Wave Theory of an Acoustic Luneberg Lens

Abstract

The wave theory of an underwater, spherical Luneberg lens is discussed in terms of a generalized wave equation, which differs from the ordinary wave equation in that it contains a term involving the gradient of the density of the lens. The special case where the density of the lens has a constant value (to a first approximation) is studied in detail. This case is of some importance since it corresponds to the compliant-tubing lens built and tested by W. J. Toulis [J. Acoust. Soc. Am. 35, 286–292 (1963)]. For this case, we determine the analytic solution of the wave equation when the lens is being irradiated by plane waves and then present the results of a numerical evaluation of that solution. In particular, we calculate the acoustic pressure field along the axis of the lens and the pressure receiving patterns. These results give us the gain and beamwidths for the range 1⩽k0a⩽30 (k0 being the wavenumber in water, a the radius of the lens). Finally, the theory is compared with Toulis's experimental results, and the agreement is found to be quite good.