Theory of Focusing Radiators

Abstract

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.

Basic Theory of Distributed Point Source Method (DPSM) and Its Application to Some Simple Problems

“…O'Neil (1949) argued that for transducers with small curvature the Rayleigh-Sommerfield integral representation (Eq. (1.16)) holds if the surface integral is carried out over the curved surface.…”

A Focussed Transducer/Scatterer Model for Ultrasonic Reference Standards

“…O'Neil [4] has shown that the Rayleigh-Sommerfeld formulation can also be used for spherically focussed probes provided that the probe is not too tightly focussed. In this case, the incident wavefield pressure, pillC, produced by the transducer is given by (Figure 1):…”

“…However, because of the double surface integrals present, this equation is not particularly suitable even for direct numerical evaluations. Fortunately, the area elements dS, dSs• can be written in the forms [4] …”

Transducer Design for High Sensitivity Ultrasonic Inspection of Titanium

“…Early work demonstrated an improved detectability of synthetic hard-alpha inclusions with low acoustic reflectivity in titanium by using high er frequency and smaller beam diameter transducers [1]. An expression which can be attributed to O'Niel [7] shows that the 0.50 amplitude contour (-6 dB) defining the beam diameter, E, is given by…”