Multidimensional wave field signal theory: Mathematical foundations

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

The Gaussian Shear Wave in a Dispersive Medium

“…However, recently Prof. Baddour has demonstrated a set of theorems, applicable to 1, 2, and 3D wave sources, that provides a solution for this type of transform. Specifically, Prof. Baddour's Theorem 5 has shown that the following results are true (Baddour 2011):…”

“…The corresponding statement to eqn (5) for complex wave numbers comes from the application of Theorem 6 (complex wave number with negative imaginary part) from (Baddour 2011), which states:…”

“…With our choice of negative real part for the complex k , Prof. Baddour's Theorem 6 (Baddour 2011), ensures that eqn (4) and (5) are still valid for complex k . Hence, for F z a Gaussian of amplitude P = 1, eqn (4) and (5), and v̂ z ( x , ω ) = iωû ( x , ω ) yield:…”

“…This paper first treats the idealized one-dimensional case of a Gaussian shear wave generated by radiation force, then moves to the more realistic but more intractable case of an axis symmetric beam in cylindrical coordinates. We make use of the unifying mathematical framework and fundamental theorems derived recently by Prof. Baddour (2011). In some cases, approximate solutions are derived where exact analytical solutions are not found.…”

An approach to viscoelastic characterization of dispersive media by inversion of a general wave propagation model

“…Then, by applying Baddour's theorem 25 and selecting the appropriate solution according to the Sommer¯eld radiation condition we obtain…”