The evaluation of ARTEMIS ambiguity functions is critical to waveform choice, the proper selection of reference signals, and the ability to estimate system performance. In the case of narrowband systems the evaluation is facilitated by neglecting Doppler-induced distortions of the transmitted modulation function. However, ARTEMIS-type sonar systems require the utilization of signals that are broadband with respect to the carrier frequency and long in duration, so that the narrowband approximation cannot be employed. This creates difficulties in both analytic and computer ambiguity function evaluations. Analytic difficulties arise from the inability to solve the appropriate integrals computer difficulties, based on direct crosscorrelation, arise from the long computation time required. This report shows how evaluation of the ambiguity functions of ARTEMIS-type FM sonar signals having high time-bandwidth products can be greatly facilitated by asymptotic approximations derived on the basis of the principle of stationary phase. Application of the theory allows many seemingly difficult ambiguity evaluation problems involving highly complicated integrals to be rapidly solved using algebraic techniques.