Finite-Amplitude Interfacial Waves

Abstract

The gravity waves at the interface between two uniform, unbounded fluids of different densities in the presence of a current or relative horizontal velocity have been considered. The fluids are supposed to be immiscible, incompressible, and inviscid; and the motion is assumed to be irrotational. This chapter discusses the properties and existence of finite amplitude two-dimensional, periodic waves of permanent form that propagate steadily without change of shape. "Two-dimensional" means that the flow field depends only on the horizontal direction of propagation. In the field of surface gravity waves, which is the limit of the present study when the density of the upper fluid is zero, it has been found that three-dimensional waves of permanent form exist and are observed experimentally. Such waves would also exist and be important for interfacial waves. For the purpose of calculating steady waves, there is no loss of generality in taking the speed of propagation parallel to the current, because an arbitrary constant transverse velocity might be linearly superposed on any two-dimensional steady wave without affecting its properties. The wave can be reduced to rest by choosing a frame of reference moving with the wave. The problem is then to calculate steady irrotational solutions of the Euler equations that satisfy continuity of pressure across a common streamline.

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