MARK A. DO ELA

THE EFFECT OF SWELL ON THE GROWTH OF WIND WAVES

Several researchers have noted a pronounced reduction in the growth of ind wa es in tanks when mechanically generated waves are added to the system. In order to isolate the cau e of this effect, which occurs for long wave slopes as low as 5 percent, a laboratory experiment wa performed to estimate the direct wind-energy input to short waves in a pure wind sea and in a wind ea contaminated with "swell." The wind input to wind-generated waves seems to be insensitive to the presence of additional long-crested paddle-generated waves. Furthermore, the changes in the de elopment of the wind sea oc­ casioned by the presence of paddle wa es argues against enhanced di ipation a the cause of the reduc­ tion in growth rate. It is hypothesized that paddle-wa e-induced detuning of the resonance conditions for weak nonlinear interactions among wind waves may be the dominant cau e. This idea is somewhat tentative; further detailed exploration of the effect of long wave slope, frequenc ,wind peed, and fetch is needed to identify the cause more positively. In the context of remote sensing of the ocean surface, the sen iti ity of ind- ea de elopment to quite gentle long wave slopes-whatever the cause-certainly strengthens the ca e for a pectral 0 ean-\ a e measurement capability in future satellite systems.

INTRODUCTION 1+ W+D, (1) It is well known that the addition of a train of long, mechanically generated \ a e (" well") to exi ting wind waves in a laboratory tank has a dramatic effect on the \ here VI( i the group peed, d ian appropriate energy density of the \ ind wa es. 1-5 This fact i clearly fetch -dependent \ ind-dri en urrent, 1 i the rate of ener­ illustrated in the set of three spectra in Fig. 1. Figure gy tran fer from the \ ind, Withe rate of nonlinear 1a shows the spectrum of a mechanically generated wave tran fer among \ a e component , D i the rate of energy train of 0.707 hertz frequency and 0.067 steepne . Fig­ dissipation, and withe radian frequency. In laboratory ure 1b is a purely wind-generated pectrum produced by experiment uch a tho e de cribed in Fig. 1, measure­ an II-meter-per-second \ ind mea ured 26 centimeter ment are made at teady tate 0 that the a erage spec­ above mean water level. In Fig. Ic, both wind and paddle tral den it at an) fet hand frequen i determined by are operated together. All three pectra are on the same the entire Up\ ind e olution of the ource function I, (linear) scale and are derived from observations at 50 W, and D: meters fetch in water 1.2 meters deep. The growth of the mechanical wa es in response to the wind and the E(w; x) = [(V, + d ) -' (I /tv + D)] dx . (2) attenuation of the wind wave are apparent when both L paddle and wind are excited together. It i noteworthy that the wind pectrum is not only dramaticall reduced The ource function are related to E and 0 are but its peak i at appreciably higher frequencie \ ith trongl fetch dependent. VI( and d rna al 0 be fetch swell than without. What are the possible cau e of this dependent but much more weakl . The fetch dependen e pronounced effect on \ ind waves by quite gentle s\ ell? of the group speed ari e through change in amplitude The development of the \ ind-wave energy pectrum di per ion, - while fetch difference in d rna be the E(w; x, t) is governed by the energy balance equation 6 re ult of de eloping boundar) la er . Thu , the ob en'ation of attenuated wind \\,a\'e in the pre ence of laboratory well may be au ed by change in the rate of wind input, di ipation or non­ lark . Donelan i a re earch s 'ienti t linear tran fer occa ioned b) the well. Change in the with Environment Canada, at the Canada group peed or drift urrent are Ie likely andidat . Centre for Inland \\ aters, Burlington, Banner and Phillip ' and Phillip and B anner~ argue Ontario L R4 6. that di ipation of the wind wave i greatly en han d in the pre ence of well. Their argum nt on rn th additi e effect of wind drift and orbital \' 10 itie at rh

18 .fohn, H opki 11'. APL Techni 'al D ige<;1. \ 'o /lime 8. \ lImber I (19,'- j 200,------1,------~ t (a) Q) ..c: Paddle only Beach ~ :~c-}I' :a = 5.47 centimeters squared E ~ ;j2 ro o :l 0- en

~ 100- - (meters) Paddle ~ Q) E Figure 2-Sketch of the wind-wave tank showing the loca­ .~ tion of the wave staffs. The asterisk indicates the location Q) u of the wave follower and x-film anemometer. The vertical ex­ aggeration is x 5 .

0 .I \. I 200 In this article, we focus our attention on wind inputs, tl (b) (ij another of the source functions, which has the con­ ..c: Wind only :a 2 venient property that it (or most of it) can be measured ~ 1/ = 16.8 centimeters squared ro directly by measuring the pressure near the surface and :l 0- the surface elevation. The experiments reported below en en are a preliminary attempt to explore the possible effect 100 2 of swell on the wind-input rate to wind-sea components Q) E of the spectrum. '';:; c::: Q) ~ THE EXPERIMENTS ~ lJJ The wind-wave flume and relevant instruments are il­ 0 lustrated schematically in Fig. 2. The wind tunnel was 300 closed, and the airflow could be driven at speeds up to I (e) 14 meters per second. The paddle waves were generated Paddle and wind by a piston-type paddle and dissipated on a beach of 1/ 2 = 15.0 centimeters squared fibrous matting with a slope of 1:8. Nine capacitance tl wave staffs of 1. I-millimeter external diameter were dis­ (ij ..c: tributed along the 80-meter fetch. The flume was 4.57 ~ 200 r- - meters wide and 3.06 meters high, and the water depth ro :l was 1.20 meters. 0- en At the station shown in Fig. 2, an electro hydraulic en 2 wave follower kept a pressure probe close to the mov­ Q) ing water surface. The pressure probe was of the disk E '';:; type designed by Elliott II and was operated with the c::: ~ 100 r- - axis of symmetry horizontal and transverse to the flow. Pressure fluctuations were converted to electrical signals by an MKS Baratron (Model 223 AH) differential pres­ sure transducer. The probe was connected directly to the positive port of the transducer and through a pneumatic low-pass filter to the reference port. Connections among probe, transducer, and filter were made with rigid metal tubing, and the complete system was mounted on the Frequency (hertz) wave follower so that the motion of the follower pro­ Figure 1-Wave spectra at 50 meters fetch in a laboratory duced no flexing of the interconnecting tubes and the wind-wave tank. (a) The spectrum of a continuous train of concomitant wave-coherent (but spurious) pressure sig­ 0.707-hertz paddle-generated waves of steepness ak = 0.067. nal. A pair of capacitance staffs 10 centimeters on either (b) The spectrum of a pure wind sea with measured wind of side of the disk provided concurrent surface elevation 11 meters per second at 26 centimeters height. (c) The spec­ trum of waves with wind and paddle excited together as in and also crude estimates of the directionality and the (a) and (b) . long-crestedness of the waves under the disk. Estimates of the pressure slope correlation, and hence the wind input I, are particularly sensitive to the phase crest of the swell that act to reduce the amplitude at between waves and surface pressure. Therefore, the which the short waves break. Wright 9 and later Plant phase distortion introduced by the probe was measured and Wright 10 explore the hypothesis of Phillips and in the manner of Snyder et aI., 12 and appropriate phase Banner and found that it is unable to account for the corrections were made. The pressure-sensing ports were magnitude of the attenuation of the short waves or its necessarily a few centimeters above the surface. The ele­ dependence on wind speed. vation above the moving surface was varied between

Johns H opkins A PL Technica l Digesl , Volume 8, Number I (1987) 19 Donelan - The Effecl of Swell on the Gro wlh of Wind Waves and 9 centimeters, depending on the strength of the wind 30r------~------~------~ and, therefore, the likelihood of a drop of spray hitting the probe and blocking its ports. Consequently, the pres­ sure measurements were corrected using the observed ex­ ponential decay with height without a change of phase. 13 The fractional energy increase per radian sew) was computed from the quadrature spectrum Qu(w) be­ 10 tween pressure p and surface elevation 11 after correct­ ing for the phase and amplitude response of the pressure measuring system: :0 ~ [Qu (w) JP'1 e k~ co ::J (3) 0- 3 Pw gE(w} CJ) CJ) E k Q) where is the wavenumber of the wave of radian fre­ E . ~ quency w, z is the height of the pressure measurement c Q) above the moving surface, Pw is the density of water, u I I and g is the acceleration due to gravity. I ~ 1 Six runs, analyzed to investigate the effect of swell on I 1 the wind input to the wind waves, are summarized in 1 / Table 1. Two different fan speeds were used. The small 1 / I / differences in measured (pitot-tube) wind speeds arise 1 / 0.3 1 / because of changes in the surface stress occasioned by 1/ the different wave fields. At each fan speed there are 1/ three runs of 0.527-hertz paddle waves with various ini­ // tial steepnesses: ak = 0.0 (no waves), 0.053, and 0.105. ,1 /

0 .1 L-____~ __-L ______~ ______~ 3 10 30 100 Table 1 - Fetch dependence of wind-sea variance. Fetch (meters)

Figure 3-The spatial evolution of the vari ance of the wind Wind Speed at Fetch (meters) sea only with (colored symbols) and without (bl ack symbols) 35 Centimeters Paddle at Intersection paddle waves. The initial steepness of the paddle waves, ak, Run (meters per Slope, with was 0.105. The triangles refer to the 11-meter-per-second wind cases and the circles to the 7-meter-per-second cases. No. second) ak Ci 'Y 381 397

381 6.86 0.0 0.0087 1.32 382 10.72 0.0 0.015 1.56 0.16 4.2 = 0.105) paddle waves. In each ca e, a straight line (or 418 7.29 0.053 power law) fit of wind- ea ariance 11 to fetch x rep- 419 11.08 0.053 resents the data ery ell: 400 7.10 0.105 0.019 0.925 6.2 0.27 11 2 = ax) 397 11.09 0.105 0.026 1.18 (4) In run 397 beyond the 50-meter fetch, the econd har­ monic of the paddle wa e (1.054 hertz) and the wi nd­ An x-film anemometer was used to measure the mo­ sea spectrum merge, making it difficult to a sign a val­ mentum flux 35 centimeters above the mean water Ie el, ue to the purely wind-sea ariance. Longer fetches are and the result was corrected, assuming a linear stress gra­ therefore omitted in that run. dient, to yield surface friction velocity. The pressure-ele­ The value of a ( ith 11 in centimeter and x in meter) vation cross spectra were computed from 2 14 samples and 'Yare tabulated in Table 1 a are the (extrapolated) converted at 20 hertz using a 12-bit analog-to-digital con­ fetches at intersections of pair of lines. verter. The x-film data were sampled at 200 hertz. In Several interesting changes in the fetch dependence are each case, all signals were filtered with matched 12- apparent in Table 1 and Fig. 3_ There i a clear increase decibel-per-octave Bessel (linear phase shift) low-pass in the exponent with wind peed for the ind- ea-onl filters, with - 3-decibel points set at the Nyquist fre­ cases, but the lines inter ect at mall (e entiall zero) quency. fetch. On the other hand, the addition of paddle a e to the wind sea without appreciably changing the ind RESULTS AND DISCUSSION speed causes a marked decrea e in the growth rate of The spatial growth of the wind sea only is illustrated the wind sea. These pairs of line (with and without pad­ in Fig. 3 for four of the runs of Table 1: the runs with dle waves) intersect at 6.2 and 4.2 meter for the 1m er wind only and the runs with wind and the steeper (ak and higher wind speeds, respecti ely. Perhap the \ a e

20 Johns H opkin APL Techni al Dige l. \ 'olume . 'umber I (/9 - ) Donelan - The Eff ec( of Swell on (h e Grow(h of Wind Wa ves develop to this point before the influence of the paddle waves changes their spatial development. Certainly, in the first few meters at these wind speeds, the waves are not yet limited by breaking and may not be steep enough for nonlinear interactions to affect the spectral balance. In fact, Kahma and Donelan 14 have shown that at a fetch of 4.7 meters the spectral peak begins to move to­

C") ward lower frequencies only when the wind exceeds o about 4 meters per second. This change in spectral de­ velopment occurs at lower fetches for higher winds and x corresponds to the point where the fastest growing wave­ """ lets (approximately 8 hertz) begin to be limited by break­ 3 ing. Does the paddle wave enhance the breaking of the wind waves, alter the nonlinear interactions among wind waves, or cause a substantial reduction in the rate of wind input to the wind waves? We now direct our at­ tention to the last part of this question. 0.03 0 . 1 0.3 Figure 4a illustrates the observed dependence of the rate of wind input, t, in terms of the ratio of the fric­ tion velocity, U * , to phase speed, c. Plant, 15 on the ba­ 30.------~----~------~ sis of the wave amplification theory of Miles,1 6 col­ (b) lected various estimates of t and showed that there was general agreement with the line shown, although the scat­ ter in t covered 5 decibels for the laboratory data. The C") 10 data in Fig. 4 are from the wind-only runs at the peak o frequency and higher frequencies (30 and 70 percent higher). Figure 4b shows the same data but plotted ver­ x sus (U(7r1 k)1 c) - 1, a form suggested by Jeffreys' ideas 17, 18 of wave amplification by wind and explored 3 in more detail by Donelan and Pierson. 19 U(7rl k) is the wind at one-half wavelength height estimated from the measured wind speed and friction velocity. The fitted line shown corresponds to 3 10 30 ( U (;Ikl - 1) U(7r/k) )2. .+ 2 t = 4.1 x 10 - 5 ( - 1 (5) c Figure 4-The measured fractional energy input per radian from the wind for the pure wind-sea cases at three frequen­ cies at and above the peak. The triangles refer to the 11-meter­ When estimates for the wind input for the cases with per-second case and the circles to the 7-meter-per-second paddle waves are added (Fig. 5), the scatter is consider­ case. The line in (a) is from Plant's 15 compilation of various ably increased, but no appreciable bias is evident. Note sources of data. that the wind-sea variance is reduced by a factor of about 2.5 (4 decibels) in the presence of paddle waves, so that spurious effects of window leakage. The paddle frequen­ if this is due to changes in the wind input, the values cy (0.527 hertz) and sampling frequency were selected of t for the paddle plus wind cases should be signifi­ so that an integer number of cycles fit within a fast Fou­ cantly less than those for the wind-only cases. In the pad­ rier transform block of data (2048 points). No such con­ dle cases, the frequencies selected are, as before, the peak trol is possible with the wind waves, and therefore a data of the wind sea and two higher frequencies, although window with very low sidelobes was applied to the blocks the wind-sea peak frequency is always greater when pad­ of 2048 points before the Fourier transforms were per­ dle waves are present. In two cases (397 and 400), the formed. coherence between pressure and elevation was too low Harris 20 has explored in detail a wide range of data to allow a reliable estimate of t at the third frequency. windows. On the basis of his analysis, the Blackman­ The results suggest that the wind-input rate to the wind Harris four-term window was applied to these data. The sea is rather insensitive to the presence of longer mechan­ window has sidelobes that are no larger than - 70 ically generated waves of small steepness. decibels and therefore is suitable for our purpose. The Figure 6 traces the spatial development of the spec­ window's 3-decibel bandwidth is 0.0186 hertz so the pad­ trum for run 400 (U = 7.1 meters per second, ak = dle-wave line spectra are slightly broadened. Spectra 0.105). The corresponding smoothed spectra for run 381 from eight consecutive blocks of 2048 samples were aver­ are superimposed to illustrate the changes caused by the aged so that the wind-wave spectral estimates of Fig. 6 paddle waves. Since the spectra cover more than six de­ have 16 degrees of freedom and a 90 percent confidence cades, care must be taken in the analysis to avoid the interval of - 2.2 to + 3.0 decibels as shown.

Johns H opkins APL Technica l Digesl. Volume 8, N umber I (1987) 21 Donelan - The Effect oj Swell on the Growth oj Wind Waves

103~-+--~------~----~-----.-----.

30 f 90% confidence

(V') 10 - 1 o 10 x 10 - 3

3

3 10 30 (UI;lkl _l)

Figure 5-The fractional energy input per radian to the wind CIl sea as in Fig. 4b but with the cases with paddle waves add­ eQ) ed (open triangles refer to the U = 11-meter-per-second cases, open circles refer to the U = 7-meter-per-second . ~ 101 c cases). The centered dot denotes the steeper paddle waves, Q) ak = 0.1 05. u

It is apparent that the spectral density near the peak of the wind-only spectrum (colored line) is greatly 10- 3 reduced (about 10 decibels) in the presence of paddle waves. At higher frequencies, however, the trend is reversed but the differences are not as pronounced. Here the wind waves with paddle are only about 1 decibel higher than the wind waves without. Doppler smearing of the spectra can introduce some differences, but in this case they are likely to be small since orbital velocities 10 - 1 added by the paddle are roughly compensated for by the reduced orbital elocities at the spectral peak. It is interesting to note that the spectral density just 10 - 3 above the peak of the paddle-modified wind sea is gener­ ally close to the spectral density at the same frequency 0.00 1.00 2.00 3.00 4.00 5.00 in the pure wind sea. But in the latter case, the peak fre­ Frequency (hertz) quency is appreciably lower so that the reduction in total wind-sea energy occasioned by the introduction of paddle Figure 6-The spatial evolution of power spectra on log-linear waves arises because the rate of progression of the wind­ scales. The solid lines are the spectra with paddle waves of sea peak (to lower frequencies with increasing fetch) is steepness ak = 0.105 and wind U = 7.1 meters per second. slowed. It is well known that the downwind growth of The colored line is the corresponding smoothed wind-only case. The fetch increases upward, x = 29.1, 39 .2, 55 .0, and the forward (low-frequency) face of a fetch-limited spec­ 69 .8 meters. trum is far greater than either theoretical or experimental estimates of direct wind input would support. Hassel­ 6 mann ,21 ,22 has demonstrated that the additional energy Since the nonlinear tran fer to the forward face i en­ flux may come from weak nonlinear interactions involv­ sitively dependent on the steepness of wa es at and abo e ing a quartet of wavenumbers and frequencies that satis­ the peak, it may be argued that the paddle wa es re­ fy the resonance conditions for gravity waves: duce the steepness of the wind wa e at breaking and consequently also suppress the nonlinear flux of energ (6) to the forward face. Howe er, as pre iou I noted, at each fetch the paddle-modified wind wa e equal or e - ceed the energy density of the pure wind ea at frequen­ where wand k are linked by the dispersion relation cies above the peak of the paddle-modified ind a es. w = (gk) Y2. Thus, the slower reduction of the peak frequenc of the

22 John H opkin A PL Te hnical Dige c, Volume . umber J (/9 - ) Donelan - Th e Effect of S well on the Gro wth of Wind Waves paddle-modified wind sea does not appear to be due to of long-wave slope, frequency, and wind speed, attempt­ weaker nonlinearities in the waves themselves. ing thereby to clarify the mechanisms for arresting wind­ Mechanisms that lead to the modulation of short-wave sea development. energy (e.g., Longuet-Higgins and Stewart23 ) and, In the context of ocean-wave forecasting, this effect hence, enhanced breaking in steep wind seas and direct of swell on wind sea is clearly important yet is not, to dissipation enhancement mechanisms (e.g., Phillips and our knowledge, included in forecast models. The cor­ Banner2) become less efficacious as the ratio of short­ rect prediction of the development of storm seas may wave phase speed to long-wave orbital velocity increases. depend on accurate swell predictions in global wave This would be reflected in Fig. 3 by increasing the slope models. The storm seas themselves eventually become with fetch in the paddle-modified cases as the increas­ swell, and it is not difficult to see that errors may ac­ ing phase speeds in the wind seas far exceed the modest cumulate and propagate from one storm to a subsequent orbital velocities of the paddle waves. one elsewhere. This would seem to strengthen the case As we have seen from the experiments described for a capability to sense directional spectra of ocean above, the rate of energy input from the wind seems to waves in future ocean satellite systems. be insensitive to the presence of the paddle waves. What remaining consequence of the introduction of REFERENCES quite gentle paddle waves can have such a pronounced I H . Mitsuyasu, " Interacti ons between Water Waves and Wind (I)," Rep. In s{. Appl. Mech. Kyushu Un iv. 14, 67-88 (1966). effect on the development of the wind sea? As a work­ 20. M. Phillips and M. L. Banner, "Wave Breaking in the Presence of Wind ing hypothesis, we suggest that the paddle waves may Drift a nd Swell, " J. Fluid Mech. 66, 625-640 (1974). act to detune the sharp resonance at the heart of the 3M. Hatori , M. Tokuda, and Y. Toba, "Experimental Study on Strong In­ teractions bet ween Regular Waves and Wi nd Waves - I," J. Ocean. Soc. weak nonlinear interactions among quartets of gravity Japan 37, 111-119 (198 1) . waves. One way in which this may occur is through the 4L. F. Bli ven, N. E. Huang, and S . R. Long, "Experimenta l Study of the Influence of Wind on Benjamin-Feir Sideband Instability," J. Fluid Mech . changes in apparent gravitational acceleration experi­ 162,237-260 ( 1986) . enced by the wind waves riding on the long waves. As ST. Ku saba and H . Mitsuyasu, "Nonlinear Instability and Evolution of shown by Phillips and Banner, 2 if the long-wave slope Steep Water Waves under Wind Action," Rep. Ins{. Appl. Mech. Ky ushu Univ. 33, 33-64 (1986). is small, the dispersion relation for the short waves is 6K. H asselmann, "Weak Interaction Theory of Ocean Surface Waves," in altered to Basic Developmen{s in Fluid Mechanics, §5.2, Academic Press, New York (1968). 7M. A . Donelan, J. H amilton, and W. H . Hui, " Di rectional Spectra of Wind-Generated Waves," Phil. Trans. R. Soc. London A 315, 509-562 ( 1985). 8M . L. Banner and O . M. Phillips, " On the Incipient Breaking of Small where the subscripts Land S refer to long and short Scale Waves," J. Fluid Mech . 65 , 647-65 7 (1974) . waves. Thus, as paddle waves pass through groups of 9 J. W . Wright, "The Wind Drift and Wave Breaking," 1. Phys. Oceanogr. 6 , 402-405 (1976). wind waves, the latter experience variations in their dis­ lOW. J. Plant and J . W. Wright, " Growth and Equilibrium of Short Gravi ­ persion relation and consequent detuning of the reso­ ty Waves in a Wind-Wave Tank ," J. Fluid Mech. 82, 767-793 (1977). II J. A. Elliott, " I nstrumentation for Measuring Static Press ure Fluctuations nance leading to nonlinear interactions and, in particular, within the Atmospheric Boundary Layer," Boundary-Layer Meteorol. 2, the development of the forward face of the spectrum. 476-495 (1972). It is noteworthy that the forward face of the wind-sea 12 R. L. Snyder, R. B. Long, J. Irish, D. G. Hunley, and . C. Pflaum, " An In strument to Measure Atmospheric Pressure Fluctuations above Surface spectrum (Fig. 6) is considerably less sharp when paddle Gravity Waves," 1. Mar. Res. 32,485-496 (1974). waves are present than otherwise. The slightly larger t3 R. L. Snyder, F. W. Dobson, J . A. Ell iott, and R. B. Long, "Array Mea­ spectral densities at high frequencies in the presence of surement of Atmospheric Pressure Fluctuations above Surface Gravi ty Waves," J. Fluid Mech. 102, I-59 (198 1). paddle waves may also reflect a detuning and consequent 14 K. K. Kahma and M. A. Donelan, "A Laboratory Study of the Minimum broadening of the nonlinear transfer region of influence. Wind Speed for Wind Wave Generation" (submitted to 1. Fluid Mech., 1986). tSW . J. Plant, "A Relationship bet ween Wind Stress and Wave Slope," J. CONCLUDING REMARKS Geophys Res. 87, 1961-1967 (1982) . 16J . W. Mil es, " On the Generation of Surface Waves by Shear Flow," J. Evidence from various laboratory experiments leaves Fluid Mech. 3 , 185-204 (1957). little doubt that quite gentle long waves, added to a pure 17 H . Jeffreys, " On the Formation of Waves by Wind, " Proc. R. Soc. Lon­ wind sea, cause substantial reductions in the growth of don Ser. A 107, 189-206 (1924). t8H. Jeffreys, " On the Formation of Waves by Wind. II ," Proc. R. Soc. the wind sea. The reason, however, is not so clear. Pro­ London Ser. A 110, 341-347 (1925) . posed mechanisms for enhanced dissipation of the wind t9M. A. Donelan and W . J . Pierson, " Radar Scattering and Equilibrium waves by long waves and wind drift do not seem to ex­ Ranges in Wind-Generated Waves - with Application to Scatterometry" J. Geophys. Res. (in press 1987) . plain the observed spectral evolution, and measurements 20F. J . Harris, " On the Use of Windows for H armonic Analysis wit h Di s­ of direct wind input to wind waves reported here ap­ crete Fourier Transform," Proc. IEEE 66,5 1- 83 ( 1978). 21 K. Hasselmann, "On the Non-Linear Energy Transfer in a Gravity Wave pear to be insensitive to added paddle waves. Spectrum. Part I," J. Fluid Mech. 12 , 481 -500 (1962). We suggest that a possible candidate for the reduc­ 22 K. Hasselm ann, "On the Non-Linear Energy Transfer in a Gravity Wave tion in the wind sea induced by paddle waves is the Spectrum. Parts 2 a nd 3," J. Fluid Mech. 15 ,273-281 , 385-398 (1963) . 23M. S. Longuet-Higgins and R. W . Stewart, "Changes in the Form of Shon detuning of resonant nonlinear interactions. We will con­ Gravity Waves on Long Waves and T id al Current s," 1. Fluid Mech. 8, tinue to examine the wind-sea evolution for the effects 565-583 (1960).

Johns Hopkins APL Technica l Digest, Vo lume 8, Number I (1987) 23