title: A fractional Fourier transform analysis of the scattering of ultrasonic waves.
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.
authors: Department of Mathematics and Statistics , University of Strathclyde , Glasgow, UK.
authors: Katherine M M
authors: KM
authors: Tant
authors: Department of Mathematics and Statistics , University of Strathclyde , Glasgow, UK.
authors: Anthony J
authors: AJ
authors: Mulholland
authors: Department of Mathematics and Statistics , University of Strathclyde , Glasgow, UK.
authors: Matthias
authors: 0000-0001-8813-7914
authors: M
authors: Langer
authors: Centre for Ultrasonic Engineering , University of Strathclyde , Glasgow, UK.
authors: Anthony
authors: A
authors: Gachagan
keywords: inverse problems; non-destructive testing; scattering theory; ultrasonics