A fractional Fourier transform analysis of the scattering of ultrasonic waves

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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 nondestructive testing 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 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.
Original languageEnglish
Article number471
Number of pages14
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2175
Publication statusPublished - 8 Mar 2015


  • ultrasonics
  • non-destructive testing
  • inverse problems
  • scattering theory


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