Integrals with a large parameter: water waves on finite depth due to an impulse

Abstract

The classical Cauchy–Poisson problem, of water waves generated by an impulsive disturbance on the free surface, is treated in three dimensions for finite constant depth. The conventional solution is in the form of a Fourier-Bessel transform. We wish to find its asymptotic behaviour at large distances r and large times t . Difficulties arise at the wavefront, where r/t is equal to the maximum group velocity. In the analogous two-dimensional problem the waves near the front are associated with two nearly coincident points of stationary phase and described asymptotically by Airy functions. In the three-dimensional problem the solution is expressed as a double integral, where there are four nearly coincident points of stationary phase. A systematic procedure is given for successive terms in an asymptotic expansion, involving the square of an Airy function and its derivatives. In both two and three dimensions it is important to use an appropriate transform of the variable(s) of integration, to achieve uniform validity of the asymptotic expansion. Calculations are performed to illustrate the utility of the asymptotic results.