Feynman's inverse problem

We analyse an inverse problem for water waves posed by Richard Feynman in the BBC documentary Fun to Imagine. The problem can be modelled as an inverse Cauchy problem for gravity-capillary waves on a bounded domain. We do a detailed analysis of the Cauchy problem and give a uniqueness proof for the inverse problem. This results, somewhat surprisingly, in a positive answer to Feynman's question. In addition, we derive stability estimates for the inverse problem both for continuous and discrete measurements, propose a simple inversion method and conduct numerical experiments to verify our results.