Transient solutions for the angular dependence of acoustical transmitters and receivers employing paraboloidal...

Time-domain solutions are presented for the angular dependence of waveforms in the far field of a point source at the focus of a rigid paraboloidal reflector, and also for waveforms at the focus as a function of the direction of a plane wave incident on the reflector. The main restriction is that the wavelength is small in relation to both the radius of the aperture and the minimum radius of curvature of the reflector, conditions which are satisfied for reflectors with appreciable gain. The solution in the far field due to a point source at the focus is related by the principle of reciprocity to the solution at the focus due to an incident plane wave. Both solutions are expressed as the convolution of an explicit expression for the unit step response with the time derivative of the pressure waveform incident on the reflector. Results are presented illustrating the angular dependence of the reflected pressure waveforms at the focus due to incident N waves and tone bursts.