Abstract Approximate procedures for solving the first-passage time problem for a linear oscillator subjected to random excitation with a time-varying mean are presented. Numerical results are given for the example of the oscillator subjected to a model representing sonic boom excitation. These results are compared with results obtained from digital simulation. The methods are applicable for lightly damped (narrow-band) oscillators. To the author's knowledge, there is no previous work on oscillator first-passage time approximations where the excitation is assumed to have a time-varying mean.