This paper presents the theory of the spherical, compliant‐tube Luneburg lens. Since a mixture of compliant tubes and water is dispersive, the compliant‐tube lens is a Luneburg lens only at a single frequency. The wave equation for the pressure field is solved for plane wave incidence. This solution corresponds to measurements made with an omnidirectional hydrophone on the lens. The wave theory is further developed to include solutions corresponding to measurements made by a dipole and cardioid hydrophone. The intensity, which is not proportional to the pressure squared, is also calculated. Numerical evaluation of these solutions is performed for a 10‐ft‐diam lens over the frequency range 500–5000 Hz (D/λ=1 toD/λ=10). This particular lens was chosen for study since it corresponds to a lens built by the Autonetics Division of Rockwell International. This lens is a Luneburg lens at 5000 Hz and is not perfect focusing for frequencies below 5000 Hz. The on‐axis gain, beam patterns, and directivity indices are calculated for the three sensors. In general, the on‐axis gain and directivity index are different. A comparison is made between measured results and theoretical results. Finally, the compliant‐tube lens is compared with a Luneburg lens (at all frequencies) and a liquid lens.