The Uniform geometrical Theory of Diffraction for elastodynamics: Plane wave scattering from a half-plane

Abstract

Diffraction phenomena studied in electromagnetism, acoustics, and elastodynamics are often modeled using integrals, such as the well-known Sommerfeld integral. The far field asymptotic evaluation of such integrals obtained using the method of steepest descent leads to the classical Geometrical Theory of Diffraction (GTD). It is well known that the method of steepest descent is inapplicable when the integrand's stationary phase point coalesces with its pole, explaining why GTD fails in zones where edge diffracted waves interfere with incident or reflected waves. To overcome this drawback, the Uniform geometrical Theory of Diffraction (UTD) has been developed previously in electromagnetism, based on a ray theory, which is particularly easy to implement. In this paper, UTD is developed for the canonical elastodynamic problem of the scattering of a plane wave by a half-plane. UTD is then compared to another uniform extension of GTD, the Uniform Asymptotic Theory (UAT) of diffraction, based on a more cumbersome ray theory. A good agreement between the two methods is obtained in the far field.

Acoustic Scattering Models from Rough Surfaces: A Brief Review and Recent Advances

“…One of the main inconveniences of the Kirchhoff approximation (KA) is its incorrect quantitative prediction of edges diffraction, since the approximation of the surface field by the geometrical field is invalid near the edges. To handle this drawback, the physical theory of diffraction (PTD) has been proposed in electromagnetism [66], acoustics [59,67] and elastodynamics [68][69][70] and compared to other uniform scattering models [67,70,71].…”

2D elastic plane-wave diffraction by a stress-free wedge of arbitrary angle

“…11b where there are no reflected L waves). This is due to the fact that the SF diffraction coefficient is a GTD-like coefficient obtained from a far-field asymptotic evaluation (3.38) and it diverges at shadow boundaries [3,13]. The FE code notably computes the reflected waves as well as the diffracted waves in angular regions surrounding the reflected poles ; consequently, reflected waves give a contribution to the FE diffraction coefficient.…”

“…11a, 11b and 11d. Some uniform methods have been proposed to handle the divergence of GTD diffraction coefficients and build a spatially uniform total field and some of them have been applied to elastodynamic half-plane scattering [13,7,8]. On Figs.…”

“…It was developed in electromagnetics by Kouyoumjian and Pathak [11] using the Pauli-Clemmow [12] asymptotic approximation of integrals. Kamta Djakou et al [13] have extended it to elastodynamics with application to the scattering from a half-plane. This method is computationally efficient but still requires a trustworthy diffraction model in order to be applied.…”

The spectral functions method for elastic plane wave diffraction by a wedge

“…Nevertheless, this ultrasonic PTD model can be time consuming for large specimen surfaces. A second solution to this problem, called the Uniform Theory of Diffraction (UTD) has been proposed in elastodynamics by Kamta Djakou et al [7] and developed for a half-plane scatterer. To apply the aforementioned UTD method to the case of a wedge, a generic and trustworthy wedge diffraction model is necessary.…”