A thin rubber layer with cavities can be used to redistribute normally incident sound in the transverse direction, where it suffers losses by anelastic absorption. This is the basis for an old idea for anechoic submarine coatings. In this paper, modern techniques for electron scattering and band gap computations for photonic and phononic crystals are adapted to model the anechoic effect numerically. Reflection and transmission matrices are computed recursively, from basic ones for layers containing periodic arrays of cavities. The shear and viscoelastic properties of the rubber material turn out to be crucial, and multiple scattering among the cavities cannot be ignored. The reflectivity of a multilayered fluid-solid structure will depend in a nonlinear way on the geometry and the material parameters. Two different techniques are used to design coatings with favorable sound absorbing properties. The first one is based on analyticity of the reflectivity as a function of the complex shear wave velocity of the rubber material, and winding-number methods are applied. With appropriately implemented error control, the existence of thin coatings with zero reflectivity at a specified frequency is actually proved. To achieve low reflectivity within a broad frequency band, however, different techniques are needed, and genetic algorithms are applied.