Long-period waves pose a threat to coastal communities as they propagate from deep ocean to shallow coastal waters. At the coastline, such waves have a greater height and longer period in comparison with local storm waves, and can cause severe inundation and damage. In this study,we considered linear long waves in a two-dimensional（vertical-horizontal） domain propagating towards a shoreline over a shallowing shelf.New solutions to the linear shallow water equations were found, through the separation of variables, for two forms of transition shelf morphology: deep water and shallow coastal water horizontal shelves connected by linear and parabolic transition, respectively. Expressions for the transmission and reflection coefficients are presented for each case. The analytical solutions were used to test the results from a novel computational scheme, which was then applied to extending the existing results relating to the reflected and transmitted components of an incident wave. The solutions and computational package provide new tools for coastal managers to formulate improved defence and riskmitigation strategies.
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