On the initial-value problem for a wavemaker

Abstract

The linearized initial-value problem for the generation of straight-crested waves in a deep, inviscid liquid in response to the prescribed motion of a piston wavemaker of finite depth is solved through integral transforms. The indicial admittance (the surface-wave response to a step-function velocity of the wavemaker) is cast in similarity form and expressed in terms of confluent hypergeometric functions for pure (no surface tension) gravity waves. This gravity-wave result, due essentially to Roberts (1987), provides an outer approximation for x [Gt ] l and gt 2 [Gt ] l ( x = horizontal distance from wavemaker and l = capillary length) but implies an infinite wave slope at the contact line ( x = 0) in consequence of the neglect of surface tension. The corresponding similarity solution for capillary waves (no gravity) with either fixed contact angle or fixed contact line is constructed and is found to be analytic in x for t > 0 if the contact angle is fixed or singular like x 4 log x if the contact line is fixed. An inner approximation for gravity waves with either fixed contact angle or fixed contact line is constructed for x = O ( l ) and gt 2 [Gt ] l . The Laplace transform of the general solution is expressed in terms of confluent hypergeometric functions, which permits a compact discussion of its analytical properties.