It was shown in Part I [J. Acoust. Soc. Amer.43, 709–715 (1968)] that the wave equation for an acoustic Luneburg lens is∇2p − (1/ρ)∇ρ⋅∇p = (nL2/c02)∂2p/∂f2, wherepis the pressure, ρ the density,c0the constant speed of sound in water andnL2 = 2−r2/a2Luneburg's index of refraction. Hereris the radial distance measured from the center of the lens andais the radius of the lens. Also, we have in general thatnL2(r) = c02ρ(r)κ(r), where κ(r) is the compressibility. In Part I, the solution of the wave equation was obtained for plane‐wave irradiation when ρ is constant. In this paper, we first obtain the solution to the wave equation for plane‐wave irradiation whenρ(r) = nL2(r)/c02κ, κ being constant. Then the general case is examined when ρ(r) is an arbitrary function of position. It is found that at low frequencies, the speed of sound is affected by the presence of ∇ρ. while at high frequencies, this effect is negligible. Consequently, a completely arbitrary choice of ρ(r), even though κ(r) is chosen so thatρ(r)κ(r) − nL2/c02, could destroy the perfect focusing property (no spherical aberration) of the lens at low frequency. Therefore, conditions are established for ρ(r) so that the lens will retain its perfect focusing property at all frequencies.