Spectra of Water Waves in Channels and around Islands

Abstract

Spectra of bounded and unbounded basins are calculated on an asymptotic approximation based on smallness of the seabed slopes. This leads to the geometrical optics theory of surface waves of Keller, which is used to extend the known, exact result for an idealized beach to the prediction of the spectra of a variety of more natural water bodies. Attention is directed to cylindrically symmetrical and two-dimensional topographies corresponding, respectively, to islands and to continental slopes and laboratory channels. Marked differences are found in the character of the spectra for these two types of topography.

Localised continental shelf waves: geometric effects and resonant forcing

“…In §3 the offshore modal structure is of horizontal scale commensurate with the scale of offshore depth variation and the slow variation in the WKBJ analysis is alongshore. This differs from the WKBJ analysis in Shen et al (1968) for non-rotating free-surface waves and for rotating, stratified edge waves in Zhevandrov (1991), Smith (2004) and Adamou et al (2007) where the alongshore profile is fixed and the waves are short compared to the scale of offshore variations. Topography varying slowly in both horizontal directions is considered for non-rotating free-surface waves by Keller (1958), short topographic Rossby waves in Smith (1970), trapped modes in quantum rings by Gridin et al (2004) and Bruno-Alfonso & Latgé (2008), trapped modes in elastic plates by Gridin et al (2005) and trapped modes in slowly-varying acoustic waveguides by Biggs (2012).…”

Stratified rotating edge waves

“…Ursell (1952) showed that the Stokes edge mode was the first member of a family of trapped modes, with the number of discrete modes depending on the slope of the bottom. Dispersion relations for edge waves over arbitrary topography were then found in the case of gentle bottom slopes by Shen, Meyer & Keller (1968), Miles (1989) and Zhevandrov (1991, hereafter referred to as Z91) among others.…”

“…Z91 shows that the results of Shen et al (1968) provide the correct leadingorder solution to the edge-wave dispersion relation for a homogeneous fluid in the small-slope limit. We could hence follow Keller & Mow (1968, § 5) which gives the appropriate method for the stratified case (the addition of rotation does not alter the dispersion relation to leading order).…”

Stability and Trapping of Slow Hydromagnetic Waves in Rotating Cylindrical Geometry

“…SHEN (1976) has applied an asymptotic technique of the type used here to the problem of surface waves in a viscous fluid. Some variant of his approach may be applicable to the present problem.…”

“…In these cases, caustics appear and may act as boundaries of the wave regions, depending on the values of c and d. Techniques for dealing with asymptotic eigenvalue problems involving caustics in the fluid domain were developed by KELLER and RUBINOw (1960), and exploited by many authors. A useful reference in this regard is the paper by SHEN et al (1968) relating to the spectra of water waves.…”