Ultrasonic field estimation for random P -and S-wavenumbers in isotropic solids using DPSM

Abstract

The measurement of elastic constants often shows randomness, which consequently affects the propagation of P- and S-waves in a solid material. In the theory of wave propagation, longitudinal and transverse waves are characterised using P-and-S wavenumbers ($k_p$ and $k_s$), which can be modelled as a random variable to simulate random ultrasonic fields. In this research work, an efficient and accurate solution technique to model random wave propagation due to random wavenumbers using Distributed point source method (DPSM) is developed. DPSM is a semi-analytical method that requires Green’s function (GF) solution for producing ultrasonic fields in homogeneous or heterogeneous solids and near the fluid-solid interface. The generation of ultrasonic fields at higher frequencies and in complex structures requires a large number of distributed point sources, thereby leading to the computation of a greater number of GFs. Therefore, the numerical calculation of random wavefields increases the computational complexity. An analytical model to approximate first-order and second-order moments of the displacement GF solutions for isotropic solid corresponding to randomly distributed P- and S-wavenumbers is proposed. The statistical moments (mean and variance) of ultrasonic fields (stresses and displacement fields) near the fluid-solid interface and in the solid half-space are calculated using the proposed theoretical model. The efficacy and accuracy of the proposed model for normally distributed wavenumbers are illustrated through two numerical analyses. Initially, the mean ultrasonic fields such as displacement and stress fields are evaluated using both the proposed analytical model and Monte Carlo (MC) simulation for 1 MHz excitation frequency. Mean and standard deviations of the total scattered wavefields are computed near the interface and along the solid medium. Further, to check the robustness of the proposed analytical model ultrasonic fields for 2.25 MHz excitation frequency are computed for the same problem configuration and the mean fields are compared with the MC simulation. The computed mean displacement GF solutions and ultrasonic fields using the proposed model for normally distributed wavenumbers match precisely with the MC simulation. Further, the standard deviation of the ultrasonic fields for normally distributed wavenumbers is estimated for different transducer frequencies.