Wave propagation on the surface of a material contains information about physical properties beneath its surface. We propose a method for inferring the thickness and stiffness of a structure from just a video of waves on its surface. Our method works by extracting a dispersion relation from the video and then solving a physics-based optimization problem to find the best-fitting thickness and stiffness parameters. We validate our method on both simulated and real data, in both cases showing strong agreement with ground-truth measurements. Our technique provides a proof-of-concept for at-home health monitoring of medically-informative tissue properties, and it is further applicable to fields such as human-computer interaction.