On the forced surface waves due to a vertical wave-maker in the presence of surface tension

TLDR: In this article, the boundary-value problem for the velocity potential of a wave-maker is solved for both infinite and finite constant depth, and the solution of the corresponding problem concerning the axisymmetric wave motion due to a vertical cylindrical wavemaker is deduced in conclusion.

Abstract: The classical wave-maker problem to determine the forced two-dimensional wave motion with outgoing surface waves at infinity generated by a harmonically oscillating vertical plane wave-maker immersed in water was solved long ago by Sir Thomas Havelock. In this paper we reinvestigate the problem, making allowance for the presence of surface tension which was excluded before, and obtain a solution of the boundary-value problem for the velocity potential which is made unique by prescribing the free surface slope at the wave-maker. The cases of both infinite and finite constant depth are treated, and it is essential to employ a method which is new to this problem since the theory of Havelock cannot be extended in the latter case of finite depth. The solution of the corresponding problem concerning the axisymmetric wave motion due to a vertical cylindrical wave-maker is deduced in conclusion.