542 



radiation stress then set up by the short-wave train 

 would act to transfer momentum to the long wave. 

 In particular, if the short wave were to reach an 

 amplitude at the crest of the long wave high enough 

 for breaking, it would give up all its momentum to 

 the long wave. This maser-like mechanism was 

 examined critically by Hasselman (1971) who showed 

 that the change in potential energy in the surface 

 layer due to Stokes' transport by the short waves 

 would give a contribution to the energy transfer 

 to the large waves which would exactly cancel that 

 arising from Longuet-Higgins ' momentum transfer 

 term. Hasselman 's analysis did not take into 

 account any transfer due to modulation of surface 

 wind stress or short wave growth rate, however. 

 [This effect has been analysed by Valenzuela and 

 Wright (1976)]. Also, his analysis concerned 

 primarily gravity waves, for which resonant inter- 

 action between wave number triads only occurs to 

 third order. For capillary-gravity waves, however, 

 the dispersion relation allows resonant interaction 

 at second order. Valenzuela and Laing (1972) have 

 developed a theory for this, and Plant and Wright 

 (1977) suggest that part of the measured excess 

 growth rate in the low gravity wave range could be 

 attributed to capillary-gravity resonant interaction. 

 Benny (1975) has also shown that under certain 

 conditions , a long gravity wave may grow in the 

 presence of small scale capillary waves; the wind 

 field was not inclioded in his analysis. 



The present paper reveals yet another possible 

 mechanism for the transfer of energy from capillary 

 to short gravity waves. The theory presented takes 

 into account the effect of shear flow modulation 

 on the local growth rate of the capillaries. It 

 is found that this variation gives rise to a modu- 

 lation of the Stokes' drift which is in phase with 

 the long-wave surface slope and therefore makes 

 possible an energy interchange with the long wave. 

 It is found that the energy transfer rate due to 

 this mechanism is positive for capillaries with a 

 group velocity higher than the phase velocity of 

 the long wave so that it can provide an increase 

 in the long-wave growth for waves in the short 

 gravity wave regime. For waves running against the 

 wind the transfer rate is found to be negative, so 

 that the presence of the capillaries would always 

 increase the decay rate of the long waves. 



2. INTERACTION BETWEEN LONG AND SHORT WAVES 



We shall consider the situation depicted in Figure 

 1 with two-dimensional surface wave of small wave 

 length. A', riding on a large-scale wave of wave 

 length, X. An asymptotic analysis will be carried 

 out under the assumption that 



X'/A << 1 



(1) 



(Prime refers to the short and tilde to the long 

 waves) . The waves are excited by a wind field 

 blowing over the water surface. Only the normal 

 stress induced by the wind on the wavy surface is 

 considered in this process, the effect of shear 

 stresses being neglected. Of particular interest 

 is whether the presence of the small-scale waves 

 could change the growth rate of small-amplitude 

 long waves. 



To arrive at the simplest possible analysis, 

 terms that are of higher order than linear in the 



long-wave slope are neglected. For the short waves, 

 only quadratic and lower-order terms in the wave 

 slope are retained. Further, it will be assumed 

 that the flow in the water is irrotational, i.e., 

 the effects of surface drift currents are neglected. 

 This allows the use of potential-flow theory leading 

 to the following boundary-value problem for the 

 velocity potential <}> in deep water: 



XX zz 



with boundary conditions 



Z t XX 



(2) 



(3) 



at z = C: <^ 



_W 

 ^ P 



-gc-*^- |-|v<f|2+ T?,.../(l+?2)^/^ 



XX X 



at z = -«■: 



(4) 

 (5) 



Here, C = C(x,t) is the surface deflection, P the 

 surface pressure due to the wind, and T the surface 

 tension. Since cubic terms are neglected through- 

 out, the denominator in the last term of (4) will 

 be set equal to unity henceforth. We now separate 

 large and small scales by introducing into the 

 equations of motion 



C + ;' 



P = p + p' 



(6) 

 (7) 

 (8) 



For the boundary conditions it is useful first to 

 transfer them to the surface of the large-scale 

 motion, z = ? , by a Taylor series expansion. Thus, 



,(x,;) 



^^(x,^) + C'$zz(x,<;) + 



i'2(x,C) + <f^(x,<;) + C (*22(x,C) 



+ «>' (x,C)) + ... 

 zz 



etc. By neglecting terms involving triple and 

 higher products one finds from (4) and (5) the 

 following boundary conditions to be applied at 

 z = I: 



(9) 



water 



FIGURE 1. Long-wave short-wave interactions in a 

 shear flow. 



