Acoustic shear waves of low frequency can be detected and measured using a phase contrast based magnetic resonance imaging technique called MR Elastography or phase measurement based ultrasound techniques. Spatio-temporal variations of displacements caused by the propagating waves can be used to estimate local values of the elasticity of the object being imaged. The currently employed technique for estimating the elasticity from the wave displacement maps, the local frequency estimator (LFE), has fundamental resolution limits and also has problems with shadowing and other refraction-related artifacts. These problems can be overcome with an inverse approach using Green’s function integrals which directly solve the wave equation problem for the propagating wave. The complete measurements of wave displacements as a function of space and time over the object of interest obtained by the above techniques permit an iterative approach to inversion of the wave equation to obtain elasticity and attenuation maps.