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

As is well-known, underwater ridges and submerged horizontal cylinders can serve as waveguides for surface water waves. For large values of the wavenumber k in the direction the ridge, there is only one trapped wave (this was proved in Bonnet-Ben Dhia and Joly [Mathematical analysis of guided water waves, SIAM J. Appl. Math. 53 (1993) 1507–1550]. We construct asymptotics of these trapped waves and their frequencies as k → ∞ by means of reducing the initial problem to a pair of boundary integral equations and then by applying the method of Zhevandrov and Merzon [Asymptotics of eigenfunctions in shallow potential wells and related problems, Amer. Math. Soc. Trans. 208 (2) (2003) 235–284], in order to solve them.