Wigner distribution of a transducer beam pattern within a multiple scattering formalism for heterogeneous solids

Abstract

Diffuse ultrasonic backscatter measurements have been especially useful for extracting microstructural information and for detecting flaws in materials. Accurate interpretation of experimental data requires robust scattering models. Quantitative ultrasonic scattering models include components of transducer beam patterns as well as microstructural scattering information. Here, the Wigner distribution is used in conjunction with the stochastic wave equation to model this scattering problem. The Wigner distribution represents a distribution in space and time of spectral energy density as a function of wave vector and frequency. The scattered response is derived within the context of the Wigner distribution of the beam pattern of a Gaussian transducer. The source and receiver distributions are included in the analysis in a rigorous fashion. The resulting scattered response is then simplified in the single-scattering limit typical of many diffuse backscatter experiments. Such experiments, usually done using a modified pulse-echo technique, utilize the variance of the signals in space as the primary measure of microstructure. The derivation presented forms a rigorous foundation for the multiple scattering process associated with ultrasonic experiments in heterogeneous media. These results are anticipated to be relevant to ultrasonic nondestructive evaluation of polycrystalline and other heterogeneous solids.

Voigt, Reuss, Hill, and self-consistent techniques for modeling ultrasonic scattering

“…If φ i (t) is the i th collected waveform, then Φ (t) = φ 2 RMS (t) = ∑ N i=1 φ 2 i (t) /N gives a squared measure of the scattered energy [16]. The quantity φ RMS was first theoretically modeled by Rose [17] and later defined by Ghoshal and Turner using Wigner transforms of the incident and scattered fields [1]. Normal incidence backscatter results in an incident wave being scattered at a scattering angle θ = π.…”

“…Normal incidence backscatter results in an incident wave being scattered at a scattering angle θ = π. For this configuration, the singly-scattered response (SSR) is given by [1] [1]. Figure 3 shows the SSR curves for the Voigt, Reuss, Hill, and two self-consistent definitions of Ξ 3333 3333 , C 0 i jkl and v L .…”

“…Scattering measurements have been correlated with grain size, grain elongation, stress, and texture [1,2,3,4,5,6]. The accuracy of the inversion depends on how well the theoretical scattering model represents the propagating wave and the microstructure.…”

Stress-dependent ultrasonic scattering in polycrystalline materials

“…Experimentally measuring stresses from the scattered response is the primary future direction of this work. This scattered response is obtained by analyzing the average incoherent scattering present in a collection of ultrasonic waveforms using either an ultrasonic pulse/echo 23,24 or pitch/ catch 25 transducer configuration. In the absence of defects or other inhomogeneities, the scattered response is dominated by scattering from grain boundaries.…”

“…There is also strong agreement between theoretical backscatter models and experimental scattering measurements. [23][24][25] Thus, no decoupling of competing effects 18 is necessary when correlating material stresses to ultrasonic scattering.…”