Water waves generated by an oscillatory surface pressure travelling at critical speed

Abstract

A certain perturbation is defined for disturbances caused by a pressure distribution oscillating and travelling on the surface of infinitely deep water without surface tension. Its transverse gradient is then considered along the water surface for a source speed and frequency that combine at a critical level. Boundedness tests become crucial because of this criticality. Quantitative asymptotic results are secured at large distances from a finite source domain. Separate travelling wave solutions hold within two downstream wedges and one upstream-extended wedge. Construction of a typical wavevector is demonstrated. Every wavevector is spatially dependent only on the polar angle. Along that pair of upstream-inclined wedge boundaries, the associted wavefuncion is halved in value and assumes a simplified form. Another wavefunction undergoes the same experience along one pair of downstream inclined wedge boundaries. However, at interior points near the remaining pair, two other wavefunctions combine into another form. This then simplifies along either boundary to a new function which dominates due to its lower attenuation rate. The group velocity vector always points in the outward radial direction. Explicit representations for it indicate a spatial dependence on the radial direction alone. Finally, long range surface approximations of the original perturbation are directly deduced from those of its transverse gradient via substitutions of Fourier transformed source factors. They hold for a class of applied pressures which is somewhat restricted by an integral criterion on the pressure function.