On interaction between intermediate-depth long waves and deep-water short waves

The interaction between a unidirectional deep-water short-wave train and an intermediate water-depth long wave is studied. The steady solutions are derived up to third order in wave steepness, respectively, using two different approaches: a conventional perturbation method employing linear phase functions to describe both long- and short-wave phases and a phase modulation method using a modulational phase function to model the short-wave phase. The two results are shown to be identical for a parametric range χ₁ coth Kdχ₃ ≤ 0.5, where χ₃ is the short-to-long wavelength ratio, χ₁ and K are, respectively, the long-wave steepness and wavenumber, and d the water depth. When χ₁ coth Kd approaches χ₃, the conventional solution converges slowly and eventually diverges for χ₁ coth Kdχ₁. The slow convergence of the conventional solution results from the approximation of a modulated short-wave phase by a linear phase formulation. In addition to the increasing modulation of the short-wave phase, amplitude and wavenumber as the water depth decreases, it is found that the modulation of the short-wave intrinsic frequency and potential amplitude along the long-wave surface become significant. Previous results about virtually non-modulated short-wave intrinsic frequency and potential amplitude are only limited to the case of unidirectional wave modulation in very deep water.