Abstract
The elastodynamic scattering behaviour of a finitesized scatterer in a homogeneous isotropic medium can be encapsulated in a scattering matrix (Smatrix) for each wave mode combination. In a 2 dimension (2D) space, each Smatrix is a continuous complexvalued function of 3 variables: incident
wave angle, scattered wave angle and frequency. In this paper, the Smatrices for various 2D scatterer shapes (circular voids, straight cracks, rough cracks and a cluster of circular voids) are investigated to find general properties of their angular and frequency behaviour. For all these shapes, it is shown that
the continuous data in the angular dimensions of their Smatrices can be represented to a prescribed level of accuracy by a finite number of complexvalued Fourier coefficients that are physically related to the angular orders of the incident and scattered wavefields. It is shown mathematically that the
number of angular orders required to represent the angular dimensions of an Smatrix at a given frequency is a function of overall scatterer size to wavelength ratio, regardless of its geometric complexity. This can be interpreted as a form of the Nyquist sampling theorem and indicates that there is an upper bound on the sampling interval required in the angular domain to completely define an Smatrix. The variation of scattering behaviour with frequency is then examined. The frequency dependence of the Smatrix can be interpreted as the Fourier transform of the timedomain impulse response of the scatterer for each incident and scattering angle combination. Depending on the nature
of the scatterer, these are typically decaying reverberation trains with no definite upper bound on their durations. Therefore, in contrast to the angular domain, there is no lower bound on the sampling interval in the frequency domain needed to completely define an Smatrix, although some pragmatic solutions
are suggested. These observations may help for the direct problem (computing ultrasonic signals from known scatterers efficiently) and the inverse problem (characterising scatterers from measured ultrasonic signals).
wave angle, scattered wave angle and frequency. In this paper, the Smatrices for various 2D scatterer shapes (circular voids, straight cracks, rough cracks and a cluster of circular voids) are investigated to find general properties of their angular and frequency behaviour. For all these shapes, it is shown that
the continuous data in the angular dimensions of their Smatrices can be represented to a prescribed level of accuracy by a finite number of complexvalued Fourier coefficients that are physically related to the angular orders of the incident and scattered wavefields. It is shown mathematically that the
number of angular orders required to represent the angular dimensions of an Smatrix at a given frequency is a function of overall scatterer size to wavelength ratio, regardless of its geometric complexity. This can be interpreted as a form of the Nyquist sampling theorem and indicates that there is an upper bound on the sampling interval required in the angular domain to completely define an Smatrix. The variation of scattering behaviour with frequency is then examined. The frequency dependence of the Smatrix can be interpreted as the Fourier transform of the timedomain impulse response of the scatterer for each incident and scattering angle combination. Depending on the nature
of the scatterer, these are typically decaying reverberation trains with no definite upper bound on their durations. Therefore, in contrast to the angular domain, there is no lower bound on the sampling interval in the frequency domain needed to completely define an Smatrix, although some pragmatic solutions
are suggested. These observations may help for the direct problem (computing ultrasonic signals from known scatterers efficiently) and the inverse problem (characterising scatterers from measured ultrasonic signals).
Original language  English 

Article number  105964 
Number of pages  11 
Journal  Ultrasonics 
Volume  99 
Early online date  25 Jul 2019 
DOIs  
Publication status  Published  1 Nov 2019 
Keywords
 Forward model
 Ultrasonic array
 Scattering amplitude
Fingerprint Dive into the research topics of 'Angular and frequency behaviour of elastodynamic scattering from embedded scatterers'. Together they form a unique fingerprint.
Profiles

Dr Jie Zhang
 Department of Mechanical Engineering  Research Fellow
 Solid Mechanics
 Ultrasonics and Nondestructive Testing (UNDT)
Person: Academic , Member