A study of the waves and boundary layers due to a surface pressure on a uniform stream of a slightly viscous liquid of finite depth

Abstract

The 2D problem of linear waves generated by an arbitrary pressure distribution p0(x,t) on a uniform viscous stream of finite depth h is examined. The surface displacement ζ is expressed correct to O(ν) terms, for small viscosity ν, with a restriction on p0(x,t). For p0(x,t)=p0(x)eiωt, exact forms of the steady-state propagating waves are next obtained for all x and not merely for x≫0 which form a wave-quartet or a wave-duo amid local disturbances. The long-distance asymptotic forms are then shown to be uniformly valid for large h. For numerical and other purposes, a result essentially due to Cayley is used successfully to express these asymptotic forms in a series of powers of powers of ν1/2 or ν1/4 with coefficients expressed directly in terms of nonviscous wave frequencies and amplitudes. An approximate thickness of surface boundary layer is obtained and a numerical study is undertaken to bring out the salient features of the exact and asymptotic wave motion in question.