Numerous phenomena in the fields of physics and mathematics as seemingly different as seismology, ultrasonics, crystallography, photonics, relativistic quantum mechanics, and analytical number theory are described by integrals with oscillating integrands that contain three coalescing criticalities, a branch point, stationary phase point, and pole as well as accumulation points at which the speed of integrand oscillation is infinite. Evaluating such integrals is a challenge addressed in this paper. A fast and efficient numerical scheme based on the regularized composite Simpson's rule is proposed, and its efficacy is demonstrated by revisiting the scattering of an elastic plane wave by a stress-free half-plane crack embedded in an isotropic and homogeneous solid. In this canonical problem, the head wave, edge diffracted wave, and reflected (or compensating) wave each can be viewed as a respective contribution of a branch point, stationary phase point, and pole. The proposed scheme allows for a description of the non-classical diffraction effects near the “critical” rays (rays that separate regions irradiated by the head waves from their respective shadow zones). The effects include the spikes present in diffraction coefficients at the critical angles in the far field as well as related interference ripples in the near field.

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