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Multidimensional wave field signal theory: Mathematical foundations

http://aip.metastore.ingenta.com/content/aip/journal/adva/1/2/10.1063/1.3596359

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### Abstract

Many important physical phenomena are described by wave or diffusion-wave type equations. Since these equations are linear, it would be useful to be able to use tools from the theory of linear signals and systems in solving related forward or inverse problems. In particular, the transform domain signal description from linear system theory has shown concrete promise for the solution of problems that are governed by a multidimensional wave field. The aim is to develop a unified framework for the description of wavefields via multidimensional signals. However, certain preliminary mathematical results are crucial for the development of this framework. This first paper on this topic thus introduces the mathematical foundations and proves some important mathematical results. The foundation of the framework starts with the inhomogeneous Helmholtz or pseudo-Helmholtz equation, which is the mathematical basis of a large class of wavefields. Application of the appropriate multi-dimensional Fourier transform leads to a transfer function description. To return to the physical spatial domain, certain mathematical results are necessary and these are presented and proved here as six fundamental theorems. These theorems are crucial for the evaluation of a certain class of improper integrals which arise in the evaluation of inverse multi-dimensional Fourier and Hankel transforms, upon which the framework is based. Subsequently, applications of these theorems are demonstrated, in particular for the derivation of Green's functions in different coordinate systems.

© Copyright 2011 Author(s).

Received 07 February 2011
Accepted 22 April 2011
Published online 24 May 2011

Acknowledgments:
This work was financially supported by the Natural Science and Engineering Research Council of Canada. The author declares that there are no competing interests.

Article outline:

I. INTRODUCTION
A. Background
B. Helmholtz equation and the multidimensional Green's and transfer function
1. The Helmholtz equation
2. Notation and sign conventions
3. SpatialGreen's function and spatial transfer function
4. Spatial transfer function in one, two and three spatial dimensions
C. Motivation for the theorems
II. ANALYSIS: THEOREMS AND PROOFS
A. Theorem 1: Spherical Bessels and a real wave number
1. Proof of Theorem 1
2. Interpretation of Theorem 1
3. Sommerfeld radiation condition
B. Theorem 2: Spherical Bessels and a complex wave number
1. Proof of Theorem 2
2. Interpretation of Theorem 2
C. Theorem 3: Bessels and a real wave number
1. Proof of Theorem 3
2. Interpretation of Theorem 3
D. Theorem 4: Bessels and a complex wave number
1. Proof of Theorem 4
2. Interpretation of Theorem 4
E. Theorem 5: Complex exponentials and a real wave number
1. Proof of Theorem 5
2. Interpretation of Theorem 5
F. Theorem 6: Complex exponentials and a complex wave number
1. Proof of Theorem 6
2. Interpretation of Theorem 6
III. RESULTS: APPLICATIONS OF THE THEOREMS TO DERIVING SPATIALGREEN'S FUNCTIONS
A. SpatialGreen's functions
1. Example of the application of Theorems 1 and 2: 3D Green's functions for the Helmholz equation via 3D inverse Fourier transformation
2. Example of the application of Theorems 3 and 4: 2D Green's functions via 2D inverse Fourier transformations
3. Example of the application of Theorems 5 and 6: 1D Green's functions via inverse Fourier transformation
B. Insights into the basic structure of the wavefields
IV. FURTHER EXAMPLES OF APPLICATIONS OF THE THEOREMS
A. Example of application of Theorem 5: Forced vibrations of a single degree of freedom undamped system
B. Example of application of Theorems 5 and 6 to Fourier tables
C. Example of application of Theorems 1-4 to Hankel and spherical Hankel transforms
D. Example of application of Theorem 1: Diffraction ultrasound Fourier theorem in 3D
V. CONCLUSIONS

/content/aip/journal/adva/1/2/10.1063/1.3596359

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1. A. Mandelis, Diffusion-Wave Fields, Mathematical Methods and Green Functions (Springer, New York, 2001).

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2. M. Slaney and A. Kak, Principles of Computerized Tomographic Imaging (SIAM, Philadelphia, 1988).

20.

20. E. Kreyszig, Advanced Engineering Mathematics (John Wiley and Sons, New York, 1993).

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21. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1964).

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22. G. Arfken and H. Weber, Mathematical Methods for Physicists (Elsevier Academic Press, New York, 2005).

http://aip.metastore.ingenta.com/content/aip/journal/adva/1/2/10.1063/1.3596359

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2016-09-28

### Abstract

Many important physical phenomena are described by wave or diffusion-wave type equations. Since these equations are linear, it would be useful to be able to use tools from the theory of linear signals and systems in solving related forward or inverse problems. In particular, the transform domain signal description from linear system theory has shown concrete promise for the solution of problems that are governed by a multidimensional wave field. The aim is to develop a unified framework for the description of wavefields via multidimensional signals. However, certain preliminary mathematical results are crucial for the development of this framework. This first paper on this topic thus introduces the mathematical foundations and proves some important mathematical results. The foundation of the framework starts with the inhomogeneous Helmholtz or pseudo-Helmholtz equation, which is the mathematical basis of a large class of wavefields. Application of the appropriate multi-dimensional Fourier transform leads to a transfer function description. To return to the physical spatial domain, certain mathematical results are necessary and these are presented and proved here as six fundamental theorems. These theorems are crucial for the evaluation of a certain class of improper integrals which arise in the evaluation of inverse multi-dimensional Fourier and Hankel transforms, upon which the framework is based. Subsequently, applications of these theorems are demonstrated, in particular for the derivation of Green's functions in different coordinate systems.

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