A fractional Fourier transform analysis of the scattering of ultrasonic waves

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6 Citations (Scopus)

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 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.
LanguageEnglish
Number of pages14
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume471
Issue number2175
Early online date18 Feb 2015
DOIs
Publication statusPublished - Apr 2015

Fingerprint

Fractional Fourier Transform
Ultrasonic Wave
Chirp
Ultrasonic waves
ultrasonic radiation
Frequency Domain
Time Domain
Fourier transforms
Excitation
Scattering
Heterogeneous Materials
Born Approximation
Defects
defects
chirp
scattering
Integrity
Forcing
Envelope
Wave equation

Keywords

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

Cite this

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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 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.",
keywords = "ultrasonics , non-destructive testing , inverse problems , scattering theory",
author = "Tant, {Katherine M. M.} and Mulholland, {Anthony J.} and Matthias Langer and Anthony Gachagan",
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AU - Mulholland, Anthony J.

AU - Langer, Matthias

AU - Gachagan, Anthony

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N2 - 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.

AB - 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.

KW - ultrasonics

KW - non-destructive testing

KW - inverse problems

KW - scattering theory

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