The diffraction of transient waves by a curvilinear boundary using the theory of generalized rays is developed. The generalized ray integrals, which represent the Fourier transform of diffracted waves, involve a double integration with respect to two wave slownesses. The inverse Fourier transform of these double integrals are computed by applying the simultaneous transformation of variables and the Cagniard method. The phase functions of the integrand provide the ray paths of incident, reflected, and diffracted waves. Since the pulses arrive at a point of observation in successive order, the theory furnishes an exact solution up to the time of arrival of the next ray and enables one to analyze, in detail, the signals recorded by the receiver.