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Introduction to the Wiener-Hopf Method

1.1 A brief history of the Wiener-Hopf method

In 1931, Wiener and Hopf (Wiener and Hopf 1931) invented a powerful technique for solving a special type of integral equation. By introducing the Laplace transform of the unknown, the integral equation was rephrased in terms of a functional equation in a suitably defined complex space. The solution method was ingenious and based on a sophisticated procedure that exploits some properties of analytic functions. It stands as one of the most important mathematical inventions for obtaining analytical solutions of very difficult problems. The number of problems that can be effectively approached by the Wiener-Hopf (WH) technique is significant. Most of these problems are very important, and often there are no alternative approaches to solve them efficiently.

In this chapter, we recall the notations and the fundamental concepts of this technique. For a comprehensive exposition of the theory, the reader can also refer to Noble (1958) and Daniele and Zich (2014). Moreover, we introduce a new methodology for the semi-analytical solution of classical WH equations, i.e. the Fredholm factorization method, which will be used in generalized form in the following chapters.

1.2 Fundamental definitions and assumptions to develop the Wiener-Hopf technique in the spectral domain

Throughout this book, we will only consider time harmonic fields with a time dependence specified by the factor e j?t (electrical engineering notations), which is omitted, where the imaginary unit is indicated by j. Applied mathematics often uses a different notation e-i?t, where the imaginary unit is indicated by i. This means that, in the natural domain, the change j ? -i transforms engineering notations in applied mathematics notations (and vice versa).

We recall that by the use of the Laplace (or Fourier transform) we define the diffraction problem in a spectral domain. In the spectral domain we need to pay particular attention to the notations used in engineering and applied mathematics (see Appendix 1.A for a discussion on the use of Fourier transforms extendable to Laplace transforms).

Due to mathematical reasons related to the WH technique, the ideal media used for describing the problems will be considered with small vanishing loss. Taking into account the engineering notation, it means that their propagation constants (for example) k has a negative (vanishing) imaginary part (see Appendix 1.A).

Throughout this book, only remote sources consisting of plane waves will be considered. As is well known, this limitation is not restrictive because, due to the linearity of the considered problems, a general source can always be expanded in terms of plane waves (Felsen and Marcuvitz 1973). In this book, a fundamental role is played by the geometrical optical (GO) contributions in the presence of incident plane waves. As these contributions are well known from elementary considerations, we assume that the reader is able to understand them.

The separation of the plus and minus Laplace transforms is fundamental to formulate the problem in terms of spectral WH functional equations, which forms the basis of the concept of the classical factorization in the WH technique.

We introduce the spectral domain ? with the following integral definitions of plus and minus WH functions that are unknown at the beginning of the procedure:

The above integrals converge for values of ? located in an upper half-plane (the first integral) or a lower half-plane (the second integral). Their analytic continuations for values arbitrary of ? define the plus and minus Laplace transforms. The singularities of the plus or minus function can be poles or branch points. For example, in the spectral domain, the GO contributions only produce poles.

While, in the presence of sources located in finite points, the plus (minus) functions are regular in the upper (lower) half-plane Im[?] = 0 (Im[?] = 0), the presence of plane waves and the losses assumed in the medium can produce non-standard poles. For a plus (minus) function, we define non-standard poles located in Im[?] = 0 (Im[?] = 0). In general, we can decompose the plus and minus functions in standard and non-standard parts:

where:

is the standard part of a plus function that is regular in the upper half-plane Im[?] = 0;

is the standard part of a minus function that is regular in the lower half-plane Im[?] = 0;

is the non-standard part of plus functions, which consists of the characteristic part of the poles located in the upper half-plane Im[?] > 0;

is the non-standard part of minus functions, which consists of the characteristic part of the poles located in the lower half-plane Im[?] < 0 .

If we consider the above Laplace transformations related to total fields, we observe that the non-standard parts of the WH unknown functions are coincident with the non-standard part of the GO contributions. Consequently, non-standard parts are known a priori:

In this book, wide use of multivalued analytic functions is demonstrated; thus, the determination of branches on Riemann surfaces is a fundamental mathematical issue.

To successfully develop the mathematical theory and applications of this book, it is useful to have technical computing software, particularly in analytic and numerical manipulations of the equations.

In particular, in this book, for wedge scattering problems, multivalued functions are defined in terms of the spectral propagation multivalued function:

where the proper (principal) sheet is defined for ? (0) = k, thus Im[? (?)] = 0 and the common selection of branches originated from the branch point ± k are the line with Im[?] = 0, the vertical lines or the lines with Re[?] = 0. A discussion on the proper and improper sheets of is presented in section 1.13.

1.3 WH equations from the physical domain to the spectral domain

WH equations arise from discontinuity problems; therefore, they take the forms of semi-convolutional equations in the physical domain:

where f(x) is the unknown, g(x) is the kernel and fo(x) is a forcing (source) term.

WH equations can be considered to be generalizations of convolutional equations:

where solutions can be obtained using Fourier transforms in the ? plane:

The semi-convolutional equation (1.3.1) can be extended in the domain to all real values of x as

where an unknown function and a plus unknown function f (x)u(x) are introduced, and u(x) is the unit step Heaviside function. The application of Fourier transform to (1.3.5) yields

where

Equation (1.3.6) is called the non-homogeneous WH equation. The homogeneous WH equation

can be defined by absorbing Fo+ (a) in the WH unknown F- (a) :

We also note that the physical support of plus unknown is x>0 and that of the minus unknown is x<0.

The standard properties of the plus and minus functions are related to the location of source poles.

1.4 WH equations in wave scattering problems

The WH equations can be classified into scalar or vector. The more general WH equations present the vector form:

where F+(?) and F-(?) are respectively plus and minus vector unknowns of order n, Fo(?) is a vector source term of order n and G(?) is the matrix WH kernel of the same order.

Now we introduce a smart property which leads us to select Laplace transforms in the deduction of WH equations.

While considering remote sources such as plane waves in layer-scattering problems, Fo(?) in (1.4.1) is constituted of plane wave contributions that results to be zero in the spectral plane defined by the double-sided Laplace transforms, although its inversion rebuilds the plane waves, Daniele and Zich (2014, Chapter 7.3). On the contrary, using Fourier transforms, plane waves are represented by distributions as e-j?ox 2pd(?-?o) that are cumbersome to be considered throughout the entire development of the WH theory.

For this reason, we deduce the WH equations for layered regions in Laplace domain by ignoring the presence of remote sources. This property can be extended to angular regions (see section 3.3). Consequently, in this framework, the...