Waveguides in a hydromechanics problem (Q2277081)

In this article we study, in linear approximation, the impact of a submerged elastic ridge on the propagation of surface waves in an ideal fluid generated by a perturbation of the floor and the free fluid surface. \textit{R. M. Garipov} studied this problem [Dokl. Akad. Nauk SSSR 161, 547-550 (1965; Zbl 0151.425)] without taking into account the elasticity of the floor. It is known that if the floor is a horizontal surface then the amplitude of surface waves diminishes proportionally to \(R^{-1}\), where R is the distance from the point of initial perturbation. Garipov's basic result states that a submerged rigid ridge generates a series of waves which correspond to a nonempty class of initial data and propagate along the ridge with an amplitude that diminishes as \(R^{-\alpha}\), \(\alpha =1/2,1/3,1/4\), depending on the form of the ridge profile. We will study the structure of the spectrum of waves propagating along the ridge. We prove that if the height of the submerged ridge is sufficiently large in comparison to the depth of the fluid over flat sections of the floor then the ridge acts as a waveguide. Then we use these results to prove the existence of waves which propagate along the ridge and diminish quicker than in the case of a rigid floor. The waveguidelike wave propagation in this system, caused by the elasticity and irregularity of the floor, allows us to prove this fact.