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Abstract
The paper is devoted to the construction of asymptotic expansions for frequencies of water waves trapped by a beach of nonconstant slope as the longshore wave number tends to infinity. The existence of such trapping modes in this asymptotic regime was proved recently in [A.-S. Bonnet-Ben Dhia and P. Joly, SIAM J. Appl. Math., 53 (1993), pp. 1507--1550]. For a beach of constant slope the formulas obtained reduce to the classical result of Ursell. Some generalizations to other geometries (waves trapped by afloating semisubmerged cylinder and by shores of a channel of finite width) are indicated. The main analytical tool used for the construction of the asymptotics is an explicit solution of a mixed boundary value problem for the Helmholtz equation in an angle with nonhomogeneous boundary conditions.
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