Acoustic scattering problems are considered when the material parameters (density and speed of sound) are functions of position within a bounded region. An integro-differential equation for the pressure in this region is obtained. It is proved that solving this equation is equivalent to solving the scattering problem. Problems of this kind are often solved by regarding the effects of the inhomogeneity as an unknown source term driving a Helmholtz equation, leading to an equation of Lippmann--Schwinger type. It is shown that this approach is incomplete when the density is discontinuous. Analogous scattering problems for elastic waves and for electromagnetic waves are also discussed briefly.