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.

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

Diffraction of water waves by a small cylindrical elevation of the bottom of a laterally unbounded ocean covered by an ice sheet is investigated by the perturbation analysis. The ice sheet is modelled as a thin elastic plate. The reflection and transmission coefficients are evaluated up to the first order in terms of integrals involving the shape function representing the bottom elevation. Three particular forms of the shape function are considered for which explicit expressions for these coefficients are obtained. For the particular case of a patch of sinusoidal undulations at the bottom, the reflection coefficient up to first order is found to be an oscillatory function of the ratio of the wavelength of the bottom undulations and that of the incident wave train. When this ratio approaches 0.5, the reflection coefficient becomes a multiple of the number of undulations and high reflection of the incident wave energy occurs if this number is large. Reflection coefficient is depicted graphically to visualize the effect of the presence of ice-cover and the number of undulations.

Wave diffraction by a small elevation of the bottom of an ocean with an ice-cover

T], study f linearzed interface wave problems for two superposed fluids often lnw) Iv,s the co)siderti,)n of dil-ferent types of singularities in one of the two fluids. In [his paper the line and poLnt singularities are investigated for the case when each fluid is of finit constant depth. The effect of surface tension at the surface of separation is i)luded. The study of internal waves in two fluids problems has attracted many authors in recent years. It is found useful to permit singularities of one type or another to occur as an idealisation of, or an approximation to certain physical situations. Problems dealing with the generation of waves at the interface of two non-mixing fluids involve the ('onsideration of singularities of different types in the fluids. In the case when bodies are present, waves may either be generated by the movement of the body or reflected from it. The two cases are essentially the same and the resulting motion can be described by the use of these singularities in a suitable way. For example, Gorgui [1] has investigated into these waves using a distribution of sources on the surface of the body. The different types of singularities that can be used in two fluids problems have been presented by Gougui and Kassem [21 and Kassem [3] in both the effects of surface tension are neglected. In [12] the authors considered the cases when the lower fluid is of finite constant depth and the upper fluid is of infinite height. While in [3] the author considered the cases when the two superposed fluids are both of finite thickness and obtained the potentials for motions resulting from mu|tipoles submerged in one of the two fluids. In this paper we give a complete survey for the basic line and point singularities when the superposed fluids are as in [3] of finite constant thickness confined between two rigid horizontal planes but we here take surface tension into consideration.

Basic singularities in the theory of internal waves

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0305004100060011

Note on the reflexion of water waves at a wall in the presence of surface tension

Problems dealing with the generation of internal waves at the surface separating two fluids involves the consideration of different types of singularities in one of the two fluids. In this paper the velocity potentials describing line and point multipoles are obtained for the case when each fluid is of finite constant depth, neglecting effects of surface tension at the surface of separation.

Multipole expansions for two superposed fluids, each of finite depth

In this paper the forms are obtained for the harmonic potential functions describing the fundamental wave-source and multipole singularities which pertain to the study of infinitesimal time-harmonic waves on the free surface of water when the effect of surface tension is included. Line and point singularities are considered for both the cases of infinite and finite constant depth of water. The method used is an extension of that which has been used to obtain these potentials in the absence of surface tension.

P. F. Rhodes-Robinson

Papers

Fundamental singularities in the theory of water waves with surface tension