Thin rubber layers with spherical cavities can be used to redistribute normally incident sonar energy in the transverse direction, where it suffers loss by anelastic absorption. This is the basis for an old idea for anechoic submarine coatings. In this report, the mechanism for the anechoic effect is studied theoretically and numerically. Observing that the reflectivity is an analytic function of the shear wave velocity of the rubber material, winding- number methods areapplied to prove the existence of, and also to design, coatings with vanishing reflectivity at isolated frequencies. The spatial distribution of the absorption losses is determined, with the major part suffered in the vicinity of the cavities for compressional spherically symmetric waves. The viscoelastic shear properties of the rubber matrix material are crucial for generating this loss. The requirements for anechoism are specified using plane-wave concepts from the invariant embedding technique. A classical monopole resonance for a spherical cavity in a solid is fundamental in order to fulfil these requirements. An energy relation is derived that relates the anelastic loss in the rubber layer to loss by scattering from a single cavity. A factor is isolated that quantifies the effects of multiple scattering, which are noticeable.