A paraxial beam solution for ultrasonic guided waves in anisotropic elastic plates

Abstract

A paraxial approximation of angular spectrum representation is proposed for the construction of ultrasonic beam solutions for guided waves in anisotropic elastic plates. The angular spectrum representation used in this work is the exact integral form describing the superposition of plane guided waves with respect to the angular direction wavenumber. The approximate integral form is derived from the angular spectrum representation using the small angle approximation. Employing an inclined in-plane coordinate system shows that the approximate integral form describes the beam solution propagating in a direction inclined relative to the wavenumber direction by a skew angle. This approach demonstrates that the skew angle can be expressed as the derivative of the wavenumber with respect to the angle. Because the amplitude function for the approximate integral form is governed by the two-dimensional Helmholtz equation, the proposed formulation can accommodate the conventional beam modes developed in the field of optics. On this basis, a paraxial Gaussian beam solution is derived for anisotropic elastic plates in an explicit form and verified based on comparison with the numerical solution for the angular spectrum representation. The present work confirms that diffraction effects on the ultrasonic beam are greatly modified by the skew angle profile.