Long wave propagation and run-up in converging bays

Analytical solutions are derived to describe two-dimensional wave evolution in converging bays. Three bay types of different cross-sections are studied: U-shaped, V-shaped and cusped bays. For these bays, the two-dimensional linear shallow water equations can be reduced to one-dimensional linear dispersive wave equations if the transverse flow acceleration inside them is assumed to be small. The derived solutions are characterized as the leading-order plane-wave solutions with higher-order corrections for two-dimensionality due to wave refraction. Wave amplitude longitudinally increases with different rates for the three bay types, whereas it exhibits weak parabolic variations in the transverse direction. Wave refraction significantly affects relatively short waves, contributing to wave energy transfer to the inner bay in a different manner depending on the bay type. The perturbation analysis of very high-order wave celerity suggests that the solutions are valid only when the ratio of the bay width to the wavelength is smaller than a certain limit that differs with bay type. Beyond the limit, the higher-order effect is no longer a minor correction, implying that wave behaviours become highly two-dimensional and possibly cause total reflection. The higher-order effect on the run-up height at the bay head is found to be small within the applicable range of the solution, and thus, the run-up formula neglecting the transverse flows has a wide validity. We also discuss the limitation of run-up height by wave breaking on the basis of a breaking criterion from previous studies.