Calibrating the methods of stationary phase and steepest descent

Abstract

Analytic modeling of radiation from sources or scattering by obstacles near submerged layered environments generally leads to Fourier-type spectral integrals with integrands that contain pole and branch point singularities. In the high-frequency range, there are spectral intervals in which the integrands are highly oscillatory, thereby making their numerical evaluation problematic, especially in the vicinity of singularities that lie on the Fourier path in the absence of dissipation. Various methods of performing the numerics are examined, including deformation of the real-axis integration path into a contour in the complex plane, on which the integrand has weak or no oscillations and is exponentially damped away from a local maximum. By comparing the corresponding steepest descent and stationary phase analytic approximations with the numerics, one may calibrate these algorithms for accuracy, especially when saddle (stationary) points and singularities are in close proximity. The physical wave phenomena associated with these alternative analytic models are discussed. Attention is given to the separation of the integrand into weakly and highly oscillatory portions, as required in the analytic approximations.