By Philip L. F. Liu
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Extra info for Advances in Coastal and Ocean Engineering
W. Dingemans 0 & A. K. 5 Fig. 1. Parameter space of the coefficients inclusive surface tension, showing where they change sign as functions of surface tension 7 (abscissa) and kh (ordinate); from Ablowitz and Segur (1981). the amplitude envelope. No soliton solutions are possible in the regions A, E and B where Xu > 0. The reader would find it worthwhile to refer to Ablowitz and Segur (1979, 1981) and Djordjevic and Redekopp (1977) for an elaborate discussion on the behaviour of the solutions in the separate regions.
W. Dingemans & A. K. k 2 (64b) 1 1 - c l + kh1 a + 2 ^ 2 (7 l+7_ (64c) It is found that the second-harmonic term £(2>2) becomes singular when 7 = cr 2 /(3 — cr2). In that case, one has second-harmonic resonance which is possible for capillary waves. Assuming that the wave numbers are not too close to the ones for which 7 = er 2 /(3 — a2), the third-order terms may be considered. Instead of Eq. (40), now the following equation is obtained for the leadingorder mean flow >(1,0), 2^(1,0) t3V (1,0) , 2 f ,, 2^ 2 ,.
92) Using a perturbation analysis with respect to the underlying current (U, (c), Turpin et al. (1983) showed that the amplitude A of the waves satisfies the equation (note that the form of the equation mentioned below is a result of multiplying icg to their original Eq. 22) and converting the pressure amplitude to surface amplitude), IAT +icgA^ + X\ATT - vi\A\2A + ijcA = 0, (93) where T = e2t, £ = e2x and r = e(J(dx/cg) — t). In Eq. (93), A is the complex amplitude of the first order elevation, identical to c^1,1) in the expansion series of Eq.