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The Quarterly Journal of Mechanics and Applied Mathematics 1991 44(3):487-506; doi:10.1093/qjmam/44.3.487
© 1991 by Oxford University Press
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TRAPPED MODES ABOVE A SUBMERGED HORIZONTAL PLATE

C. M. LINTON and D. V. EVANS

( School of Mathematics, University of Bristol, University Walk Bristol BS8 1TW )

It has been proved that at certain frequencies trapped surface-wave modes exist above a submerged long horizontal cylinder of fairly general cross-section, but the proof does not extend to the case of a submerged horizontal plate (Jones (6, p. 682)). In this paper a numerical technique is developed which provides strong supporting evidence for the existence of trapped modes in this case also. In particular the frequencies of oscillation of such trapped surface-wave modes are approximated by first constructing an inhomogeneous integral equation of the first kind and then converting this into an infinite system of linear algebraic equations, for which the coefficient matrix is positive definite, which can be solved by truncation. It is shown how a long-plate approximation can be used to relate these trapped-mode frequencies to the reflection coefficient for the problem of a wave travelling above a semi-infinite plate being totally reflected at its end, a problem which can be solved exactly using the Wiener-Hopf technique. This approximation turns out to be extremely accurate for most parameter values and provides a very quick method for the calculation of the trapped-mode frequencies.


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