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B > (159b) S VJJ £ 2 (B) Ms (5,$) 16k3 \ IK B d3 dxdy2 dB* dx (159c) d^)B- -£)*• • " > % (159d) • dx z=0 48 M. W. Dingemans & A. K. Otta Trulsen and Dysthe (1996) now assumed a wider bandwidth to exist, lAfel k 0(e^2) kh = 0{e~1'2). and (160) With e1'2 being the new expansion parameter, the new slow spatial and time variables are e1'2x, el'2t and also the vertical coordinate is somewhat faster than before: e1'2z. With a similar expansion procedure as before, now up to fifth order in the expansion parameter e1'2, the resulting equation is, in variables with dimension, S(B) = C2(B)+N2(B,$)+£3(B), (161a) the difference being only in the linear dispersive terms.

903). A few notable differences between the situation with and without surface tension are listed below, • With surface tension included, it becomes possible that cg > gh. This is easiest seen by considering a series expansion for small kh of cg from Eq. (64c) which yields, cg = ^gh(l+j) (l + ^ ) = 0{{khf). (67) As 7 > 0, we have cg > \fgh in the shallow-water approximation. b Notice that in Eq. 14) of Djordjevic and Redekopp (1977), a printing error occurs: the first term between curly brackets has 2c instead of 2c-y in the numerator.

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. , 1990), C(x, t; X, T) = l-A(X, T) exp[iX(x, t)] + CC , *(x,z,t;X,z,T) = - i g A £ T ) C O S ^fc; Z)] exp[ tX (x,t)] + CC, (180a) (180b) with x(x,t)= dx'k(x')-cjQt, (180c) where the carrier wave frequency u is determined by: w = u)r + k • U and J^ = gk tanh kh, (180d) and fc is given by: k = Vx{x,t).

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