Wave focusing by submerged islands and gravitational analogues

Abstract

We study water waves propagating over a smooth obstacle in a fluid of varying depth, motivated by the observation that submerged islands in the ocean act as effective lenses that increase the amplitude and destructive power of tsunami waves near focal points. We show that islands of substantial height (compared to the water depth) lead to strong focusing in their immediate vicinity, and generate caustics of either cusp or butterfly type. We highlight similarities and differences with focusing of (high-frequency) gravitational waves by a neutron star. In the linear regime, the comparison is made precise through an effective-spacetime description of the island-fluid system. This description is then put to practical use: We identify caustics by solving the Raychaudhuri equation (a transport equation) along rays of the effective metric. Next, the island-fluid scattering processes are examined in detail (i.e., deflection angle, phase shifts, scattering amplitudes) using numerical simulations and analytical techniques, including the eikonal approximation and its generalisation in the form of the Gaussian beam approximation. We show that the techniques capture the key features of the simulations. Finally, we extend the eikonal approximation to the dispersive regime, demonstrating that the essential features are robust in dispersive settings. This paves the way for future exploration in a controlled laboratory set-up.