J. Fluid Mech. (1993), vol. 252, pp. 565-584 Copyright 0 1993 Cambridge University Press 565 Surface waves on shear currents: solution of the boundary-value problem By VICTOR I. SHRIRA P. P. Shirshov Institute for Oceanology Russian Academy of Sciences, Krasikova 23, Moscow 1 172 18. Russia (Received 15 July 1991 and in revised form 31 July 1992) We consider a classic boundary-value problem for deep-water gravity-capillary waves in a shear flow, composed of the Rayleigh equation and the standard linearized kinematic and dynamic inviscid boundary conditions at the free surface. We derived the exact solution for this problem in terns of an infinite series in powers of a certain parameter E, which characterizes the smallness of the deviation of the wave motion from the potential motion. For the existence and absolute convergence of the solution it is sufficient that E be less than unity. The truncated sums of the series provide approximate solutions with a priori prescribed accuracy. In particular, for the short-wave instability, which can be interpreted as the Miles critical-layer-type instability, the explicit approximate expressions for the growth rates are derived. The growth rates in a certain (very narrow) range of scales can exceed the Miles increments caused by the wind. The effect of thin boundary layers on the dispersion relation was also investigated using an asymptotic procedure based on the smallness of the product of the layer thickness and wavenumber. The criterion specifying when and with what accuracy the boundary-layer influence can be neglected has been derived. 1. Introduction Water waves in oceans and other natural basins almost always propagate on shear currents, rather than in still water. The theory of the interaction between waves and steady currents has many different aspects, among them a number of open ones (see the review works of Peregrine 1976; Peregrine & Jonsson 1983; Jonsson 1989; Craik 1985). In this paper we shall focus our attention upon one of the most fundamental open questions, which is the inevitable first step in any study of wave-current interactions, namely the boundary-value problem. The boundary-value problem we are interested in follows naturally from the fundamental equations of inviscid hydrodynamics if one considers small mono- chromatic perturbations to the steady horizontally uniform shear flow U(z). It implicitly gives the vertical structure W(z) of monochromatic perturbations with wave vector k, and their phase velocity C in terms of the basic flow profile and the wave vector. The problem reduces under certain additional assumptions discussed in g2.1 to the boundary-value problem prescribed by the Rayleigh equation (C- 42) ( W”-k2 W) + W’ w = 0, (1.1) w= 0, (1 4 zero boundary condition at the lower horizontal boundary z = - H,