The propagation of LF and VLF terrestrial radio waves is simulated on the computer with a precise theory of spherical wave functions of integer order. Anisotropic reflections from an electron and ion density profile of the ionosphere are accommodated by the analysis. Variations in the profile and the magnetic field with distance along the propagation path can be introduced ad hoc. The analysis method is incorporated into a computer program which is written in a flexible manner so that it can be applied to a variety of scientific studies of low-frequency ionospheric wave propagation. Calculations of current interest are presented to illustrate use of the technique. Amplitude perturbations of approximately 15 decibels are described at 60 kHz as a result of an electron density profile perturbation. It is also found that the 26 kHz signal is comparatively insensitive to a variation in magnetic dip angle when propagation occurs in the daytime near the magnetic equator. Thus, using values of the magnetic field vector at the center of the propagation path is a good approximation in certain circumstances.