The acoustical impulse response of a line source of finite length above an absorbing plane

It was shown in previous publications that the impulse response of an infinite line source, as well as a point source in the three-dimensional space above an absorbing plane, can be expressed in closed form. However, line sources of a finite length are much more practical. In this study, the impulse response of a finite line source above the absorbing ground is derived and presented in a closed form, which is the main result of the present work. The corresponding analytical solution is composed of three distinct signals. Those are the direct signal, the mirrored signal, and the "pressure tail." The direct signal is the line-of-sight signal. The mirrored signal is the signal emitted from the image source. It is weighted with the reflection coefficient of the infinite line source, which is a surprising mathematical result. The direct signal and the mirrored signal are both of finite duration. However, the third signal-the pressure tail-starts at the end of the mirrored signal and continues indefinitely for a finite surface impedance. In general, this last signal is relatively weak; it can be considered as the reverberation of the source caused by the infinite plane.