A semi-numerical model for near-critical angle scattering

Area of Science:

• Physics
• Mathematics
• Wave Scattering
• Numerical Analysis

Background:

• Many physical and mathematical phenomena involve integrals with oscillating integrands and multiple critical points.
• Evaluating these complex integrals, especially those with coalescing criticalities, poses a significant computational challenge.

Purpose of the Study:

• To develop a fast and efficient numerical scheme for evaluating integrals with oscillating integrands featuring coalescing criticalities.
• To apply and validate the proposed scheme in the context of wave scattering problems.

Main Methods:

• A novel numerical scheme based on the regularized composite Simpson's rule is proposed.
• The method is demonstrated by analyzing the scattering of an elastic plane wave by a stress-free half-plane crack.

Main Results:

• The proposed scheme efficiently evaluates challenging integrals with branch points, stationary phase points, and poles.
• It accurately describes non-classical diffraction effects near critical rays, including far-field spikes and near-field interference ripples.

Conclusions:

• The developed numerical scheme provides an effective solution for evaluating complex oscillatory integrals.
• This advancement aids in understanding intricate wave phenomena and diffraction effects in various scientific fields.