Standing waves, localised near the shoreline of a water basin, and asymptotic quasimodes

In this work, formal asymptotic solutions of a problem for linear water waves in a bounded basin are constructed. The solutions have the form of asymptotic quasimodes and are used for the description of standing water waves localised near the shoreline. Such short-wavelength quasimodes exist only for a discrete set of frequencies, which are determined by means of a quantisation-type condition. Some numerical results are also addressed.