Comparison of three transient solutions for the axial pressure in a focused sound beam

A comparison is made of three simple transient solutions for the axial pressure in a focused sound beam. Each solution is derived on the basis of different approximations. The acoustic source is assumed to be an axisymmetric spherical cap, the vibration of which generates a uniform normal velocity component on its surface. In case 1, the Rayleigh integral (which is strictly valid only for planar sources) is applied to the velocity condition on the curved surface. In case 2, the velocity condition on the curved surface is replaced by a pressure distribution in a plane, and the Rayleigh integral is applied to the approximate planar source condition. In case 3, the parabolic approximation is applied to both the Rayleigh integral and the source condition. The solutions for cases 1 and 3 were derived previously by others, whereas the transient solution for case 2 is a new result that generalizes an existing solution for monofrequency sources. For small aperture angles, the three approximations yield similar results in the focal region and beyond, with significant differences arising only near the source. Calculations are performed for tone bursts and N waves.