Internal waves

Vibrations of the interface of two or more heavy liquids having different densities. If the density distribution of the liquid is continuous, an internal wave is understood to mean the vibrations of the surfaces of equal densities.

Let a layer of a liquid of density $\rho_1$ and thickness $h$ overlie the surface of another liquid of infinite depth and density $\rho_2$, where $\rho_2>\rho_1$. The open surface of the liquid and the interface between the two liquids may form standing waves of length $\lambda=2\pi/k$ of two types. The vibration frequency $\sigma$ of the waves of the first kind is given by the formula

$$\sigma^2=gk,$$

while the frequency of vibrations of the second kind is given by the formula

$$\sigma^2=gk\frac{\rho_2-\rho_1}{\rho_1+\rho_2\operatorname{cotanh}kh}.$$

For a given wave length of the second kind the frequency of the vibration is small if the difference between the densities of the two is small. The amplitude of the vibrating interface waves is many times larger than that of the waves on the open surface, and is

$$\frac{\rho_1}{\rho_2-\rho_1}e^{kh}.\label{*}\tag{*}$$

The rate of propagation of the progressive waves generated by the vibrations of the first kind is

$$c^2=\frac{g\lambda}{2\pi}.$$

The rate of propagation of the progressive waves generated by the standing waves of the second type is much smaller:

$$c^2=\frac{g\lambda}{2\pi}\frac{\rho_2-\rho_1}{\rho_1+\rho_2\operatorname{cotanh}2\pi h/\lambda}.$$

The amplitude of the progressive waves of the second kind at the interface is much larger than that of the waves propagating on the open surface. This amplitude ratio is given by formula \eqref{*}.

References

[1]	W. Kraus, "Innere Waben" , Leningrad (1968) (In Russian; translated from German)

References

[a1]	H. Lamb, "Hydrodynamics" , Cambridge Univ. Press (1932) pp. Chapt. IX MR1317348 Zbl 36.0817.07