Let (Ω,d) be a complete metric space and let µ be a Borel measure on Ω. Under certain fairly general assumptions about the metric and the measure, we use lattice theory to construct an isometric copy (Ω^e,d^e) of the space (Ω,d), which is called its wave model. The construction is motivated by applications to inverse problems of mathematical physics. We show how the wave model solves the problem of reconstructing a Riemannian manifold with boundary from its spectral data.