Long waves such as tsunamis can be trapped by islands due to wave refraction,and these trapped waves will cause huge damage even in the sheltered shoreline of the island.That all waves propagating into the topography and finally reaching the coastline are called perfect trapped modes,while any waves escaping from the topography are called leaky modes.Whether these long waves can be trapped is dependent on the depth profile of the island.This paper presents analytic solutions of the ray path for waves propagating into the circular island with power function profiles.Wave height distributions over the island are further investigated based on the principia that crowded rays correspond to large wave height and sparse rays correspond to small wave height.The trapped mechanism for water waves over the island is revealed based on their ray paths.Furthermore,the perfectly trapped criterion is derived,that is,when the slope gradient at the topography toe is greater than twice the ratio of the water depth to the radial distances,all wave rays propagating on the island will finally reach the coastline,and the waves are perfectly trapped.
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