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Reflection and transmission coefficients for water wave scattering by a sea‐bed with small undulation
Author(s) -
Martha S.C.,
Bora S.N.
Publication year - 2007
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.200610317
Subject(s) - reflection coefficient , reflection (computer programming) , ripple , conservative vector field , perturbation (astronomy) , transmission coefficient , scattering , physics , optics , mathematical analysis , bragg's law , transmission (telecommunications) , mathematics , mechanics , diffraction , quantum mechanics , electrical engineering , voltage , computer science , compressibility , programming language , engineering
The scattering of a train of progressive waves by a small undulation of the bottom of a laterally unbounded sea is considered with two dimensional linear water wave theory. Assuming irrotational motion, a simplified perturbation analysis is employed to obtain the first order corrections to the velocity potential by using the Green's integral theorem in a suitable manner and also to the reflection and transmission coefficients in terms of integrals involving the shape of the function $c(x)$ representing the bottom undulation. Two particular forms of the shape function are considered and the integrals for reflection and transmission coefficients are evaluated for these shape functions. Out of these two cases, we focus mainly on a patch of sinusoidal ripples at the bottom and find that the reflection coefficient up to the first order is an oscillatory function of β which is twice the ratio of the wave numbers of the wave and the ripple bed. When this ratio becomes 0.5, i.e., when β = 1 (the situation when Bragg resonance occurs), the reflection coefficient becomes a multiple of the number of ripples, and high reflection of the incident wave energy occurs if this number is large. We present the result in graphical form for one, two, and six ripple(s) in the patch. The results find absolute agreement with available results. We also show the graphical comparison of the reflection coefficient against the wave number for one, three, and five ripple(s).