Near-field scanning in the time domain on a spherical surface— A formulation using the free-space Green’s function

Abstract

Two formulations for determining the characteristics of an unknown source of acoustic waves using the measurement of its field at its near zone are presented. The measurement in both cases is to be performed on a spherical scan surface which encapsulates the source. The first is for an ideal probe which measures the field at its location. The knowledge of the field is sufficient; its normal derivative is not required. In the second formulation a realistic probe is considered. This time it is required only that the probe has an axially symmetric receiving characteristic. With this formulation, the time functions which characterize the source are found using only the signal at the output of the probe. Both formulations are such that they are not specific to the scan surface radius. Furthermore, they are entirely in the time domain, requiring no inverse Fourier transformations left to be performed.

Time domain near‐field to near‐field transformation using a spherical‐multipole approach

“…We derive new recurrence relations which ‐ for each order of the multipole expansion ‐ lead to a very simple and numerically efficient calculation of those (additional) time domain spherical‐multipole amplitudes which are relevant in the near field only from those ones dominant in the far‐field. Similar recurrence relations for radiating fields have been presented by Wilcox [1956] for the frequency domain whereas the algebraic 3‐term‐recurrence relations presented by Koc et al [2001] relate different orders of the time domain multipole amplitudes of the scalar field.…”

“…In this context, Hansen also developed a method to efficiently solve the relatively complicated integrals with spherical Hankel functions in the denominator of the integrands by evaluating the corresponding contour integrals in the complex plane. For scalar (acoustic) fields [ Koc et al , 2001] directly started from the time domain spherical‐multipole expansion of the radiated acoustic field to evaluate the multipole amplitudes from measurements of the acoustic pressure on a spherical surface enclosing all sources. Similar to Hansen's method the approach also exploits the orthogonality of the spherical harmonics on the measurements surface.…”

Fast evaluation of transient acoustic fields

“…We note here that the temporal interpolation scheme is likely to be the largest source of error in the algorithm: if the coefficients of the expansion are computed using Eqs. (4) and (5) the resulting pressure field is exact, other than for errors caused by truncation of the expansion, at values of time t ¼ s j þ r=c. Interpolation between these time points, however, introduces errors from interpolation and/or aliasing.…”

“…In the time domain, they are not so common, but they have been used for field computation in a number of physical applications including electromagnetism 1,2 and acoustics. 3,4 In acoustics, the spherical harmonic expansion has been used as a means to characterize a source using physical data, 4 by measuring the acoustic pressure on a spherical surface containing the source, and applying deconvolution operations to find the coefficients of the expansion. In the electromagnetism literature, 2 the coefficients have been computed directly from the source distribution in order to generate an expansion that can be used to evaluate the radiation field of an antenna.…”

Time domain scattering of scalar waves by two spheres in free-space

“…(8) -(9)are in similar form with the frequency domain relations of[3]. Using direct matrix -deconvolution given[4], find the diagonal matrices 0 a and 0b from Eqs. (12)and(13), respectively; and then using Eqs.…”