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Trapped modes and scattering for oblique waves in a two-layer fluid

Published online by Cambridge University Press:  24 July 2014

M. I. Romero Rodríguez  and
P. Zhevandrov
M. I. Romero Rodríguez
Affiliation:
Escuela Colombiana de Ingeniería ‘Julio Garavito’, Autopista Norte No. 205-59, Bogotá, Colombia
P. Zhevandrov*
Affiliation:
Facultad de Ciencias Físico-Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Ciudad Universitaria, 58030 Morelia, Michoacán, Mexico
*
†Email address for correspondence: pzhevand@umich.mx
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Abstract

The system describing time-harmonic motions of a two-layer fluid in the linearised shallow-water approximation is considered. It is assumed that the depth is constant, with a cylindrical protrusion (an underwater ridge) of small height. For obliquely incident waves, the system reduces to a pair of coupled ordinary differential equations. The values of frequency for which wave propagation in the unperturbed system is possible are bounded from below by a cutoff, to the left of which no propagating modes exist. Under the perturbation, a trapped mode appears to the left of the cutoff and, if a certain geometric requirement is imposed upon the shape of the perturbation (for example, if the ridge is a rectangular barrier of certain width), a trapped mode appears whose frequency is embedded in the continuous spectrum. When these geometric conditions are not satisfied, the embedded trapped mode transforms into a complex pole of the reflection and transmission coefficients of the corresponding scattering problem, and the phenomenon of almost total reflection is observed when the frequency coincides with the real part of the pole. Exact formulae for the trapped modes are obtained explicitly in the form of infinite series in powers of the small parameter characterising the perturbation. The results provide a theoretical understanding of analogous phenomena observed numerically in the literature for the full problem for the potentials in a two-layer fluid in the presence of submerged cylinders, and furnish explicit formulae for the frequencies at which total reflection occurs and the trapped modes exist.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Geophysical and Geological Flows: Topographic effects Waves/Free-surface Flows: Wave scattering
Type
Papers
Information
Journal of Fluid Mechanics , Volume 753 , 25 August 2014 , pp. 427 - 447
Copyright
© 2014 Cambridge University Press 

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