Rapid computation of waves diffracted by an absorbing plane with precise error bounds

Abstract

The current study is motivated by a recent paper [L. N. Trefethen and J. A. C. Weideman, “The exponentially convergent trapezoidal rule,” SIAM Rev., 36, 385–458 (2014)]. It is well known that waves diffracted by an absorbing plane due to a point or line source can be expressed in Fourier integrals. At low frequencies and short ranges, numerical solutions may be obtained by evaluating the integral, for example, by the fast field formulation (FFP). However, it is found prohibitive expensive to use FFP at high frequencies and long ranges. On the other hand, the solutions can be efficiently evaluated along the steepest descent path at all situations. A common approach involves expansion of the kernel function about the saddle point and application of the pole subtraction method. By retaining the first term of the Maclaurin series, this method gives a fairly accurate analytic solution. This paper explores the trapezoidal rule for evaluating the Fourier integral along the steepest descent path with a prescribed...