Exact solution for the acoustical impulse response of a line source above an absorbing plane

The solution of the two-dimensional (2D) wave equation excited by a Dirac delta function above an infinite plane is the 2D impulse response of the half-space. Equivalently, it can be considered as the sound field generated by an infinite line source in three-dimensional (3D) space. For a point source in 3D, it is possible to express the resulting impulse function in closed form if the infinite plane is characterized by a masslike or pure absorbing behavior. Based on the 3D impulse response above an absorbing plane, a closed form solution for a line source over absorbing ground is found. This solution consists of the direct pressure contribution plus a reflected signal. The analytical expression for the reflection factor R is the main result of the work. R is a simple algebraic function of impedance, time, and spatial coordinates. The acoustical field generated by a line source with a general time dependence can be constructed by convolution. For a Gaussian signal, the resulting pressure is computed numerically. For a rectangular pulse, an analytical solution in 3D and 2D is derived. The deviations from the numerically obtained results are very small.