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Chapter 2
Wedge Diffraction: Exact
Solution and Asymptotics
The relationships us = Ez and uh = Hz exist between the acoustic and electromagnetic fields in the 2-D wedge diffraction problem. Here, Ez and Hz are the components
(of vectors E and H) that are parallel to the edge of the wedge.
2.1CLASSICAL SOLUTIONS
Diffraction at a wedge with a straight edge and infinite planar faces is an appropriate canonical problem to derive asymptotic expressions for the edge waves scattered from arbitrary curved edges. In the particular case of the wedge, which is a semi-infinite half-plane, the exact solution of this canonical problem was found by Sommerfeld (1896), who constructed so-called branched wave functions. Analysis of this work performed in Ufimtsev (1998) shows that Sommerfeld also developed almost everything that was necessary to obtain the solution for the wedge with an arbitrary angle between its faces. However, he missed the last step that led directly to the solution. This more general solution was found by Macdonald (1902) with the classical method of separation of variables in the wave equation. Later on, Sommerfeld also constructed the solution of the wedge diffraction problem by his method of branched wave functions and derived simple asymptotic expressions for the edge-diffracted waves (Sommerfeld, 1935).
Because the wedge diffraction problem is the basis for the construction of PTD, its solution is considered here in detail. First we derive this solution in the form of infinite series and then convert it to the Sommerfeld integrals convenient for asymptotic analysis. The material of this chapter, with the exception of Sections 2.6 and 2.7, is a scalar version of the theory developed by the author for electromagnetic waves (Ufimtsev, 1962).
Fundamentals of the Physical Theory of Diffraction. By Pyotr Ya. Ufimtsev
Copyright © 2007 John Wiley & Sons, Inc.
33
TEAM LinG
34 Chapter 2 Wedge Diffraction: Exact Solution and Asymptotics
Figure 2.1 A wedge is excited by a filamentary source located at the line r = r0, ϕ = ϕ0.
The geometry of the problem is shown in Figure 2.1. A wedge with infinite planar faces ϕ = 0 and ϕ = α is located in a homogeneous medium. It is excited by a cylindrical wave. The source of this wave is a radiating filament with coordinates
r= r0, ϕ = ϕ0. This is a two-dimensional problem where ∂/∂z ≡ 0. The field outside the wedge (0 ≤ ϕ ≤ α) satisfies the wave equation
u + k2u = I0δ(r − r0, ϕ − ϕ0) |
(2.1) |
and the boundary conditions |
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us = 0 |
(2.2) |
or |
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∂uh/∂n = 0 |
(2.3) |
on the faces ϕ = 0 and ϕ = α. In the case of electromagnetic waves, these boundary conditions are appropriate for the perfectly conducting wedge, and function us
represents the z-component of electric field intensity E, while function uh is the
z-component of magnetic field intensity H. In the case of acoustic waves, condition (2.2) relates to the acoustically soft wedge, and (2.3) to the acoustically hard wedge.
Outside the immediate vicinity of the source, the field u satisfies the homogeneous wave equation
u + k2u = 0. |
(2.4) |
For the two-dimensional problem, the Laplacian operator is defined by
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1 ∂2 |
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Using the classical method of separation of variables, we set in Equation (2.4)
u = R(r) (ϕ) |
(2.6) |
TEAM LinG
2.1 Classical Solutions |
35 |
and substitute this u into Equation (2.4). After simple manipulations, the latter can be separated into two equations:
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dϕ2 |
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The function and the separation constants vl are determined from the boundary conditions.
In the case of soft boundary conditions, according to Equations (2.2) and (2.8),
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α , l = 1, 2, 3, . . . , |
(2.9) |
and, in the case of hard boundary conditions, in accordance with Equations (2.3) and (2.8),
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l = 0, 1, 2, 3, . . . . |
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The Bessel and Hankel functions |
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represent the solution of the radial equation (2.7). The Bessel functions Jvl (kr) can be used in the region r ≤ r0, because they are finite at the edge r = 0, and the Hankel functions are appropriate in the region r ≥ r0, because they satisfy Sommerfeld’s radiation condition at infinity:
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(2.12) |
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Hence, the solutions of Equation (2.4) can be written as |
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1 al Jvl |
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TEAM LinG
36 Chapter 2 Wedge Diffraction: Exact Solution and Asymptotics
Figure 2.2 The integration contour L in the Green theorem (2.15).
These expressions satisfy the boundary conditions, as well as the reciprocity principles; that is, they do not change after interchanging r and r0, ϕ and ϕ0.
The unknown coefficients al and bl can be found by applying the Green theorem
∂u |
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L ∂n dl = S u ds, ds = r dr dϕ |
(2.15) |
to the fields us and uh in the region S bounded by the contour L shown in Figure 2.2. This contour consists of two arcs r = r0 − ε, r = r0 + ε and two radial sides
ϕ = ϕ0 − ψ , ϕ = ϕ0 + ψ .
Substitute us,h into (2.15), take into account that, according to Equation (2.1),
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u = −k2u + I0δ(r − r0, ϕ − ϕ0), |
(2.16) |
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and let ε tend to zero. It is clear that |
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S us,h ds −→ 0 |
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(2.17) |
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as the Bessel and Hankel functions are finite at r = r0 > 0. |
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As a result, the Green formula for us,h transforms into |
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ϕ0 ψ |
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r0 ε |
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dϕ lim |
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δ(r − r0, ϕ − ϕ0)r dr, |
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The two-dimensional Delta-function in polar coordinates equals |
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δ(r − r0, ϕ − ϕ0) = δ(r − r0) |
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TEAM LinG
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2.1 Classical Solutions 37 |
therefore |
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ϕ0 |
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∂us,h |
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+ψ δ(ϕ − ϕ0)dϕ. (2.20) |
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r0 0 |
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r0 0 |
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Equation (2.20) is valid for arbitrary limits of integration. This is possible if the integrands in the left and right sides are equal to each other:
∂us,h |
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∂r |
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r0 0 |
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I0δ(ϕ − ϕ0). |
(2.21) |
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This equation can be used to determine the unknown coefficients al and bl in the expressions (2.13) and (2.14).
To do this, we substitute us of Equation (2.13) into Equation (2.21), multiply both sides by sin vt ϕ, where vt = tπ/α, and integrate them over ϕ from 0 to α. Note that
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α sin vl ϕ sin vt ϕ dϕ |
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and obtain
with l = t
with l =t t = 1, 2, 3, . . . ,
α |
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Jvl (kr0) |
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Hv(1l |
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Jvl (kr0) = I0. |
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The expression inside the brackets is the Wronskian
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W [Jv (x), Hv(1)(x)] = Jv (x) |
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Hv(1)(x) − Hv(1)(x) |
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From Equations (2.23) and (2.24) it follows that |
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(2.24)
(2.25)
In the case of the hard boundary conditions, we carry out similar manipulations. Substitute uh of Equation (2.14) into Equation (2.21), multiply both sides by cos vt ϕ, and integrate over ϕ from 0 to α. As a result, we obtain
bl = εl |
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= ε3 = · · · = 1. |
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Thus, the coefficients al and bl are found, and the functions us,h are completely determined. Now we can write the final expressions for the total field excited by the external filamentary source.
TEAM LinG
38 Chapter 2 |
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Wedge Diffraction: Exact Solution and Asymptotics |
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In the case of soft boundary conditions, the field us is described by the following |
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In the case of hard boundary conditions, the field uh is determined by |
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These expressions relate to the excitation of the field by a cylindrical wave, with a source term I0δ(r − r0, ϕ − ϕ0) around the wedge in the region 0 ≤ ϕ ≤ α, 0 ≤ r ≤ ∞. They can be modified for excitation by a plane wave.
2.2 TRANSITION TO THE PLANE WAVE EXCITATION
For the Hankel functions with large arguments (kr0 → ∞), one can use the asymptotic expression
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ei(kr0 |
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2 vl − |
4 ) ≈ H0(1)(kr0)e−i |
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π kr0 |
The field Equations (2.27) and (2.28) can then be rewritten for the region r < r0 as
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e−i 2 vl Jvl (kr) sin vl ϕ0 sin vl ϕ, |
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TEAM LinG