SUMMARY A trajectory-based solution of Maxwell's equations is derived using the method of multiple scales. This time-domain technique utilizes an asymptotic expansion in terms of the ratio of the wave front length-scale to the length-scale of the heterogeneity. The approach provides an alternative to an expansion in terms of frequency and allows one to model electromagnetic wave propagation over a wide frequency band. At the lower end of the frequency band, the trajectory-based solution reduces to a previously derived diffusive solution. Similarly, at higher frequencies one obtains the ‘delta-like’ solution associated with hyperbolic wave propagation. However, the solution is also valid at intermediate frequencies which cannot be characterized as either diffusive or hyperbolic. A numerical illustration demonstrates the importance of both conduction and displacement currents at frequencies between 10 and 100 MHz. The amplitudes computed using the trajectory-based approach compare well with analytic results for a homogeneous whole-space. Using the technique I am able to model observations from a broad-band (3–300 kHz) experiment at the Richmond Field Station in California. In addition, ground penetrating radar waveforms in the 5–200 MHz range, gathered at the Boise Hydrogeophysical Research Site, are matched using the results of a radar velocity tomogram.