Traveling internal waves in a two-layer shallow medium with variable bathymetry and current

In the linear approximation, we examine one-dimensional problems of long internal wave propagation in a stationary flow of two-layer fluids with free boundary in a channel of variable depth and width. We use the shallow-water approximation and assume that liquids in the layers are ideal, immiscible, and have a small relative density difference inherent to natural currents. The conditions the flow must satisfy for wave propagation without reflection are found and analyzed. It is shown that there are three classes of such flows, and the characteristic properties of each of them are studied and compared with those found earlier in a similar problem for surface waves. A general analysis of the problem is illustrated by a few particular solutions. The results obtained can be of interest for understanding natural phenomena in which internal waves play a significant role.

Automated summary

In this article, the linear approximation is used to study the propagation of long internal waves in a two-layer fluid with a free boundary in a channel of variable depth and width. By using the shallow-water approximation and assuming ideal, immiscible liquids with a small density difference, the conditions for wave propagation without reflection are determined and analyzed. Three classes of flows that satisfy these conditions are identified and compared with previous results for surface waves. Several specific solutions are presented to illustrate the general analysis of the problem. The results have implications for understanding natural phenomena involving internal waves.