Acoustic and elastic wave propagation is considered in continuously stratified layers. Upon Fourier transformation, the analysis is developed for time-harmonic waves. Acoustic waves are eventually taken to be described by a one-dimensional Helmholtz equation which is shown to be equivalent to a Volterra integral equation. This allows a natural application of the method of successive approximations to initial-value problems. A fundamental system of solutions so obtained is then applied to the investigation of the reflection–transmission process produced by an obliquely incident plane wave impinging on a layer which suffers from jump discontinuities of the material parameters. The reflection and transmission coefficients are determined. For definiteness, the low-frequency limits and the particular case of homogeneous layers are examined in detail. The whole procedure, with an appropriate setting, is then applied to the equation for elastic waves.