K.M.M. Tant, A.J. Mulholland, M. Langer and A. Gachagan,
A fractional Fourier transform analysis of the scattering of ultrasonic waves,
Proc. R. Soc. Lond. Ser. A 471 (2015), 20140958, 14 pp.
Abstract:
Many safety critical structures, such as those found in nuclear plants,
oil pipelines and in the aerospace industry, rely on key components that are
constructed from heterogeneous materials. Ultrasonic non-destructive testing (NDT)
uses high-frequency mechanical waves to inspect these parts, ensuring they
operate reliably without compromising their integrity.
It is possible to employ mathematical models to develop a deeper understanding
of the acquired ultrasonic data and enhance defect imaging algorithms.
In this paper, a model for the scattering of ultrasonic waves by a crack is
derived in the time–frequency domain.
The fractional Fourier transform (FrFT) is applied to an inhomogeneous wave equation
where the forcing function is prescribed as a linear chirp, modulated by a Gaussian envelope.
The homogeneous solution is found via the Born approximation which encapsulates
information regarding the flaw geometry.
The inhomogeneous solution is obtained via the inverse Fourier transform of a
Gaussian-windowed linear chirp excitation.
It is observed that, although the scattering profile of the flaw does not change, it is amplified.
Thus, the theory demonstrates the enhanced signal-to-noise ratio permitted by the
use of coded excitation, as well as establishing a time–frequency domain framework
to assist in flaw identification and classification.