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By Philip L. F. Liu

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114c) may be neglected. This simplifies the substitution oidQ/dz in Eq. (114a). In that case, Eq. dB* + iT-B B dx 6B* dB_ dx kB dx z=0 (115) Nonlinear Modulation of Water Waves 37 which is also identical to Eq. (10) of Trulsen and Dysthe (1996). As also discussed in Lo and Mei (1985), these equations are derived under the condition that kh = 0{(fca) - 1 }

82) to ID is readily obtained by supposing fc0 is directed along tree a;-axis and ignoring all y-dependence, dA ldcgd2A dA\ +C lK ^) + „^J ,2 IAI2A 2dt^+AG*-k0"0KlAlA 1 f d fdcg\ i } dA fidcq v A = 1 \ „ , N (86a) Mtek£)-' \te-{2i£-2 ) > and d24> d ( d\ 9h d ( fc0 2. „l2\ A W - d~x { £) = dx {^ n w2 d ( u p . 3. M. W. Dingemans & A. K. Otta Shallow-water limit In the shallow-water limit, kh —> 0, we obtain, under the condition that the Stokes number is small, e/(kh)2

1990), Liu et al. (1992) and Dingemans et al. (1991). In this section, we primarily consider the second-order formulation for the wave amplitudes. In third-order, the NLS-equation formulation is obtained. For that case, we refer to Dingemans et al. (1991). 1. Formulation of the equations We consider a train of modulated linear waves propagating over a slowlyvarying bottom. In first instance, we also include an ambient current field in the considerations. Because the length scales of the resulting wave groups are much larger than those of the carrier waves, slow variables are introduced by: X = j3x and T = j3t, /3«1.

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