A method of steepest‐descents integration suitable for use when a pole may approach near to one of the saddle points in the complex plane of integration is applied to the problem of predicting the field resulting from a point source near to the interface between two semi‐infinite media. It has been shown that it is possible to obtain asymptotic approximations for the field above or below the interface to any desired degree of accuracy in a way that is more straightforward and less algebraically cumbersome than that which is employed when using the subtraction of the pole technique. Approximation for the case where the receiving medium has a high refractive index produces the expected result whereby the field below the interface is related to that incident on the interface according to Snell’s law. Measurements are made of the phase and amplitude of the sound field below the surface of a 0.5‐m‐thick layer of glass fiber material, relative to those of the field at the surface due to the point source close to the surface. Experiments are reported using continuous, swept‐tone burst and spark‐discharge sources. The measured data are compared with theoretical predictions. The predictions require knowledge of the complex propagation constant within the material. Use of an empirical formula together with measured flow resistivity is found to produce overestimates of attenuation, whereas use of a theory for propagation in a rigid porous medium based upon a classical capillary‐pore approach is found to enable tolerable agreement with measured data. This confirms directly the locally reacting nature of the boundary over the frequency range of the measurement.