High-frequency asymptotics of waves trapped by underwater ridges and submerged cylinders

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, G 0 (x, ) = 1 2 ln r is the fundamental solution of the Laplace operator and the normal derivative is taken at the point ( , ). Thus, in the leading term, the left-hand side of (19) coincides with the integral operator arising from the exterior Neumann problem…”

“…Let us analyse the structure of the integral operators entering (17) to (19). First, due to the convergent expansions for the derivative K ′ 0 (r) = −K 1 (r), we have by formula 9.6.11 from Abramowitz and Stegun 33…”

“…Thus, solving (19) for and substituting in (17), (18), we finally come to the system of two equations for and̃:…”