Waves Trapped by Submerged Obstacles at High Frequencies

Abstract

As is well known, submerged horizontal cylinders can serve as waveguides for surface water waves. For large values of the wavenumberkin the direction of the cylinders, there is only one trapped wave. We construct asymptotics of these trapped modes and their frequencies ask→∞in the case of one or two submerged cylinders by means of reducing the initial problem to a system of integral equations on the boundaries and then solving them using a technique suggested by Zhevandrov and Merzon (2003).

Water waves trapped by thin submerged cylinders: Exact solutions

“…We do this by means of the same technique as in [11,18,21]. This enables us to obtain exact solutions in the form of series of powers of ε and ε ln ε, where ε characterizes the "thinness" of the cylinder.…”

Water waves trapped by thin submerged cylinders in a two‐layer fluid: Discrete eigenvalue

“…This resulted in finding exact solutions in the form of series in powers of and ln , where characterises the "thinness" of the cylinder, by means of a technique similar to previous studies. 13,[16][17][18][19][20][21] The leading term of the series coincided, of course, with the result of McIver 10 in the case of symmetric cylinders. The goal of the present paper is to extend these results to the case of a two-layer fluid, which was studied numerically in Linton and Cadby, 22 where a two-layer fluid of infinite depth with a submerged circular cylinder was considered, and in Saha and Bora 23 where the case of finite depth was studied.…”

“…In Garibay and Zhevandrov, the problem of waves trapped by a submerged thin cylinder with fairly arbitrary cross‐section in a one‐layer fluid without any symmetry conditions was considered. This resulted in finding exact solutions in the form of series in powers of ε and , where ε characterises the “thinness” of the cylinder, by means of a technique similar to previous studies . The leading term of the series coincided, of course, with the result of McIver in the case of symmetric cylinders.…”

“…Here, n is the outer normal to Ω 1 , dl is the arc element of the boundary at the point ( , ), G(x, ) = − 1 2 K 0 (k, r), r = √ x 2 + 2 . In (16), the normal derivative of 1 on Γ F is equal to by (4), is equal to zero on Γ by (3), and is equal to − on Γ I by (6); the values of 1 on Γ F , Γ, and Γ I are , , and (15), respectively, and 1,2 are completely defined by (14), (16) as stated.…”

“…The procedure of obtaining the system for , and is quite standard (one has to calculate the limit for (16) as (x, y) tends to a point of the boundary, cf Romero Rodríguez and Zhevandrov 13 and Garibay and Zhevandrov 15 ). Using the formulas for the Fourier transform of the function K 0 and its derivative, one finally comes to the following system for̃, and (we omit these tedious calculations):…”