Journal of Applied Mathematics
Volume 2006 (2006), Article ID 53723, 24 pages
doi:10.1155/JAM/2006/53723
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
Department of Mathematics, Khalisani College, Hooghly 712 138, India
Received 31 May 2006; Revised 25 October 2006; Accepted 22 November 2006
Copyright © 2006 Arghya Bandyopadhyay. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.