On the Barber dispersion relation for internal waves

We investigate the dispersive behaviour of internal waves supported by a density gradient in a fluid which has an upper boundary surface. By identifying a realistic model for the density gradient which leads to an analytically tractable internal wave eigenvalue equation we are able to verify the general applicability of, and to characterize the deviations from, the simple dispersion relation proposed recently by Barber. We also consider the effect of a linear background density variation superimposed on the localized pycnocline; this establishes a minimum frequency and wavenumber consistent with the propagation of the internal wave and leads to significant deviations from the Barber result.