Topographic Filtering and Reflectionless Transmission of Long Waves

Abstract

The equation governing the passage of linear monochromatic, long waves over variable topography can be transformed into a Schrödinger equation. There are several transformations accomplishing this. First, a ?naive? transformation (in which only the horizontal coordinate is stretched) yields a potential energy function (?potential?) that is nonvanishing, even if the slope in topography vanishes. Second, a transformation in which also the surface elevation field is stretched leads to a potential that does vanish outside the sloping region. The latter has the property that it displays scattering against a background of adiabatic variations. For smooth bottom profiles, typical for the continental slope, it is shown that the potential has a positive lobe, the top of which acts as a ?topographic cutoff frequency.? This lobe is missed by piecewise-linear topographies. Despite that the topography, in general, acts as a high-pass filter it is shown that some particular, smooth bottom profiles exist for which long waves, obeying certain conditions, can pass reflectionless.