Water Waves Generated by a Slowly Moving Two-Dimensional Body, Part II

Abstract

A solution is obtained, valid asymptotically as speed approaches zero, for the waves generated behind a two-dimensional surface_-piercing body moving ahead at constant speed U. The method of matched asymptotic expansions is used. There are two regions of interest: (i) In a thin surface layer behind the body but not contiguous to it, the generalized WKB method is used to determine the wave motion, except for a constant multiplicative factor. (ii) In a region behind the body and close to it, an integral equation is formulated and solved. This near-field solution can be determined completely from the condition that there must be a stagnation point at the intersection of the free surface and the body surface. Matching to the far-field solution then determines the unknown factor in the far-field solution. No radiation condition is available, since the thin surface layer of waves behind the body is completely isolated from any possible corresponding layer upstream. An asymptotic formula for wave resistance is found, in which the resistance is proportional to C10U48 , where C is the body curvature at the intersection of the body and the undisturbed free surface. If C = O , the power of U in the resistance formula is higher than 48; its value depends on what is the lowest non_zero derivative of body shape at the intersection. It is speculated that, for an analytically vertical body surface in some neighborhood of the intersection, the wave resistance is proportional to exp(-1/U) as U approaches zero.