Theory of Focusing Radiators

An approximate theory has been derived describing part of the sound field due to a concave spherical radiator, vibrating with uniform normal velocity; the radius a of the circular boundary is assumed to be large relative to the wave-length and large relative to the depth of the concave surface. The theory describes the distribution of pressure, particle velocity, and intensity along the axis of symmetry and in the vicinity of the focal plane, perpendicular to the axis at the center of curvature. It is shown that the ratio of the intensity at the center of curvature to the average intensity at the radiating surface is nearly equal to (2πh/λ)2 where h is the depth of the concave surface and λ is the wave-length. This ratio can be made very large by suitable choice of dimensions, and the focusing is then very sharp. The point of greatest intensity is not at the center of curvature but approaches it with increasing kh = 2πh/λ, and the greatest intensity is not much greater than the intensity at the center of curvature except when kh is small. In the central part of the focal plane, at angle θ from the axis, the pressure is approximately proportional to (2/ka sinθ)J1(ka sinθ), which is equivalent to the directivity function of a flat circular piston of radius a, for the region at large distance from the piston. The calculations are in reasonable agreement with G. W. Willard's experimental data for a 5-mc concave quartz crystal, when allowance is made for the non-uniform normal velocity of the crystal.