Abstract
Many important physical phenomena are described by wave or diffusion-wave type equations. Recent work has shown that a transform domain signal description from linear system theory can give meaningful insight to multi-dimensional wave fields. In N. Baddour [AIP Adv. 1, 022120 (2011)], certain results were derived that are mathematically useful for the inversion of multi-dimensional Fourier transforms, but more importantly provide useful insight into how source functions are related to the resulting wave field. In this short addendum to that work, it is shown that these results can be applied with a Gaussian source function, which is often useful for modelling various physical phenomena.
Highlights
In recent work, Baddour[1] considered a certain class of a certain class of improper integrals which arise while using multidimensional Fourier transforms with multidimensional wavefields
The integrals of interest are of the following form φ(x) x2 − k2 jn (xr φ(x) x2 − k2 Jn x dx
The proofs in Ref. 1 use the boundedness of φ(x) to permit the use of Jordan’s lemma. In this short note we extend the approach to φ (x) = e−ax[2] so that the same results can be used, without the use of Jordan’s lemma
Summary
Baddour[1] considered a certain class of a certain class of improper integrals which arise while using multidimensional Fourier transforms with multidimensional wavefields. The integrals of interest are of the following form These integrals arise in the evaluation of inverse Fourier transforms in 3D, 2D and 1D respectively. These integrals were used to prove the derivative-free photoacoustic shell theorem[2] and can be used in applications ranging from photoacoustics to shear waves.[3,4,5] In equations (1) to (3), jn(x) are spherical Bessel functions of the first kind of order n and Jn(x) are Bessel functions of the first kind, of order n, r is a positive real variable and k2 is the wavenumber that may be real or complex.
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