Numerical calculations with an exact analytical time-domain Green’s function for the van Wijngaarden wave equation

Abstract

An exact analytical time-domain Green’s function is obtained for the van Wijngaarden wave equation when two of the constant coefficients satisfy a certain equality. This analytical expression enables numerical assessments demonstrating that lossy time-domain Green’s function calculations with the inverse fast Fourier transform (IFFT) produce larger errors than expected for most temporal values, even when the IFFT is evaluated with several million points. This analytical time-domain Green’s function also provides the foundation for efficient numerical calculations of spatial impulse responses, which are demonstrated for a 1 cm radius circular piston radiating in water at axial distances of 1 cm, 10 cm, and 1 m. As the axial distance increases, the difference between the lossy and lossless spatial impulse responses also increases, which is particularly evident in the paraxial region. However, the difference between lossy and lossless spatial impulse responses diminishes as the observation point transitions from an on-axis location to another position with the same axial distance just outside of the paraxial region. The observed differences between the lossy and lossless spatial impulse responses transitioning out of the paraxial region are explained by comparing the relative temporal extents of the lossless spatial impulse response and the lossy time-domain Green’s function.