Elsevier

NDT & E International

Volume 42, Issue 3, April 2009, Pages 210-214
NDT & E International

Time reversal study of ultrasonic waves for anisotropic solids using a Gaussian beam model

https://doi.org/10.1016/j.ndteint.2008.09.010Get rights and content

Abstract

Time reversal (TR) of ultrasonic bulk waves in fluids and isotropic solids has been used in many applications including ultrasonic NDE. However, the study of the TR method for anisotropic materials is not well established. In this paper, the full reconstruction of the input signal is investigated for anisotropic media using an analytical formulation, called a modular Gaussian beam (MGB) model. The time reversal operation of this model in the frequency domain is performed by taking the complex conjugate of the Gaussian amplitude and phase received at the TR mirror position. A narrowband reference signal having a particular frequency and number of cycles is then multiplied and the whole signal is inverse Fourier transformed to obtain the time domain signal. The original input signal is seen to be fully restored by the TR process of MGB model and this model can be more generalized to simulate the spatial and temporal focusing effects due to TR process in anisotropic materials.

Introduction

The origin of the time reversal (TR) concept traces back to time reversal acoustics [1], [2], [3]. In time reversal acoustics, an input bulk wave can be exactly reconstructed at the source location if a response signal measured at a distinct location is time-reversed and reemitted to the original excitation location. This phenomenon is referred to as TR of bulk waves and has been used in many applications including ultrasonic nondestructive evaluation and underwater acoustics.

While the TR method for bulk waves in fluids and isotropic solids has been well established [4], [5], the study of the TR method for anisotropic solids is relatively new. The spatial and temporal focusing effect due to the time reversal mirror (TRM) in anisotropic solid was first studied by Zhang et al. [6], [7] by using a ray method. The beam focusing effects will be different depending on the wave propagation direction due to the anisotropy dependence of the time reversal process of propagating waves. Thus, it is necessary to examine whether an original input signal is fully restored at the source location before the TR method is applied for anisotropic media.

In this paper, the full reconstruction of the input signal is attempted through the TR process of ultrasonic bulk waves in anisotropic solids. To achieve this goal, a modular Gaussian beam (MGB) model is employed to simulate the TR process of the longitudinal wave propagation in anisotropic solids. The MGB model provides an efficient formulation for ultrasound propagation, because its properties can be described in analytical matrix form even after propagation through general anisotropic media and after interactions with multiple curved interfaces. It is shown that complete reconstruction of the original input signal can be achieved by the TR process of MGB model.

Section snippets

MGB model for anisotropic solids

We describe a MGB approach for ultrasonic beam propagation shown in Fig. 1, where a single Gaussian beam is radiated from a circular source and travels in solid media composed of two anisotropic solids and an interface. We assume the beam propagation along symmetry directions of anisotropic solids and a normal interface with respect to the beam path. Thus, the x1x3 plane in Fig. 1 constitutes a symmetry plane and the x3-axis represents one of the symmetry directions. For the geometry of Fig. 1

Time reversal simulation for anisotropic solids

In this section, a numerical experiment is executed to demonstrate the applicability of the time reversal method to elastic wave phenomena in anisotropic solids. As an example of the use of this method, we consider a unidirectional graphite/epoxy composite whose properties are assumed to be transversely isotopic: C11=C22=15, C12=7.7, C13=C23=3.4, C33=87, C44=C55=7.8, C66=3.65 GPa and ρ=1.595 g/cm3.

Eq. (4) shows that the slowness surface curvatures are key parameters needed to define the

Conclusions

The MGB model was used to simulate a simple TR process for anisotropic materials and it was shown that complete reconstruction of the original input signal could be achieved by the TR process of this model. The present MGB model can also be applied to simulate other TR processes for anisotropic solids: longer propagation distances, broadband pulse shape for the input and reconstructed TR signal waveforms, arbitrary propagation directions with multiple curved interfaces.

Ultrasonic inspections

Acknowledgement

This work was supported by the Korea Science and Engineering Foundation (KOSEF) Grant funded by the Korea government (MOST) (No. 2007-00467).

References (16)

  • M. Fink

    Time reversal of ultrasonic fields: Part I. Basic principles

    IEEE Trans UFFC

    (1992)
  • N. Chakroun et al.

    Time reversal processing in ultrasonic nondestructive testing

    IEEE Trans UFFC

    (1995)
  • M. Fink

    Time-reversed acoustics

    Scientific American

    (1999)
  • M. Fink et al.

    Acoustic time-reversal mirrors

    Inverse Problems

    (2001)
  • C. Draeger et al.

    Acoustic time reversal in solids

    J Acoust Soc Am

    (1997)
  • B. Zhang et al.

    Time reversal self-adaptive focusing in anisotropic elastic solid medium

    Acoustical Physics

    (2003)
  • B. Zhang et al.

    Theoretical and experimental study of time reversal in anisotropic medium

    2004 IEEE Ultrason Symp

    (2004)
  • L.W. Schmerr et al.

    Ultrasonic NDE systems: models and measurements

    (2007)
There are more references available in the full text version of this article.

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