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By Philip L. F. Liu

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The depth is slowly varying, h = h{(32x) and /? is supposed to be proportional to e where e is the wave slope which conforms with the common assumption in the NLS scaling. Because j3 = A/A is the modulation parameter and the modulation of the carrier wave gives rise to a wave group, A may be seen as a measure for the horizontal extent of the wave group. Because the group velocity is a function of the depth h and therefore also a function of e2x, the following multiple scales are introduced now, dx {[• Z = e2 t (70) Note that the role of r and £ is reversed compared to the constant-depth case.

The essential difference between the NLS equation (26) and (76) is the term —ifiiB in the right-hand side of Eq. (76). Another difference is that the coefficients Ai and Ui in Eq. (76) are functions of £. Equation (76) describes the evolution of wave packets propagating over an uneven bottom under the condition that reflection can be neglected and ^Note that (1 — a2)2 occurs in the last term between curly brackets in the expression for V\ and not (1 — a)2 as given by Djordjevic and Redekopp (1978, Eq.

Narrow-band approximation in nonlinearity only The limited bandwidth for which the NLS and the modified equation (mNLS) as given by Dysthe (1979) are derived hampers the application to real waterwave problems as noted amongst others by Trulsen and Dysthe (1996). These authors enhanced the extent of the bandwidth. Both the NLS and the mNLS equations are derived under the conditions that: lAfcl k 0(s) and kh = 0(s~l) with e = ka . (158) The resulting equation, valid up to 0(e4) has been given in Eq.

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