C,(B) and B, (6), Bo(B) as they are used for the numerical computation. The original values are given in parenthesis, if they differ from the used values. Table 5. Values of 'resistance-factors" C(p) and "dissipation-factors" B(g) for different values of p in the stages 8<1.0. 8 0.05 0.15 0.25 0.30 0.40 0.45 0.55 0.65 0.75 0.85 0.95 1.00 C,(B) 16.75 13.38 10.65 9.48 Co(B) 7.50 6.71 5.39 4.42 3.71 3.21 2.88 2.78 .103 (8.58) (7.30) (5.50) (B 1 LO.25 8.60 7.85 .103'12+50) (9775) (8.00) (7.87) B, (8) 6.38 5S75 .103 (6.50)(8.78) 4°73 4-01 3447 3.06 2.75 2.60 The very rapid development of the sea in the earliest stages at 3<1/3 may be shown by an example for a wind velocity v = 16 m/sec. The results given in section 7 demonstrate that the state pB=1/3 is attained from a fetch of only 8.3 km. But a fetch of 300 km is nece- ssary to attain the state Bp = 0.81 = B.,; ‘That means that at any rate 300 km is necessary for the development of the sea to the stage 81 | B = Bn? that is, for the development of the first characteristic wave in the sea, which was called the "short sea" or "sea" and characterized by a phase velocity smaller than the wind velocity. From a fetch of 300 km, this "sea" or "Ba7wave" would have attained its maximum height, length and final steepness. But beyond this state, the growth of the sea continues, and longer waves are gener- ated by the wind. The complex sea will be nearly fully arisen when the fetch is 500 km. These results will be discussed later on in more detail, but they show that the uncertainties involved in our assumptions for the state B<1/3 play only a minor role, when the development of the complex sea with "ages" of the waves B>1/3 is considered. The condition for fully developed sea at any given wind velocity is A =D. With this and the equations (34) and (36), B,, is related in a fixed manner to the wind velocity v (see also [8]). But from equations (37) Be a m ook an ) M(1) and according to (38) M(1) = u(p)e2? (1-8) If we introduce this in (57), we have 8r@ gn? M1) ent Bo (B) zi rove v8 ? where If(1) according to (36) may be written * -2r ( B ) W(1) = 0.1825-1074y9/2 ¢ 82 Because B,, = f(v), M(1) must be a function of v too, if we assume Ba to be constant at all wind velocities. If M(1) represents a function of v only, it has to be determined in such a way that there are no contradictions to the relation Bin = f(v) as determined for example, in a previous paper [8]. With er(p 7 1) = = f(y) (60) na 182.5 -invy , it follows from (59) that M(1) = 107’y3[em7¢ sec7+] . (61) If numerical values for constants are introduced, and (61) is considered, we get from (57a) By(B) = 2.6-107387 . (62) Thus with (60), B,(8) is represented as a function of B. Since C5(8) according to (28) only is a function of 8, the difference C, - By = F(f). Equation (60) relates the phase velocity @ = BV of fully developed "seas" (B,-waves) to the wind velocity (v given in cm/sec), and yields practically the same values for v>2 m/sec as calculated previously [8]. Table 6 shows the values 8, given by (60) and the values 6, as published in {8}. Table 6. £6, = f(v) at different wind velocities given by formula (60) and in an earlier report [8]. a a a ee + B,,660) 0.48 0.56 0.61 0.66 0.70 0.74 0.81 0.88 0.94 1.00 Bt 8] 0.425 0.53 0.60 0.66 0.70 0.74 0.81 0.88 0.94 0.995 v m/sec 2 4 6 8 10 12 16 20 24 28 83 It is possible that the ratio Bet is not strictly constant for all wind velocities. Probably, it is smaller at lower wind speeds, but this question has to be checked by observations. In this con- nection, Chapter I of this report, where this particular question has already been discussed, may be referred to. With very weak winds, a value 8, 1/3 is to be expected when v = 1.23 m/sec. If at a wind velocity v = 1.5 m/sec the value B, = 0.38 as given in [8], it fol- lows from equation (60) that os = 1.30. The differences are perhaps only of minor significance. 6) Generation and growth of longer waves. Within the fetch area where the waves are generated there always exists a large number of wave trains with different lengths and heights, traveling with the wind or at small angles to the wind di- rection. Fluctuations of both the wind velocity and direction may thereby contribute to the formation of short crested irregular waves, and it does not seem astonishing that a complicated pattern of ir- regular wave motion results from interference and criss-crossing. But experience shows that special systems of larger waves always can be recognized in the wave mixture. They dominate the sea surface as characteristic waves and are the striking features of the wind driven undulations. So far the discussion of wave generation has dealt with waves called “sea" or "B waves." When these waves have attained a certain maximum length and height, or a certain "age" B, and steepness 6,, they apparently do not continue to develop, but remain constant as relatively steep waves, breaking from time to time. However, the complex sea in this stage is not fully developed. If the wind con- 84 tinues, the sea surface pattern changes its appearance considerably. Symptoms of fully or nearly fully developed sea are striking fluct- uations of wave heights and periods, groups of waves and the occur- rence of outsize waves with long stretched crests. In stormy weather the sea begins to “roll." To all appearance longer waves emerge at the rough sea surface probably independent of the fully arisen Ba 7Waves y and distinguished by lesser steepness but with propagation velocities which may exceed the wind velocity. By interference of these wave systems typical phenomena of complex wave motion result. In Chapter I, an attempt was made to explain the observed strik- ing fluctuations of wave periods and heights in the complex wave pattern by coincidence of three characteristic waves, called the "sea," the "intermediate wave" and the "longer wave." In the fol- lowing discussions these dominating waves may also be noticed by their "ages," that is, by the ratio "phase velocity : wind velocity" in fully arisen state, that is "B -wave," "8(1)-wave" and "8, *-wave." It seems probable that in the mixture of wind generated ocean waves certain undulations are favored by the wind with respect to the energy transfer. They have a maximum growth, and having attained their maximum values of height and length, they are maintained by a steady energy supply as long as the wind remains unaltered. The first characteristic wave, the relatively steep Ba7wave travels with a phase velocity, that is always slower than the wind velocity. Probably we have to presuppose the existence of these waves, in order that longer waves with a higher amount of energy, and faster than the wind can be generated. The short but steep Pras waves cause, with superimposed smaller waves, the broken appearance 85 of the sea, and a definite roughness of the sea surface. Thus they contribute essentially to the resistance coefficient and therefore to the effective horizontal stress, which acts at the rough air-sea interface. If longer waves emerge at this rough interface the total horizontal stress will do a net amount of work even if the longer waves travel faster than the wind (see equation (6) with (4a), or A, in (9)). Thus, a certain "waviness" of the air-sea interface will be maintained against dissipation as long as the wind velocity remains constant. However, the mechanism governing the tangential transfer of energy to waves which may travel with 6 > v is not yet completely explained. If the surface mass transport velocity 18-0 from Stokes' theory is taken into account, (equation (42)), a transfer of energy from wind to waves would be possible even if the waves move faster than the wind (equation (6) and A, in (9)). This idea, first used by Sverdrup-Munk [1] seems plausible, and is fit to over- come the difficulties. But if the difference between the wind velo- city and the horizontal component of particle velocity is introduced in the expression for Te and if the variation in shear due to the variation of wind velocity and due to the variation of the water particle velocity at the wavy surface is considered, apparently no energy is added even on the basis of Stokes' theory, if the waves move with 6 >v. The result of an analysis of Schaaf and Sauer [15] would limit the growth of the waves to the region where the wave velocity is less than the wind velocity. From the expression for A, given by these authors, it follows that no energy is added by tangential shear stress, if the wave velocity exceeds about 75% of 86 ' the wind velocity. But, in fact, coservations show that longer waves with 06 >Vv are generated in the wind region (see [23]). Probably these waves have been generated under conditions not covered in the analysis of Schaaf and Sauer. Special attention must be called again to the fact that our knowledge of the actual distribution of T, (and Ce). or of the actual difference of wind velocity-water particle velocity at the real ocean surface with its complicated wave pattern is too meager, and it seems very difficult at present to estimate the effect of normal stress components and tangential stress components separ- ately. The actual rough sea surface may offer entirely different conditions than are assumed in mathematical treatments of the prob- lem, where only a simple sine-wave profile and simple water particle displacements are considered. A special mechanism therefore may be considered in this con- nection, which probably plays no unimportant part in the generation of long waves which are faster than the wind. The idea is that short period waves in the generating area may disappear, transfer- ring their energy to other waves. This special kind of energy trans- fer seems to be possible only in complex wave motion, and its mech- anism perhaps may be explained in the following way. Consider the fully arisen steep "sea" or Ba 7wave traveling always with @ v. Every time the low crest of the long wave disturb- ance overtakes a crest of the steep Bm Wave » the water particle velocities of both wave motions are added, and according to the di- mensions of the two waves, the horizontal particle speed will increase 87 more at the surface, than in deeper water. Thus, the horizontal particle speed will not only increase at the crest of the Bm7Wave , but at the same time the vertical velocity gradient will become steeper too. Both effects support an extensive breaking of the crest of the superimposed steep Ba Wave. Now the long wave over- takes the breaking crest and approaches the next By 7crest in the leeward direction, where after some time the same thing happens, and so on. Thus, as long as the faster traveling long wave over- takes a train of undisturbed B,y7waves (with originally maximum steep- ness or) breakers may occur at the crest of the long wave disturb- ance, and every new breaker is placed to the leeward of the previous one, looking in the direction of wave propagation. It may be men- tioned that perhaps this may be offered as an explanation for the "law of breakers" by K. Wegener [24], who states: "Die See bricht so, dass sich eine brechende See immer vor die vorhergehende setzt." Furthermore, the foam patches (at higher velocities) will orient themselves in rows, extending to leeward and gradually disappearing at the windward end. The particle speed of the By 7 waves at a given wind velocity is greater than the particle speed of any other wave with 6 > v anda maximum steepness given by (17), as shown in the following Table 7 by the values u. at different wind velocities. The maximum hori- fe) zontal component of particle speed at the surface is given by Uy = T60 . The table shows On and Gis at different wind velocities for the fully arisen p,-wave, as well as for the B(1)-wave and ae -wave. From this, amplitudes of particle speed u, are computed. At a wind 88 Table 7, Steepness (5), phase velocity (@ ). and amplitude of horizontal velocity component u_ at the sea surface for three characteristic waves. v m/sec a LO 16 20 24 28 on 0.083 0.067 0.0555 0.049 0.0445 0.0407 By Twave, oO m/sec 4284, °9500° 12.95, 17.6 122.6 27.9 Uy m/sec O74 9 1.648 ° 2.26 27 e106 3.56 8(1)-wave (@ (1) m/sec 5 10 16 20 24 28 Coane m/sec 0.64 1.28 2.05 2.56 3.06 3.56 Ga wave (Or B/Sec. geal rao.3 29,9 27.4 39209 «= 38. 6, =0.0222Lu, m/sec 0.48 0.86 1.54 1.92) 72.3 25 4 7 velocity of v = 16 m/sec, for example, the sum of u, for the p -wave and the B(1)-wave would be 4.31 m/sec. When the crests of the Ba waves break, the waves lose energy and diminish in height. Part of this energy is dissipated and lost with respect to wave motion. But it seems reasonable to assume that another part of the energy, given off by the By wave will be trans- ferred to the underlying long wave. Because these breakers always occur at the crests of the long waves where the particle displace- ment is in the direction of wind and wave propagation, the particle velocity originally connected with long wave motion will be speeded up by the forward rushing water masses of the breakers.* Here the breakers do a net amount of work analogous to the work done by the wind in the case of waves with © O. When in the generating process of complex wave motion the first characteristic waves (B,,7waves ) attain their maximum steep- ness, the wave length of these waves, or their propagation velocity respectively, is given by 8, (equation (60))and their height H by (17). In this stage of wave development, a certain amount of energy in the difference A - D is left, and this remainder is used for generation of longer waves until an equilibrium state is attained between the acting forces, that is, between the drag, the normal pressure forces and the dissipative forces. Without additional in- crease of the effective resistance of the sea surface, and therefore without an additional increase of the total drag at a given wind vel- ocity, this stage A = D will be approached by nature in the easiest way when waves which proceed with a phase velocity equal to the wind velocity (@ = v3 8(1) =1) are generated. Let us assume that the sea surface takes on a waviness which corresponds to the wave length of §(1)-waves, originally beginning with very small disturbances which may always be present. By tak-= ing up energy from wind, the height H(1) of these disturbances will increase until it attains its maximum value. This maximum height, H(1) > is given by the maximum steepness, according to (17) 6(1),, = 2n eT = 0.0406 . (63) Thus (2) fw ACd)s ante and, (63a) The total wind forces at the sea surface have not changed dur- ing the generation of these "intermediate waves" or B(1)-waves. As 90 long as there is a positive amount of energy A - D left, the waves will grow further even when reaching the state OC ne But they cannot exceed the maximun steepness given by (63), and there- fore they will grow by increasing their wave length. Later on, we shall see that the further development of waves beyond the state B(1), 6(1),, manifests itself in an increase of wave length, where- as the height remains nearly the same. Thus, the longer waves (p*-waves) rather soon increase in length, and attain propagation velocities that exceed the wind velocity. With this newly generated waviness the normal pressure components of wind force are out of phase by the phase difference 7m. The further development of the waves therefore will be increasingly delayed, the more their wave lengths increase and the faster these waves travel. Finally the state A = D will be reached approximately when the longer waves travel with a phase velocity of o* = 1.37v. The work done by these nor- mal pressure forces, which act with a negative sign at the long B, "Waves, together with the dissipation, balance the work done by tangential stresses on complex wave motion in this state. It is to be expected that with increasing height of the 8(1)- waves, interference phenomena appear between these waves and the 37 Waves , which depend upon the wind velocity and upon the state of development of the B(1)\waves. As a consequence, from time to time, particularly high waves with considerable steepness will occur, forming higher and more spacious "breakers" than in the preceding stages of wave development. These increasing breakers imply an eddying of more extended and deeper water masses of the surface layer, and therefore will be succeeded by an increase of the ot turbulent state of surface layers. It seems reasonable to assume that the turbulence, that means the eddying of the surface layers as far as they are concerned in the wave motion, increases with the development of height of the 8(1)-wave. By (38) the coefficient of eddy viscosity in the stage of fully arisen Ba waves was given as M(B) In the stage of fully developed £(1)-wave with a height given by (63), we have M(1) as given by (37) when Bp = 1. For the increase of eddy viscosity during the stage of 6(1)-wave development, we assumed M(1) to be a function of wave height H(1) = 2a(1), and put a M(L) = Maden aa (64) where a(1),, is the maximum amplitude of the B(1)-wave. As an example, the following table shows the increase of the coefficient of turbulence (eddy viscosity) with increasing 8(1)- wave at a wind velocity of v = 10 m/sec: a(1)/a(1)_, fo) 0.25 le OL “0n75tw T08 M(1) egs 36251 4669 avi60028! 077.142 9950 The increase of the £(1)-wave with increasing fetch for stationary wind conditions (Case B) will be, considering ce = 0", and ACL) = const... & gE 5 2D dx 7 or = gpa os =A-D. (65) Taking 8(1) = 1, we have with (27) 92 -rp [y(p.) +s'me ™1 - 84) le'v p\ = (66) A C5 (B,)p'v> The dissipated energy according to (64) and (32) is 2 2r(1-8).a/a(1) D = 2M(1)° go = 2u(B )9-e Ta? e e a (67) where a means the amplitude of the increasing B(1)-wave, a(1),, its maximum value given by (63) and Aq.) = £2 ve (68) the wave length. With a, = A(lne™ = ACA. (69) (67) can be written 2 2r(1-6 )aS= D = eng Gere Meme A(1)n (70) Since @ =v and B(1) = 1, equation (65) takesthe form —————(]- =P a (71) }.gpva $8 = c,(p,)o'v) = Br@g —8—5 ulp_ de ACE according to (66) and (70). By division with o'v3 and considering (68) we get 3 2 rg 2g-M(B_ da =(1-B8 Ja eee. 2 dare pies | habe 2n m 2 pte a dx = Colby) pty? a (72) or tx = FEE, . a (7) C(p_-) - ——~% a? —3- (1 - 8) 2 Ba A a 2) aes Ban a With regard to the later numerical calculation, it is more convenient ie) to introduce the dimensionless wave steepness 2a oe cen in place of the amplitude a. From equation (68) we have pe - Ar ve Ge and therefore ax = 4 a es re 6 (1) d 6(1) (74) co(B,) - amg 5(1)? M(B) exp E (1 - 0(2)] p'v In an analogous manner we get in the case of an unlimited fetch with increasing duration of wind action (Case A) according to equation (45), and the condition CAG) 2 9, the differential relation 4 =(1- -p,,)6(1) a em-5(1) dees = C5(B,)p'v> - On g6(1)° M(B, ye? (75) dt = fe. = 1 acs ei HC NEE Lal esa cee Moc Me it on i 2 A > (76) Co(B,) - aes 6(1) M(B.) exp E (1 = pon) With ee x = 8Mrgn 2 e7-TB,, the fully developed state of complex sea is given by equation (34). This happens when the long waves (8° -waves) are fully arisen and A =D. When considering the growth of the £*-waves up to the state Pat = 1.37, it is necessary to take into account that with increasing 94 wave length the height of these waves may also grow. But the results will show that the increase in wave height is only of minor sig- nificance, and this result confirms experience at sea. For Case A where the waves are considered to grow with the duration of wind action along an wnlimited fetch, and for Case B where the waves are considered to grow along the fetch under the action of a wind of sufficiently long duration, we get analagous to (57) and (58) for the state p*>1 Case A at = 2 Se ee ag 5 1 ast By = B ae ae Ky 3 1/3 < 6¢B,, (82) B=1/3 a) Sa, SX? th 6(1) 46(1) + ees Owls ‘(pj euae yaheuts Seen pea ecy eee C,(8, ae 6(1 Bia exp[(r/n)(1-8,,)6(1 ] B(1) $ 0<6(1)<8(1),, 5 (83) t 2.2 Bm a2 g-2FB* * ae BF eh ea SaDaUAeye (a ties p* + Kz 1<8%< By" 5 (84) The "factors of energy supply " C,(8), Co(p) and c3(8) are given by (20), (28) and (30), "the factors of energy dissipation" 96 B,(B), B,(8) and B,() are given by (55), (57), and (80). Let the density of the air be p' = 1.251073 and the density of the sea water p = 1.028. The other constants, which act in the formulas are already explained and their numerical values are given in the preceding text. Ky» Ky and K3 in (82), (83) and (84) are constants of integration, taking into account a duration parameter gt/v as an additional value at the lower limit of integration. Case B: Growth of the sea with a constant wind of unlimited — SS SS oe ee eS ee B=1/3 3 w) B= Fy orp? Seg egy 4B BSI (85) Bm 3 -2rp (2ST) Eee, eee e =o Is : 2) opt 2 ea om Ee ae = 5G Ap + K's 1/3<6 = 0.788" 3; B<1/3 , (91) and from (17) 5 = 4m e7TP g? = 1,350°7F p°3 1/3<8< Ba (92a) and s = 4m oT P*g a? = 1,350 TP*pn®; 1c pecs, (92b) Equation (92b) has a maximum value when p* = 1.191. That means, 98 in the state of development of complex wave motion at sea, the long wave at a certain stage o”°* = 1.19lv grows up to a height a little larger than its height in the stage of fully developed sea, where c ~ 1.37ve But the differences are not very important, likewise the growth of the wave height from the fully developed £(1)-wave to the state of fully arisen B,, ~wave is of minor importance. Hence, it follows that in the stage of "long wave"=generation the energy supply by wind is used almost completely for an increase of the wave length. (Compare the heights H in Table 10 for these stages of wave development.) Tables 8 and 9 represent some numerical results of the theory which may be supplemented by the graphs in figures 15 through 19, and figure 26. These tables and graphs are suitable for determin- ing the characteristics of complex wave generation by means of data from adequate, synoptic weather maps, for given wind and fetch conditions. Numerical examples: From Tables 8 and 9 or graphs, at wind velocities of v = 5 m/sec, 10 m/sec, 16 m/sec, 20 m/sec, and 24 m/sec, the following character- istics of complex sea, represented in Table 10 can be found. These examples show that the minimum fetch (x) or duration (Ce) needed for generating fully arisen sea rapidly increases with increas- ing wind velocity. At v=5 m/sec, the complex sea with the three characteristic waves would be fully developed over a fetch x,= nls Fs) km, or over an unlimited fetch it would take t, = 2.25 hours from the time the wind starts to blow (with constant velocity). At v = 10 m/sec, the minimum fetch would be x, = 106 km and the duration oF Table 8. 1) ex/v", et/v end gH/wfor different values of B = 6/v in the stuges of develooment 8 <1/3 and 1/3<8< 1. 8 0.10. 0.20% 0.30. 0. 0.40 0.45 0.50 0. 0.60 ex/v- 0.55 23.5 180 385 657 1040 1560 2260 3160 et /v 22 = 309-Ss«1520 2750 «4240 »=S «6020. «Ss 8270-10880 14000 g@H/v- 0.0078 0.0311 0.0700 0.0925 0.1108 0.129 0.147 0.163 0.1785 B G.65U0790 1 0% 0.80 210285 _» 0 goacitoy 1.00 ex/v° 4270 6100 8050 10800 14050 17500 21200 26000 et/v 17600 23000 28400 35400 43500 52000 60000 70000 @H/v* 0.1920 0.206 0.217 0.227 0.236 0.243 0.248 0.254 2) a = f(t) and a = f(x) for the growth of 8(1)-waves at different wind velocities v. (a = H/2 = wave amplitude) v=2m/sec : H=0.105 m a (cm ) 1 2 + Dac HorsHi a) 0.1.052m rab ssds > spade (Bet saiek.S+ NI pS b 1900814 ex/v° «525 t(min)’ “Olaoebeer ore MOBI n © 2602s eer 13 <6 et/v 1050 eH/v- 0.254 v=5 m/sec :.-H = 0.65 m a (cm fe) 10 15 20 2 ee For H = 0.65 m zm) mo 839° W697 see tse “oc2 2087 ex/v° 819 Simin) 0 Oreo lOc 2h ae Gale ria.y et/v 1650 eH/v° 0.254 100 vy = 10 m/sec : H =2.6 m a(em 20 40 #4260 80 100 120 130 For H = 2.6m 1 at different wind velocities between v = 2 m/sec and v = 28 m/sec. Gr 160. A6O5:V1.10'. 1.15) 1.20 125.8 10) 1 55 ex/v- 1355 1433 1513 1598 1688 1790 1920 2435 gt/v 6250 6402 6553 6702 6855 7022 7228 7995 eH /vo Oe2>4 0.62585 0.261 0.2625 0.265 0.2625 0.2615 0.259 0.255 v = 5 m/sec. x B 070. #05 10) <1. 15. 1ee0, eo: eee oes gx/v- 3469 3611 3764 3929 4110 4329 4619 5429 gt/v 13650 13927 14210 14500 14810 15170 15590 16885 gH/v- 0.254 0.2585 0.261 0.2625 0.263 0.2625 0.2615 0.259 v = 10 m/sec. Br beO eel sO5., 1.10 1.05). 62052 1.25 0 35 ex/v- 7311 7591 7889 8201 8530 8930 9440 10610 gt/v 25400 25945 26500 27050 27630 28260 29070 30860 @H/¥ 20254. 0-258 5y0,. 261, 0.2625 -0.263,0.2625..0.2615 0.259 v = 16 m/sec. ic U5083 1.05) /1,10. 1.15 1.20 1525 01 55 calm ex/v- 13210 13720 14215 14700 15220 15800 16560 18060 et/v 40560 41558 42470 43350 44220 45190 46380 48610 eH/v~ 0.254 0.2585 0.261 0.2625 0.263 0.2625 0.2615 0.259 0.258 102 v = 20 m/sec. g* HOM J0Se MialOema Se IecO.ne lee. 21,30. 0% gx/v> 18550 19274 19945 20600 21280 22030 22940 24620 et/v 52800 54210 55450 56600 57750 58940 60400 62950 GH/v 0.254 052585 0.261 0.2625 0.263 0.2625 0.2615 0.259 v= 24 m/sec. at AsO: 2 05 el lO eho eel0. 1.2 sO ee ex/v° 23700 24750 25640 26460 27230 28080 29130 31000 gt/v 63780 65820 67480 68960 70300 71660 73310 76130 GH/¥" 0.254 0.2585 0.261 0.2625 0.263 0.2625 0.2615 0.259 v = 28 m/sec. ot WO. a O5. Wel Oln eel > wl oOL 125 11.50 1.35 ex/v- 29100 30540 31690 32700 33660 34700 35950 37900 gt/v 76200 79020 81150 82930 84600 86300 88200 91200 @Hi/e- 0.25% 0.2585 0.261.0.2625 0.265 0.2625 0.2615 0.259 Teble 9. Minimum fetch and minimum durstion sat different wind velocities for generating the characteristic wave motion to the stage of fully srisen Bm - Waves, 8(1) - waves, snd fully arisen complex ses (2m). (Fetchx and duration t sre given by gx/v- and et/v) Bm-wave v(m/sec)2 7: To oo ee 1G is ss0 ens Bin 0.425 0.57 0.65 0270 0274 0.775 0.81 0.845 0.88 0.94 0.995 gx/v" 830 2650 4270 6100 7600 9350 11500 14000 16400 20700 26000 gt/v 5200 12000 17600 23000 27500 32200 37200 43000 48500 58500 70000 103 B(1) - weve (complex). v(m ec 2 5 a5 10 12 14 16 18 20 24 28 ex/v- 1455 3469 5290 7311 8980 10900 13210 15920 18550 23200 29100 gt/v 6250 13650 19650 25400 30180 35200 40560 46800 52800 63780 76200 er (Bo = 1.35) - wave (complex). v(m/sec) 2 5 LoD 10 12 14 16 18 20 24 28 ex/v° 2435 5429 7940 10611 12780 15200 18060 21320 24620 30500 37900 gt/v 7995 16885 24000 30860 36480 42380 48610 55800 62950 76130 91200 Table 10. Chsrscteristics of complex sea st different wind velocities. H = wave height, A = wave length, T = period. i) =o m/sec (2 8 = Os7 H (m) A(m) T(sec x/v> t/v neurone, fa By-wave 0.43 Be2 elo? 2650 12000 6.75 iy B(1)-wave 0.65 16.0 3.20 3469 13650 8.8 1.9 BY “mi 1.55°0.66 "29.52 » 4.33 5429 16885 13.8 2.25 2) v=10 m/sec : 8B, = 0.70 H (m) A(m) T(sec x/v° et/vo. x(k) t(h B,-wave oles SiS 4.48 6100 23000 61 6.4 B(1)-wave 2.6 64 6.4 Fale 25400 73 vane Bes 1555 22651174 8265 10611 30860 106 8.6 104 3) v = 16 m/sec : By = 0.81 H (m) A(m) T (sec) ax/v- gt/v x (km) t (hrs) Bm-wave BAe) 107 8.3 TEES OO 47200 300 IS 18) B(l)wave 6.6 163 10.2 13210 40560 345 18.4 B* = 1.35 6.75 298 13.8 18060 48610 17D. eee 4) v = 20 m/sec : &, = 0.88 H (m) _A(m)_T (sec) ax/v° et/v x (km) t (hrs) By 7 wave o.0 198 11.3 16400 48500 670 LO IPS: B(1)-wave 10.4 256 eS 18550 52800 758 30.0 Esa ee51e.6 1000 km are necessary. Thus, example 4 (Table 10) shows that at v = 20 m/sec (about 8 to 9 Beaufort) the minimum fetch for the generation of complex sea is x, = 1050 km or nearly m 600 nautical miles; at v = 24 m/sec (example 5) X,, = 1800 km or about 1000 nautical miles. These computed fetches are in agreement with the statements of experienced wave and sea observers (V. Cornish, Graf von Larisch). V. Cornish estimates the minimum fetch for the generation of fully arisen "storm waves" to be about 600 to 1000 nautical miles (see H. Thorade [22], page 293). If the wind blows constantly, the du- ration th for developing these storm waves would be at v = 20 m/sec, 1.5 days (t., 2.2 days (t tm tn 35.6 hours); and at v = 24 m/sec, it would take 51.8 hours). Furthermore, the examples represented in Table 10 show that the development of complex sea after the generation of fully arisen By waves continues relatively quickly. It has already been stated in Chapter I that the observations at sea indicate a rapid develop- ment of longer waves at the rough sea surface when the fetches ex- ceed certain minimum stretches. Under these conditions character- istic interference patterns of complex wave motion emerged as the striking features of the sea. To get an idea of the results of the theoretical computation, conditions at a constant wind velocity of v = 16 m/sec are considered in more detail. The application of the presented graphs and tables leads to the following characteristics of complex sea generation at this wind velocity: The first dominating wave (6 wave ) appears 111 fully arisen with ©, = 0.8I-v = 13 m/sec at the end of a fetch x = 300 km (Table 10). (Considering the growth with time t over an unlimited fetch, or over a fetch long enough, it would take 16.8 hours, if the wind starts to blow over an undisturbed water surface.) The development of these Bn waves with increasing fetch and duration is illustrated in figures 20 and 22. Fig. 20 shows the height and the period of the Ba-waves, as functions of the distance from the coast, for various wind durations. When the wind has blown, say, for 5 hours, a very rapid increase in wave height out to a distance of 62 km from the coast is found. The steepness of these waves is considerable, and the steepness graph in Fig. 21 gives for a dura- tion of 5 hours and a fetch of 62 km H/, = 0.0855. The wave height of the waves is 4.28 m, and their period (see curve T in Fig. 22) is T = 5.65 sece (A= 50m.) Beyond 62 km the waves are similar at a duration of 5 hours, but still in the growing state, whereas inside of 62 km a steady state has been reached. That means, at any given point along the fetch from x = 0 to x = 62 km the waves do not change with increasing duration of winé action, while beyond 62 km the waves continue to grow for a length of time which depends upon the fetch. Thus, for example, a steady state is to be found after a duration of 10 hours inside of a fetch of about 150 km. If the wind continues to blow with constant velocity (16 m/sec), the fully arisen $8 -wave with ©, = 0.81 v, T = 8.3 sec, A = 107 m, H = 5.9m appears after 16.8 hours, and inside of the fetch x = 300 km a steady state is attained as given by the curves H and T in figures 20 and 22. Correspondingly, the other graphs may be used for determining the state of the sea, that is, the steepness graph (Fig. 21) and the wave age graph (Fig. 23). (Similar graphs and tables for other 112 *soejuns 2e4en paeqanystpun ue JaAO MOTQ Of peazteqs oos/l OT = A fo puTM e& Jaye uoTyemp JO saMoy yUetesjTp ye. (x YoeT) SUTT 4seOd Wor eoueystTp JO suoTyoUNy se (eAeM=— g) JZUSWAOTeASep ees xeTdwod JO saseq4s 4sdTJ 9y4 UT (Lb) POFAed puke (H) FUSTOY eAeM "°O? “4TH Ssa;awojiy ul ‘x ‘yoja4 oo¢ 002 (oFoy fo) N sinoyuQoG oO wave period, T (sec) © sunoyug 2 sunou o's sunoy GZ sinoy O| sinoysc| 9aS/WOI[=A AJIDOJ9A PUIM suajaw ul‘ ‘yuBlay aAOM ee, wind velocity v=|16 m/sec 7.5 hours |!O hours 15 hours wave steepness, A ,in% fo) 100 200 300 Fetch,x,in kilometers Fig. 21. Wave steepnes, H/A , as a function of x and duration (hours) at a winc velocity v = 16 m/sec, corresponding to figure 20. wind velocities will be given in a following report.) If the duration of wind with v = 16 m/sec exceeds 16.8 hours, and if the fetch is longer than 300 km, the development of complex sea will continue, but for the B -wave a "steady state" is attained over any fetch or for any duration. Longer waves develop, but the Ba 7waves remain alwayS present as the first dominating waves, break- ing from time to time, and being continuously regenerated by energy supply from the wind. Table 10 (example 3) for v = 16 m/sec shows that the fully de- veloped 8(1)-wave would appear already at a fetch of x = 345 km (mini- mum duration t = 18.4 hours). Its dimensions are T(1) = 10.2 sec, A(1) = 163 m3 H(1),, = 6.6 m. The growth of this wave up to fully arisen state is given in Table 63 2)°for-y = 16m/sec. 114 (sec) ? T ? wave period *asoejans Jeqjem peqingstpun ue pue Yyoyesy peytwuT[un ue aeAo ieee OF paqaeqs oes/M 9T = A, JO puyM & Jaqze 4 UOT ZeINp go suotyouny se (eAeM=- g) YueudoTeAsp eaem xoeTduoo JO segeqs 4siTJ ayy UF (1) pofaed pue (H) 4uSTey oven *ze °3tyq sunoy ul ‘y*uolounp S| Ol S 0 das/wW 9| =A AyID0]9A Pulm ssdjaw ui ‘HW ‘yyB1ay aAOm He) wind velocity v=|16m/sec ° @ 15 hours 10 hours {e) N“ 7.5 hours o a b 5.0 hours 2 oO 2.5 hours wave age, v o °O o ~ Os os —_} 2 fo) 100 200 300 Fetch,x, in kilometers Fig. 23. Wave age, O6 /v, as a function of fetch x and duration (hours) after a wind of v = 16 m/sec started to blow over an undisturbed water surface. By interference of the 6, -wave and the p(1)-wave, characteristic variations in wave height and period occur in complex wave motion, but the form of these interference phenomena is only temporary if the wind continues to blow over longer fetches, because the relative dimensions of the dominating waves will be changed with further development of the sea. If the fetch is as long as x = 472 kn, the "long wave" with 116 B* = 1.35 arises after a duration t = 22.1 hours (example for v = 16 m/sec in Table 8, 3), and Table 10). With this, the sea has al- most attained its final stage, and this stage is not signified only by this B*-wave, but also by the presence of the other two dominat- ing waves, that is, by their interference pattern. Strictly speak- ing, the final state is attained when Bn = 1.37 has developed, but in this state A - D = 0 and this happens theoretically for x (or t) +oo. For practical purposes p* = 1.35 will approximate the fully arisen state, because the variations from §* = 1.35 to Ba = 1.37 may be neglected.. For the growth of the long waves with increasing fetch and duration see figures 15-19 and figure 26. In the fully arisen state of (wind-driven) complex sea three characteristic waves dominate the surface pattern. The smaller waves which are superimposed contribute only to a certain roughness of this main pattern. But this roughness is very important for energy transfer by wind, and therefore for maintaining the main pattern of the sea. In our example, v = 16 m/sec, the fully arisen complex sea is characterized by the following dominating waves: Batwave =: A, = 107 m H) = 5.9 m (T, = 8.3 sec) B(1)-wave : A3 = 163 m H3 = 6.6 m (T, = 10.2 sec) BL -wave =: Ao = 307 m H, = 6.8 m (T, = 14.0 sec) Some theoretical interference patterns of complex wave motion for fully developed sea at v = 16 m/sec are represented in Figures 7 and 8 in Chapter I. They show theoretical "wave records" and the variations in space and time as they could be forecasted for a fixed point on the sea surface. The computed variations of time intervals al bys between succeeding crests ("periods" in complex sea) at a fixed point may be compared with the diagram in Fig. 3 of Chapter I, which represents the results of observations. At a wind velocity of v = 16 m/sec the characteristic "periods" vary over a range of about 8.5 seconds, that is, between 6.5 sec and 15 sec. With the appearance of typical interference phenomena and groups of waves, a certain regularity is to be expected in the occurrence of high breakers (see Chapter I). —_—_— | Se Special attention was paid to observations on waves when the "Heidberg" sailed or anchored under the protection of windward shores. This opportunity arose during a longer stay at San Miguel (Azores) and in many cases at the coasts of Venezuela and Colombia, in the northern parts of the Caribbean Sea, and in the Straits of Florida, and in the Gulf of Mexico. The "Heidberg"-observations at limited fetches comprise the range of wind velocity from 1.7 m/sec to 13 m/sec with two single measurements at 16 and 18 m/sec wind velocity and fetches of 400 and 450 m, respectively.. There are included some observations at ~ low wind speeds made at the German coast of the North Sea (Islands of Sylt and Amrum) in the summer of 1950, and an observation at v = 7 m/sec made in Vineyard Sound during the author's stay at Woods Hole in the summer of 1951. The results of these observations are presented in an appendix to this report, because measurements of the growth of waves are still rather scarce. The table contains in some cases measurements of the wave height H, besides the observ- ed characteristic period (or periods) T at different fetches-.and 118 and wind velocities. The propagation velocity 6 and the wave length A are computed, by means of Gerstner's formulas, from the period T. The fetch is given by the distance from the coast to the ship's position in bearing the direction of wind. Fig. 24 represents these new observations, augmented by older estimates which were collected and used by H. U. Sverdrup and W. U. Munk [1], and by two observations of V. Cornish (see H. Thorade [6]). For comparison with theoretical results, these data are plotted into the fetch-graph Bp = 9 = . The observations are in fair agree- ment with the theoretical eyes In the region of low fetch para- meters, the observed values fp for all wind velocities fit the theo- retical curve very well, but for higher fetch parameters, the single observations spread out into the region on the right hand of the curve, as it is to be expected when the Ba waves have attained their fully arisen state. Theoretically, this is indicated by the straight lines which branch off at certain fetch parameters de-= pending upon the wind velocity. Thus, in this region we have obser- vations at fully developed By Waves , but, besides these waves, B(1)-waves and finally 8, *-waves may be observed, if the fetch is long enough. The growth of these longer waves at different wind velocities is represented in the fetch-graph by curves which branch off to the top of the graph. In the fully developed state, the observations of 8 values (computed from observed "periods" at given wind velocities) range between the upper straight line and the lower straight line for the given wind-velocity, but a "piling up" of observations is to be expected around the lower line (Bas the upper line with p,.* = 1.37, and p = 1. The distribution of P-values as computed from observed periods" T in fully developed 19 ‘sjoquiAs Aq SuoljDAsasqo ‘sansno Aq UMOUS SGIYSUOI}O]/a4 JOIIJaIOBYY ‘SJ9JaWOIOd jDUOISUaWIPUOU BHuISsN ‘A AyIDO{9A PUIM PUD X YOJaj JO UOIJIUNY D SD O AyID0}9A BADM yz ‘big zf xB Sleaz9s¢ » ¢€ 2 vOlegr9s + ¢ 2 ‘Oless9s pp € 2 2leau9cb € 2 Olegz9¢ » ¢€ 2 OOlegr9se ¢ 2 lee 28S/wS @ 99S/w 010 29S/WOIV 988/y 02 8]0H Spoomae 60 238S/woZs = ysiuiOOW 228/w ez . RS inte > Yysiuiody ‘| $94d}aj $0046 10} [1] YUNW-_dNIpPsarS Aq Pajda||09 e uosqi9 + (1S-0S61) ,649q9p!19H,, SWo | (ydos9 yo}a4 ) $4 (2“s)Az¢ 120 sea (represented in Fig. 5, Chapter I) is in fair agreement with the results for high fetch parameters, but these observations are not plotted on the diagram. The data used for comparison of theo- retical results in the graph are special observations made aboard the "Heidberg" at well known fetches as mentioned above, and also some other data used by Sverdrup=-Munk. But the difficulty with the latter data is that it is not known what the given f-values, or periods, mean (average values or maximum values of observed o (or T) at a given v). Most of them are placed into the part of the graph with ex/v> >10", where, depending on the wind velocity, a fully arisen or nearly fully arisen sea is to be expected. The observations of Stanton and Gibson at low fetches are separately indicated. The "Heidberg"-observations, and well defined other data for higher fetch parameters in the region of the straight lines are stated to- gether with the wind velocity in m/sec. The curves in figure 24 give the stage of growth, and the straight lines the fully arisen state of dominating waves. Thus, on the right hand of the curves fully developed waves are to be expected. The other graphs (figures 25 and 26) are to be interpreted similarly. The representation of data at low fetches by means of 5/6 = 0 (ex/v") in Fig. 25 shows the different relations at different wind velocities.” The complete set of "Heidberg"-observations of partially developed waves was divided into three parts, comprising the observations at wind velocities lower than 7.5 m/sec, between 7.5 m/sec and 15 m/sec *T is the period of waves observed at limited fetches, and T.| is the period of the fully developed 3 -wave as given by formula (fa) in Chapter I. The ratio T/T. corrésponds to the ratio 6/6 _, where O = gT/2r and c= et _/an. 12m ‘sjoquiXs kq pasooipul aso suoionsasqo ‘Ky001aA puim??Sugz pun Sy Q| 9%,S=0 10 (2 %G)P = -% diysuoojas josyasoayy ayy MOYS Saasnd aaIys YJ “SB4D42} MO] JO OJOP PaAsasqo jo uUOyDJUasasday ¢zBi4 Ol Ss 6809S 6 ¢€ 2 Ol v» 6819S » € z Ol € 68.96 vw € rd */y6 z Ol 2 60.96 » € 4 Ol 68l9G vb ¢€ Ol 2 O 6819S » € 2 S/wl-veA S/w9l-vl=A S/wlsa S/w SGIA>G2L S/wGl>A UOJUDIS ysiuso9 310H Spoom , 5489P!9H,, + SUOI}DAJOSGO 122 298/WO} 2a8/w9} 208/WOZ oes/Wr7 208/we2 “sjoquwAs hq SUOI,DAJAaSGO ‘SAaAIND Aq UMOYS SdIYyYSUO!jOJa1 |091,91094 | 'SuajawosOd ;OUOISUaWIpPUOU BHulsn ‘A Ay190/aA PUIM PUD xX Ydjaj ¥O UOI}DUN} OD SO H jyBiay aAomM 92614 2"Ab yunw-dnipsaas Aq pajyoe|j09 © 61aqpiaH ‘SWo :SUOIJDAJ@SGO A A yp Hs 123 and higher than 15 m/sec. They are marked in Fig. 25 by different signs. The observations at a given fetch-parameter scatter over a certain interval of the ratio o/o ,,, but there is a distinct separ- ation between lower and higher wind speeds of such a kind, so that at a given value ex/v", the ratios o/o. at light winds are higher than at moderate and strong winds. The three curves are the theo- retical relationships between 1/6. = (ex/v~) at. 5,-10,; and'20 m/sec wind velocity. When the fetch parameter has attained a certain value, depending upon the wind velocity, the ratio 6/o, or T/A becomes unity. That means that these waves have attained their maxi- mum value of & or T as given by oe. or Lea for the Ba wave in the fully developed state. Fig. 26 represents a comparison of measured wave heights H at different fetches and wind velocities with computed heights using the dimensionless form eH/v- = O(ex/v"). At low fetch parameters the computed heights appear a little too high compared with the observa- tions, but it must be kept in mind that the computed values for the stage B<1/3 are maximum values. By the empirical relationship 5 = 2p = 0.124 (equation (18)), the steepness of the waves in their earliest stages of development probably has been assumed a little too high (the value 2p = 0.10 seems to be more representative). But these early stages of wave development are only of minor significance with respect to the later development of the sea. * *Supplement after completion of this report: New observations by U. Roll [25] also indicate a very rapid increase of wave steepness at very low fetches, similar to the broken curve of Fig. 12 of this re- port, and in agreement with the observations of the "Heidberg" at low B-values. Probably it would be a better approximation to use the broken curve in Fig. 12 when considering the relationship 6 = f(g). It 14 It may be called to the user's mind that these results refer to pure "wind sea" which is under the influence of a steady wind. In practical wave forecasts, intervening "dead sea" or swell has to be taken into account, if necessary. It may be mentioned in this connection, that the estimates of the decay of the waves require a consideration of the turbulent state of the sea surface layers. For similar conditions with respect to air resistance, the decay of the waves (including swell) is different depending on whether these waves travel through a calm region of the ocean with a smooth sea surface or whether they have to cross storm areas where heavily breaking wind-generated seas and a considerable state of turbulence are to be expected. is to be expected that the results of this report will not be changed essentially by taking this slightly changed empirical relationship between the steepness and the ages of the waves, but the discontinuities at about 6 = 1/3 may disappear (see the remarks on page 79). In the following report on methods of practical wave forecasting, this slightly changed empirical relationship can be taken into account. 125 Acknowledgements. The preversation of this paper has been msde possible by the sponsorship of the Office of Naval Research at New York University. My sincare thanks sare due to Professor B.Haurwitz for critical read- ing the paper in manuscript form and for helpful advice during my stay in the United States of America since July 1951. I greatly appreciate the help of Dr.W.J.Pierson for administrative advice,and the assistance of Mr.W.Marks who corrected the English of this report. I wish to extend my thanks to the firms F.A.Bolten,Hamburg, and the Hamburg-America-Line,Hamburg, which afforded the opportunity to make extensive observations at sea by permission to participate in a voysge from Hamburg to the West Indies and the Gulf of Mexico aboard M.S."Heidberg". During my stay at Woods Hole Ocesnographic Institution in summer of 1951 I had the onportunity to look into the results of some un- published wave records obtsined by Dr.H.R.Seiwell.For this stay and the kind permission to make use of these records I am very much obliged to Admirsl EK.H.Smith and Dr.C.0.D'Iselin of the Woods Hole Oceanographic Institution. The text of this report was typed by Mrs.Sadelle Wladaver,und the figures were prepared by Mrs.Gertrude Fisher,to whom I am indebted for their careful work. 126 References [1] Sverdrup, H. U., and W. H. Munk, 1947: Wind, Sea and Swell: Theory of Relations for Forecasting. H. 0. Pub. No. 601, U. S. Navy Dept. Hydr. Office. [2] Schumacher, A., 1939: Stereophotogrammetrische Wellenauf- nahmen. Wiss. Erg. d. Deutschen Atlant. Exped., "Meteor," 1925-1927, Bd. 75 He 2, Lietee 1. (Mit Atlas). [3] Schumacher, A., 1950: Stereophotogrammetrische Wellenauf- nahmen mit schneller Bildfolge. Deutsche Hydr. Ztschr., Ietoly Yc [4] Cornish, V., 1934: Ocean Waves and Kindred Geophysical Phenomena. Cambridge University Press, London. [5] Krilmmel, 0., 1911: Handbuch der Ozeanographie. Bd. 2, Stuttgart. [6] Thorade, H., 1931: Probleme der Wasserwellen. Probl. d. Kosm. Physik, Bd. XIII and XIV, Hamburg. [7] Bigelow, H. B., and W. T. Edmondson, 1947: Wind Waves at Sea, Breakers and Surf. H. 0. Publ. No. 602, Washington. iT] [8] Neumann, G., 1950: Uber Seegang,Dunung und Wind. Deutsche Hydr, Zeitschr., Bd. 3, pp. 40-57. " [9] Neumann, G., (In press): Uber die komplexe Natur des See- ganges, I. Teil: Neue Seegangsbeobachtungen im Nord- atlantischen Ozean, in der Karibischen See und im Golf von Mexiko (M.S. "Heidberg," Okt. 1950-Febr. 1959). Deutsche Hydr. Ztschr. [10] Seiwell, H. R., 1948: Results of Research on Surfece Waves of the Western North Atlantic. Pap. Phys. Oceanogr. and Meteorol., ve. X, no. 4. [11] Seiwell, H. R., and G. P. Wadsworth, 1949: A New Development in Ocean Wave Research. Science, v. 109, pp. 271-274. [12] Neumann, G,, 1949: Die Meeresoberflache als hydrodynamische Grenzflache und das Windfeld uber den Wellen. Annalen der Meteorol., Hamburg, Heft 5/6. it [13] Neumann, G., 1948: Uber den Tangentialdruck des Windes und die Rauhigkeit der Meeresoberflache. JZeitschr. f. Meteorol., Potsdam, Jahrgang 2, Heft 7/8. [14] Neumann, G., 1949: Die Entstehung der Wasserwellen durch Wind. Deutsche Hydr. Zeitschr., Bd. 2, Heft 5. 127 [15] [16] [17)} [18] [19] [20] [21] [22] [23] [24] baa Schaaf, S. A., and F. M. Sauer, 1950: A note on the tangential transfer of energy between wind and waves. Trans. Amer. Geophys. Union, XXXI, June 1950. Prandtl, L., 1942: Flhrer durch die Strémunglehre. Verlag, F. Vieweg ue. Sohn, Braunschweig. Thorade, H., 1931: Probleme der Wasserwellen. Probl. ds Kosm. Physik, ve XIII und XIV, Hamburg. Seite 49. Jeffreys, H., 1925: On the formation of water waves by wind, Proce Roy. Soc., Ser. A., CVII, Math. Phys. Sc. London, und ebenda, «C, London, 1926. Motzfeld, H., 1937: Die turbulente Strémung an welligen ee Se ee Ss - SS SS - Prandtl, L., 1932: Meteorologische Anwendung der Str&n- ungslehre, Beitr. Phys. fre Atm. 19, page 192. Lamb, H., 1907: Lehrbuch der Hydrodynamik, Deutsche autorisierte Ausgabe nach der 3 engl. Auflage, Seite 699. Thorade, H., in Lehrbuch der Navigation. Neumann, G., 1951: Uber Seegang bei verschiedenen Windstlrken (Bemerkungen zur Frage der Seegangs-Vorausberechnung) "Hansa," Zeitschr. f. Schiffahrt, Schiffbau, Hafen, 88; No. 21, 1951, page 801 with reference to results obtained by L. Schubart and W. Méckel. Wegener, K., 1937: Seegang und Dunung, Annalen der Meteorologie, Jahrgang 65. Roll, U., 1951: Neue Messungen zur Entstehung von Wasserr wellen durch Wind, Annalen der Meteorologie, 4. Jahrgang, Heft 1-6. 128 3.8 Appendix Observations of wave periods and heights st limited fetches. Island of Amrum (depth of water 20 - 40 cm). “a 0.60 0.75 0.83 0.82 0.83 0.50 o 0.935 7 1.30 1.28 1.30 0.78 0/7 Vv 0.207 O.216 0.260 0.285 0.271 0.205 South coast of San Miguel (Azores) 7 2.1 3.3 4.06 1.6 1.4 5 Del 1.4 0.9 0.6 0.62 0.9 0.65 0.9 0.88 o 5e2/ 5.15 6.35 26> 2.19 2.34 3.27 2.19 1.40 0.93 O.97 1.40 ce 1.40 1.37 C/V 0.605 0.43 Piel 041359 0.137 0.45 0.63 0.20 O2es5 0.558 0.206 Oven O17 0.50 0.36 129 H F/v° vo A HL On02 “Aas5 “Os01eG: » MO St iese7 0.05 44.2 O07 0.88 Dail 60.0 O206 94.0: 0.029. 1:05 . 5.7 86.6 OnO2 6.9 On0145 0.39 Bie: H R/vo ve A H/ 6160. ALANS) 461 Ona02 IAG) 8.9 1990 1569 0.30 15.6 0.015 3.00 10.0 738 738 O25 Bil: 0.020 5.00 Se O.15 106 0.041 1.26 Bae.) 385 163 112 86 1660 760 South coast of San Miguel (Azores). continued. Vv sn) eo aon 3.7 Dae 4.3 2.9 4.4 54 Ome 9.8 8.0 6.5 TO Vv 10.02 10.2 F 390 5200 2000 480 430 550 550 500 3500 305000 285000 250000 7300 i be Ba 88) F 315000 300000 Ak 0.4 5.0 4.8 (8.0) 3.6 (7.6) 2.6 (4.8) 2.8 (5.0) o O/v 0.364 0.527 e571 0.337 0.261 0.333 0.321 0.311 0.578 0.763 0.765 Lece 0.70 1.47 0.622 EeiG 0.623 12 H F/v° v A 1350 0.25 8320 4540 0.39 1.10 5350 0.072 " 1.00 420 0.15 298 0.080 1.30 652 259 1200 28800 29200 49400 1730 2260 Caribbean Ses (approaching La Gusira). Aly o 4, 10. N Es ds S 67 8 ) 56 tony O/ Vv 0) 0459 0.765 0.74 0.95 130 H F v v 1407 29700 16295 28300 H/A 10.0 H/A Ceribbesan Sea (aporosching La Gusira). continued. v F 9.8 285000 9.7 370000 7-3 250000 6.5 8.0 10.3 7400 33200 F 9250 9250 9250 9250 9250 9250 T < 2.0 1.8 (4.8) o WO MAF WOOF 6/Vv 0.755 1.00 0.765 PA Coal H 28 1.26 1.155 0.975 F ve 29000 38600 45800 1720 5100 Off La Guairsa (Venezuels). 6O/v 0.488 elo 0) e446 0.525 0.394 0.61 0.505 Lel/ Oe5r 131 H 0.40 0.60 r/ Vv 2260 2330 3850 1220 1440 870 2 0.096 0.078 A 6.2 725 H/A H/A 6.5 8.0 Vv F 4.7 3000 4.5 3000 Sie 50000 3.3 5300 325 377000 Vv F Vee. 4600 13.0 222000 8.0 34100 9.5 *295000 9.9 295000 10.1 295000 Vv F 9.6 222000 9.5 111000 9.0 111000 Off Port su Prince (Haiti) om O/7 Vv H F ve vo H/A Te Jo 0.415 1360 2.03 0.452 1480 53455 0.643 18400 baie 0.567 4860 1.32 0.377 30200 Off Cape Haitien (Hsiti) or G/v H F v vy A _H/A 4.06 0.333 V2 310 0.079 10.6 1943 a By 0.86 13200 4.56 0.57 0.7 485 0.092 13.4 5.2 6.55 0.69 32600 SAS 1.0 13.2 1.58 74 0575 200) 50200 ~-0.200 35.0 567 745 0.735 2.0 - 29000 0.200 35.8 pies) 14.8 1.46 C2 Bahsma Chennel Co 6/v H Vv Vv H «+7 0.675 32000 0.938 1.41 6.4 9.0 2.6 6.71 0.705 12300 0.9 1.145 4.0 ney} 6.32 0.692 13700 8.6 3.2 0.96 1.46 132 Bshame Channel (continued). v F m o O/v H 12.6 40700 5 4 8.42 0.670 2.0 - (11.0) 217.1 1.36 3.0 10.7 78000 B07 S790 0.760 “2 = 3 9.0 260000 52057 56/0 0.632 10.1 260000 4.60 7.18 0.710 Habana (Harbor) Vv F al o O7 Vv H ee) 700 1.1 Tere On ole 5.8 300 0.9 1.40 0.241 5.0 300 0.8 area, 0.250 OLS 6.0 300 0 1.56 0.26 0.15- 0.20 5.0 300 O.0>5 1.55 0.266 0.10- OielS 5D 350 Os 72s tbete 0.3c 0.10 6.3 1500 1450 2.03 Oe522 pris) 50 0.64 1.00 Oe 7 5 0.08 Puerto Mariel (Cubs) Vv F ly o O/Vv H 3.9 650 0.66 1.03 0.264 5.0 700 0.95 1.48 0.296 4.2 600 O.88 ¢l.37 0.326 133 2 Pes (Bay and Harbor) iy ve wv 427 280 340 F/v v A 2500 0.124- 45.5 O2185 6800 0.17 - Owed 41500 25000 F/v° vo A 2351 89 120 0.059 1.00 83.5 0.049 1.56 120 0.049 1.15 285 05080" 0.el 378 14.8 0.64 H/ te 6.6 Leo H/A Puerto Msriel (Cubs) (Bay and Harbor). continued. v F uy o O/v H v v x H/A 8.2 1150 1.30 2.03 0.248 S.2 1250 1.42 eae 0.270 0.40 186 0.058 Sel D 1267 4.0 2000 1.30 2S 0.510 1250 5.0 2000 1.30 2.03 0.410 0.25 805 0.098 2.63 9.5 3.7 3100 1.18 1.84 0.496 2260 TO65 2000 2.00 ye alye 0.297 182 12 2000 1.98 5.09 0.423 375 8.3 2350 1.85 2.89 0.348 341 Off Puerto Msriel v F T o O/v H km vo A H/A 4.5 ALOO: ugheooe «e341 _-0.556 2020 Ce As, 4.5 7200. 1260, 2.61 0.625 0.35 3700 0.170 5.05 6.9 5 4 18000 rear S.e/ 0.607 6160 Gulf of Mexico v F T o O/v HH _gF/v* vA H/A 10.6 370000 4.5 7.00 0.660 32900 12.2 400000 Dai bade d OO 0.738 2.8 26400 O.18& 51.5 5.4 Notation? v = wind velocity in m/sec F = fetch in meters f = observed periods in seconds. H = wave height in meters. H/A = Steepness in per cent. © = phase velocity in m/sec; @ = 5 rT: A = wave length in meters; A= = 7°, 134 Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. 10. xD les ie. 13. 14. a irs LIST OF FIGURES Frequency distribution of observed wave periods T (sec) at different wind velocities. For explanation of T,, T5 and T3 see text. (Steady wind conditions. Frequency distribution of characteristic periods at different wind velocities (fully developed sea) Periods of characteristic waves (T) as functions of wind velocity v(m/sec). (Observations are indteated Dy SYMbOUSs ) «606. wl sve «@ 6 se os «6 Ss Observed heights of characteristic waves at GQivtoerenc owinGmeverOC1ULes. drs. «i wis! «0 su.0n ete ssstce Wave pattern of fully developed waves at a wind velocity of 16 m/sec. Computed "wave records" for a time interval of 260 seconds at two different localities. "Wave record" (b) is located 550 m GOWNWLNG ePLOM. (Caria crbe aia eve, Vex ea; te hs cles teen teenie Profiles of wave patterns, showing the variations of dominant waves with space and time at a wind walocity of 16 °m/ Sec -205 (assoc. 2 (4S oats 6 ems Profiles of wave patterns, showing the variations of dominant waves with space and time at a wind vellocity Of 410 mM/seG 2-5 Guts os oust ae eS ee Schematic wave profile with a rough surface (% ) and wind force components (7,7 +t) of the effective stress 7. (@ = velocity of wave propagation in GQirection of the wind velocity v . «ss « « « « + * Streamlines of the air over a ray surface, and distribution of wind components 7 y and 7; in Schematic representation . . . s«e-+ces2#s-2 ect @ Wave steepness 5 = H/XA plotted against the ratio o/v = 8. Assumed relationship shown by full line . Relationship between the coefficients of eddy viscosity M(cm-1 g sec-1) and the wind velocity v (m sec~l) at different stages of sea development (wave age B). Values of the coefficients of eddy vESCOSity=shown ‘by Symbols s))'s 4 2% or-Pss 8 6 5 8 Dimensionless quantities C(f) and B(f) as BUNCCTHONS "Of “wave age (she ts es SP 8 See. te Fetch graph. Wave velocity as a function of fetch x and wind velocity v using nondimensional PANAMSUCT Ss 6155! ° 20. Coe eg. 24. 25.6 26. Page Duration graph. Wave velocity as a function of duration t and wind velocity v, using nondimensional parameters e e e e se e e e e e e e es e e e e e e es e e 107 Duration graph. Wave height as a function of duration t and wind velocity v using nondimensional Parameters