An evaluation of the Kirchhoff approximation in predicting the axial impulse response of hard and soft disks

To test the ability of the Kirchhoff approximation for estimating the various components in the near-field impulse response of a circular disk, the predictions from a time domain formulation of the Helmholtz–Kirchhoff solution [Trorey, Geophys. 35, 762–864 (1970)] are benchmarked against results obtained via the Fourier synthesis of highly accurate frequency domain solutions [Kristensson and Waterman, J. Acoust. Soc. Am. 72, 1612–1625 (1982)]. In these numerical experiments, a collocated point source and receiver lie on the symmetry axis of an acoustically hard (rigid) or soft (pressure release) disk. A time-domain analysis is carried out in order to unambiguously evaluate the Kirchhoff approximation for different components of the scattered field. It is found that, while Helmholtz–Kirchhoff predicts the correct reflected component, it fails to accurately predict the strength of the diffracted component. The magnitude of the error depends on whether the disk is soft or hard and on the source/receiver height above the disk. The error in the diffracted component exceeds, in some cases, 100%. Furthermore, it is observed that the Helmholtz–Kirchhoff approach does not include the secondary or multiply diffracted arrivals which are more pronounced for the hard disk.