Estimation of accuracy of the asymptotic solution near the shore

In order to obtain some rough estimates of the error involved in taking the asymptotic solution for the free surface, the various results obtained by Peters have been used and the sea bed is taken as having a constant slope. An analysis has been carried out to ascertain whether or not the solutions sinusoidally varying with respect to the distance from the shore can be taken as the actual solution with a certain degree of accuracy. Though the present analysis is concerned with the solutions at the free surface, it is possible to obtain asymptotic solutions sinusoidal with respect to the variable x, the distance from the shore, valid throughout the region occupied by the fluid when depth is variable.