Time reversal study of ultrasonic waves for anisotropic solids using a Gaussian beam model
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 x1–x3 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).
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