252 Chapter 10 Diffraction Interaction of Neighboring Edges on a Ruled Surface
with N = α/π and
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For large arguments |x| |
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w(s, ϕ) ≈ −
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One can show that the field u1s(R1, ϕ1) + us(t) and its normal derivative are con-
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tinuous at the |
shadow boundary (ϕ |
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1), the function (10.21) reduces to the ray asymptotic (10.20). |
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to mention the review paper by Molinet (2005) related to the |
excitation of two-dimensional |
edge waves by the creeping waves and whispering- |
gallery waves propagating over convex and concave scattering surfaces, respectively.
10.3DIFFRACTION OF ELECTROMAGNETIC WAVES
In a general case, a wave diffracted at edge L1 can be considered as the sum of two
waves with orthogonal polarizations, that is, with the components Ht and Et . Because they play the role of the waves incident on edge L, we denote them as Htinc, Etinc. The
wave with component Htinc can be represented as Equation (10.1) and its diffraction at edge L is calculated in the same way as in Section 10.1. The wave with component Etinc can be represented as Equation (10.14) and its diffraction at edge L is calculated as shown in Section 10.2.
These calculations result in the following ray asymptotics:
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eikR+iπ/4 |
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sin2 γ0 |
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2π kR(1 + R/ρ) |
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∂Einc(R1 |
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Eteq(ζst ) = − i2kR10 sin γ01 sin γ0 |
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(10.29)
(10.30)
(10.31)
(10.32)
10.3 Diffraction of Electromagnetic Waves 253
and
inc |
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eikR10+iπ/4 |
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2π kR10(1 + R10/ρ1) |
In view of Equations (7.149) and (7.150), one can rewrite the ray asymptotics in terms of the components parallel to the tangent to the edge at the diffraction point ζst :
(t) |
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eikR+iπ/4 |
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(ζst )g(ϕ, 0, α) |
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sin γ0 |
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Their comparison with Equations (10.4) and (10.20) reveals the same relationships between acoustic and electromagnetic diffracted rays as those established in the previous sections:
uh = Ht , |
if uhinc = Htinc, |
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at the diffraction point on the scattering edge.
The uniform asymptotics for the diffracted wave (valid for the directions ϑ = π − γ0, 0 ≤ ϕ ≤ α) are described by the electromagnetic versions of Equations (10.6) and (10.21):
Eϕ(t) = −Z0Hϑ(t) = Z0Htinc(ζst )Dh(χ , ϕ, γ0) |
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R(1 + R/ρ) |
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ik sin γ0 |
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of acoustic waves and its normal derivative are continuous
+ (t)
10.1 Show that the field u1h uh
Here, the superscript “t” means that this wave (diffracted at edge L) is radiated by
(t) = (0) + (1)
the surface current, j j j . Together with the incident wave (diverging from edge L1), they form the total field. One can show that the total field and its normal
derivatives are continuous at the shadow boundary for the incident wave (ϕ = π ).
√
Away from this boundary ( kχ sin γ0 cos ϕ2 1), the uniform asymptotics (10.38), (10.39) transform into the ray asymptotics (10.29), (10.30).
Asymptotics (10.38), (10.39) written in the terms of the components Et and Ht completely agree with those of Equations (10.6) and (10.21), and confirm the relationships (10.36), (10.37) between the acoustic and electromagnetic waves.
PROBLEMS
254 Chapter 10 Diffraction Interaction of Neighboring Edges on a Ruled Surface
= (t)
at the shadow boundary (ϕ 0). Functions u1h and uh are defined by Equations (10.1) and (10.6), respectively.
10.2Show that away from the shadow boundary, function (10.6) for acoustic waves transforms asymptotically into the ray approximation (10.4).
10.3Show that the field u1S + uS(t) of acoustic waves and its normal derivative are continuous at the shadow boundary (ϕ = 0). Functions u1S and uS(t) are defined by Equations (10.14) and (10.21), respectively.
10.4Show that away from the shadow boundary, function (10.21) for acoustic waves transforms asymptotically into the ray approximation (10.20).
10.5Show that away from the shadow boundary (ϕ = π ), function (10.38) for electromagnetic waves transforms asymptotically into the ray approximation (10.34).
10.6Show that away from the shadow boundary (ϕ = π ), function (10.39) for electromagnetic waves transforms asymptotically into the ray approximation (10.35).
(m) us,h
Chapter 11
Focusing of Multiple Acoustic Edge Waves Diffracted at a Convex Body of Revolution with a Flat Base
The theory presented below is based on the papers by Ufimtsev (1989, 1991).
11.1 STATEMENT OF THE PROBLEM AND ITS CHARACTERISTIC FEATURES
This problem is illustrated in Figure 11.1, which shows a convex body of revolution excited by the axisymmetrical incident wave
constant along the edge.
The axis of symmetry (z-axis) is a focal line for elementary edge waves/rays. With respect to the observation points P on this axis, each diffraction point at the edge is a point of the stationary phase. The elementary rays propagate in the directions of the edge-diffraction cones, which transform (in this particular case) into the meridian planes. Because of that the directivity patterns of elementary edge rays are expressed in terms of the Sommerfeld functions f and g, as shown in Sections 7.6 and 8.1. The functions f and g are defined in Equations (2.62) and (2.64), and they describe the field generated by the total scattering sources j s,htot = js,h(0) + js,h(1) induced near the edge. Analysis of this field is the main objective of the present chapter.
Here, we ignore the exponentially small multiple edge waves created by the creeping waves (running over the front part, z < 0, of the object) and take into account only the multiple diffraction of edge waves propagating over the flat base. The denotation will be used for the field of multiple edge waves, where the index m = 2, 3, . . .
indicates the order of diffraction.
Fundamentals of the Physical Theory of Diffraction. By Pyotr Ya. Ufimtsev
Copyright © 2007 John Wiley & Sons, Inc.
255
TEAM LinG
256 Chapter 11 Diffracted at a Convex Body of Revolution with a Flat Base
Figure 11.1 Body of revolution excited by the source Q. Focusing of diffracted edge waves occurs at points P on the z-axis. (Reprinted from Ufimtsev (1989) with the permission of The Journal of Acoustical Society of America.)
The first-order (primary) edge waves excited directly by the incident wave (11.1) are determined by the integral expression (8.4) applied to the circular edge. Due to the symmetry of the problem, the integrand in Equation (8.4) is constant and the integration over the edge results in the expression
uspr |
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ikR |
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= u0eikφi · a g(ϕ, ϕ0 |
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(11.2) |
R |
where a is the radius of the edge and α is the external angle between the faces of the edge. This asymptotic expression is valid for any point of observation on the focal
line outside the scattering object, under the condition kR |
1. |
Multiple edge waves us,h(m) with m = 2, 3, . . . (arising due to diffraction of waves running over the flat base) can be found with application of Equations (10.3) and (10.19), where functions Fs,htot transform into functions f and g. For calculation of the edge waves running over the flat base, one can utilize the ray asymptotics (10.4) and (10.20), where one should set γ0 = π/2, R = 2a, and ρ = −a. On their way to the
opposite point of the edge, these waves intersect the focal line and acquire the phase
√
shift equal to (−π/2), which is a direct consequence of the factor 1/ 1 + R/a = 1/√1 − 2a/a = i.
Now one can proceed to the calculation of multiple edge waves.
11.2MULTIPLE HARD DIFFRACTION
According to Equation (10.3), the field created by the (m + 1)-order edge waves on the focal line can be represented in the form
u(m+1) |
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eikR |
u(m) |
(ζ )dζ |
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g(ψ , 0, α)u(m) |
eikR |
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Here, uh denotes the m-order edge wave propagating along the flat base to the opposite point ζ at the edge, where it undergoes diffraction and creates the elementary
TEAM LinG