The steady solution of unidirectional short-(wavelength) and long-(wavelength) wave interaction in deep water was derived using a conventional perturbation approach and a modulational wave model approach respectively by Zhang et al (1993a, hereinafter named as ZHY). Their results revealed that for the interacting waves of well-separated wavelength scales, conventional perturbation approaches may encounter convergence difficulty resulting from large modulation in the short-wave phase, and the difficulty can be overcome by directly formulating the modulated short-wave phase in the phase function. In certain cases of studying wave-wave interaction, the water depth is only intermediate with respect to the long wave though still deep enough with respect to the short wave; for example, a wave field outside the surf zone consisting of two or more distinct wave trains well-separated in the frequency domain (Thompson 1980), and the modulation of mechanically generated short waves by long waves in a wave tank (Spell et al. 1993). It was shown that the modulation of short waves by long waves increases when the water depth becomes shallow relative to long wavelengths (Longuet-Higgins and Stewart 1960). To accommodate the effects of finite water depth on wave modulation, the study of the modulational wave-phase model is herein extended to allow for water depth that is deep with respect to a short wave but intermediate to a long wave.
Following the work by ZHY (1993a), the steady solution of the interaction between a deep-water short wave and intermediate water depth long wave is derived up to third order of wave steepnesses using a conventional perturbation approach in the rectilinear coordinates and a modulational wave model approach in the orthogonal curvilinear coordinates, respectively.