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Michael V. Vesnik,
Institute of Radioengineering and Electronics of Russian Academy of Sciences

Received August 9, 1999

A new method of solving 2-D diffraction boundary problems for perfectly conducting half-infinite scatterers is presented. The method is based on constructing a universal integral representation of the diffracted wave with use of the theory of conformal transformations. This diffraction wave in integral form being summarized with geometrical optics components of incident and reflected waves represents a solution of Helmholt's equation boundary problem. A rigorous solution for half-infinite scatterers with polygonal envelope is received. The asymptotic of the integral representation is constructed in a compact form, convenient for further applications.


To the present moment rigorous analytical solutions of diffraction theory boundary problems are received with use of either variables separating method or Wiener - Hopf method with it's modifications and the number of such solutions is not a high one. This paper presents a new method which allows to increase a number of rigorous solutions in 2-D diffraction theory.

2. Statement of the problem

Consider the 2-D perfectly conducting half - infinite body of an arbitrary shape disposed in (x, y) plane and exited by an incident wave . In terms of geometrical optics (GO) the incident and reflected waves and have "light - shadow" boundaries (Fig. 1).

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Fig.1. and regions.

The solution for the diffracted wave u is constructed in the form of an integral representation in the plane of complex variable a:




the sense of remaining designations will be explained later.

3. Construction of an integral representation and an asymptotic

Assume that y indicates the value of a complex variable and thus the (x, y) plane is a complex one. Using the theory of a complex variable one can find one-to-one conformal transformation of the region w=(u, v) to the region z=(x, y). Here w represents the upper half-plane of the complex plane (u, v) and z represents the exteriority of the scatterer in the complex plane (x, y):



The inverse to (2) function is:



and even if the expression (3) does not exist explicitly then fixing z one can uniquely find the corresponding value of w from (2). Therefore we suppose (3) to be known as well.

Consider a case of excitation by plane wave coming from the directing angle in the z region. The expression for the wave in polar coordinates will be:



but the method presented is suitable for other types of excitation too.

In polar coordinates (2) becomes:


Here , are the functions of polar coordinates. In accordance with (3) the inverse functions and :


For the observation point with coordinates in z region and coordinates in w region from (5) and (6) we get:



Let us introduce new complex values a , b , , and fix one-to-one relationship between them:

, , ,


where is the real value. The curve is shown at Fig.2:

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Fig. 2. region.

Let us now define a new function on the set of complex values a (and at the same time - on the sets of complex values b , or ):



Let us designate the earlier introduced regions z and w as and and regions and - as . Let us introduce second sheets of complex surface , , and to be disposed over the corresponding first sheets. Now instead of the single (primary) value related to the single sheet we get two values related to the first or second sheets correspondingly: and , and , , and , and instead of , , and .

Now determine that the incident GO field exists only in the first sheet and the reflected GO field exists only in the second sheet. Define the relationship between the coordinates of the second sheets with the same formulas (5), (6) and (8) that the one of the first sheets.

To this point the values without the index "1" or "2": , , and were related to the single regions z, w, and which we considered before. From this point we shall stand that their domain of definition relates to both sheets "1" and "2".

Let us now perform a reflection of all the points of the region from the upper half-plane to the lower one symmetrically with respect to the horizontal axis. Then, "pasting" to along the horizontal axis which corresponds to the boundary of the scatterer we compose from them a full plane which will be named as (Fig. 2). The same operation we make with the regions and thus receiving a new region .

Consider now the integral representation (1). The paths of integration in a - plane "1" and "2" are traced in such way that the corresponding points in the region fall to the shadow boundaries of the functions and in region if . In (1) from (8) when , when . The direction of moving along the paths "1" and "2" is shown at Fig. 2.

It may be shown that the integral representation (1) meets wave equation and radiation condition. It is also continuous fraction by the value in region . Thus is the divergent part of the solution of our boundary problem (sometimes it is being named as the diffracted wave).

Let us now build a function V:

- at the "lighted" places of region and

- at the "shadowed" places of region.


Here P is the or . As the V function is a continuous fraction on all the region including the boundary of the scatterer the function U:



meets boundary conditions by Dirichle or Newmann correspondingly and thus represents the required solution of our boundary problem. The solution is rigorous for the scatterers with polygonal contour. For other scatterers an additional mathematical investigation of solution accuracy is required.

To receive an asymptotic of integral representation (1) considering it is more convenient to use b coordinate. If the parameter is large and varies slowly, i.e. , from (8) one can receive the derivative:

; .


Then, as , from (1) we get:


The final expressions will be:





The asymptotic for the case of two "light - shadow" boundaries is:



where and - are the saddle points of function (9) - look [1].

Viewing (15) on can see that it represents a sum of terms connected with the saddle points disposed on GO shadow boundaries of incident and reflected waves. The shadow boundaries diverge from one point on the surface of the scatterer and locate in different sheets.

4. Solution constructing algorithm

So, there is the statement of the problem: 2-D perfectly conducting scatterer of an arbitrary shape exited by the plane (or other) incident wave. The solution of boundary problem is required. In correspondence with the method presented the following algorithm may be offered:

1. The first step should be done is to find the conformal transformation (2) which defines the paths of integration in a region. As soon as it is done one can construct the solution in integral form (1). To find the asymptotic the next steps should be done:

2. Finding the saddle points of function (9).

3. Finding the (12) and (14) in the saddle points .

4. Finding the solution in correspondence with (11).

5. An example of method application for the case of edge diffraction

As an example of method application let us find the known solution for the case of plane wave (4) scattering by perfectly conducting edge with external angle radian. The results of performing all the steps of our algorithm will be the following:

1. Conformal transformation (1): . Inverse function: from which for the observation point we get: and from (8) follows:

2. Saddle points of the function (9): , .

3. The derivative (12): , the function (14):

From this one can easily get (15) which fully corresponds with the known solution for the edge [2].

6. Conclusion

In this paper a new analytical method of solving 2-D diffraction problems is presented. The solution for the case of plane wave scattering by half-infinite body is constructed with use of this method. The effectiveness of the method has been illustrated in [3] with receiving a rigorous solution for the case of plane wave scattering by half-infinite plate of finite thickness. It seems to be no limitations preventing to apply the same method for the case of other types of exiting waves and for other types of 2-D scatterers, for instance, multiply connected or of the limited size.

In addition to the generosity of the method presented one can note its essential effectiveness and convenience because if conformal transformation (2) is known then the constructing of analytical formula (15) for asymptotic solution may be done immediately after calculating the derivative (12).

The author wishes to acknowledge Prof. V.A. Kaloshin for useful discussions.


[1] F. Frank, R. Mises. Differential and integral equations of mathematical physics.

[2] Pauli W. On asymptotic series for functions in the theory of diffraction of light. Physical Review, 54, No. 11, 924-931, 1938.

[3] M.V. Vesnik. "Analytical Solution to a Boundary Value Problem for the Helmholtz Equation",
Journal of Communications Technology and Electronics, Vol. 45, No. 1, January 2000

Author: Vesnik Michael V., e-mail: vesnik@mail.cplire.ru

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