Ufimtsev P. Fundamentals of the physical theory of diffraction (Wiley 2007)(348s) PEo
.pdf282 Chapter 13 Backscattering at a Finite-Length Cylinder
and
Ex(1)right = E0x |
a |
|
eikR |
2π |
0sin ψ Fθ(1)(ψ , θ , φ) · θx |
|
|
|
|
|
e−i2kl cos ϑ |
|
|
||
2π |
R |
π |
|
||||
|
|
(1) |
|
(1) |
i2ka sin ϑ sin ψ |
||
+ cos ϑ cos ψ [Gθ (ψ , θ , |
φ) · θx + Gφ (ψ , θ , φ) · φx ]1e |
dψ , |
|||||
(13.60) |
|||||||
Ey(1,z)right = Hx(1)right = 0. |
|
|
(13.61) |
Equations (13.59) and (13.61) show that no cross-polarization takes place in this problem, due to its symmetry with respect to the plane x = 0. In the direction ϑ = π − 0, which is the focal line for EEWs, these equations predict the field
Ex(1)left = E0x |
2 |
f (1) |
|
2 |
, |
2 |
, |
2 |
− g(1) |
2 |
, |
2 |
, |
2 |
|
|
R |
e−i2kl , |
(13.62) |
||||
|
a |
|
|
π |
|
π |
|
3π |
|
π |
|
|
π |
|
3π |
eikR |
|
|
|||||
|
|
f (1) |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
Ex(1)right = E0x |
4 |
0, 0, |
2 |
|
− g(1) 0, 0, |
|
2 |
|
|
R |
ei2kl . |
(13.63) |
|||||||||||
|
a |
|
|
|
|
|
3π |
|
|
3π |
|
|
|
eikR |
|
|
|
|
The above equations allow the complete calculation of the total scattered field (13.45). They involve elementary functions, Bessel functions, and the one-dimensional integrals (13.58) and (13.60), which can be calculated numerically. However, one can avoid this direct integration by introducing approximations similar to (13.22), (13.23) and (13.26), (13.27).
The idea of these approximations is as follows. Away from the focal line (ka sin ϑ 1), the asymptotic evaluation of the integrals (13.58) and (13.60) leads to the following ray asymptotics:
Ex(1)left |
a |
1 |
|
|
[ f (1)(1)ei2ka sin ϑ −iπ/4 |
|
||||||||||
= E0x |
|
|
|
|
√ |
|
|
|
|
|||||||
2 |
|
|||||||||||||||
|
|
π ka sin ϑ |
|
|||||||||||||
|
+ f (1)(2)e−i2ka sin θ +iπ/4] |
eikR |
ei2kl cos ϑ |
(13.64) |
||||||||||||
|
R |
|||||||||||||||
and |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
(1)right |
= E0x |
a |
|
f (1)(3) |
e−i2ka sin ϑ +iπ/4 eikR |
e−i2kl cos ϑ . |
|
|||||||||
Ex |
|
|
√ |
|
|
|
|
(13.65) |
||||||||
2 |
|
R |
||||||||||||||
|
π ka sin ϑ |
|
These are the electromagnetic versions of the acoustic asymptotics (13.18) and (13.25). They reveal again the equivalence relationships existing between the acoustic and electromagnetic diffracted rays,
Ex(1) = us(1), |
if u0 = E0x . |
(13.66) |
As shown above, the focal asymptotics (13.17) and (13.62) for acoustic and electromagnetic waves are different. However, they are small quantities of the order
TEAM LinG
13.2 Electromagnetic Waves 283
(ka)−1, compared to the basic component E0(0x)disk scattered from the left base (disk). Therefore, the approximation (13.28) derived for acoustic waves can also be used for calculation of electromagnetic waves, but with the relative error of the order (ka)−1.
In this case, it is just sufficient to replace in Equation (13.28) the quantity u0 by E0x and us(1) by Ex(1).
13.2.2 H-Polarization
The incident wave
Hxinc = H0x eik(z cos γ +y sin γ ), |
Hyinc,z = Exinc = 0 |
(13.67) |
generates the uniform currents |
|
|
jy(0)disk = −2H0x e−ikl cos γ eikρ sin γ sin ψ , |
jx(0)disk jz(0)disk = 0 |
(13.68) |
on the left base of the cylinder (Fig. 13.1), and |
|
|
jz(0)cyl = −2H0x sin ψ eik(z cos γ +a sin γ sin ψ ), |
jx(0)cyl = jy(0)cyl = 0 |
(13.69) |
on its cylindrical part (−l ≤ z ≤ l, π ≤ ψ ≤ 2π ). One can show that these currents radiate the field
Hx(0) = uh(0) |
(13.70) |
under the condition u0 = H0x , where function uh(0) is determined in Section 13.1.1 and represents the acoustic field scattered at a rigid cylinder. Therefore, the PO curves in Figures 13.5 and 13.7 for the backscattering of acoustic waves from a rigid cylinder also display the backscattering of electromagnetic waves from a perfectly
conducting cylinder. |
|
|
|
|
|
|
|
|
|
|
|
The nonuniform |
|
currents induced |
in the vicinity |
of the left (z |
= − |
l) and |
|||||
|
|
|
|
(1) |
(1)left |
(1)right |
|
|
|||
right (z = l) edges radiate the field Hx |
= Hx |
+ Hx |
|
, which is determined |
|||||||
according to Section 7.8 as |
|
|
|
|
|
|
|||||
|
a eikR |
|
|
|
|
|
|
||||
Hx(1)left = H0x |
|
|
|
ei2kl cos ϑ |
|
|
|
|
|
|
|
2 |
R |
|
|
|
|
|
|
||||
|
2π |
{sin ψ [Gθ(1)(ψ , θ , φ) · φx − Gφ(1)(ψ , θ , φ) · θx ] |
|
|
|||||||
× |
0 |
|
|
|
|||||||
− cos ϑ cos ψ Fθ(1)(ψ , θ , φ) · φx }ei2ka sin ϑ sin ψ dψ , |
(13.71) |
TEAM LinG
Chapter 14
Bistatic Scattering at a
Finite-Length Cylinder
14.1 ACOUSTIC WAVES
The geometry of the problem is shown in Figure 14.1. The diameter of the cylinder is d = 2a, and its length is L = 2l. The incident wave is given by
uinc = u0eik(y sin γ +z cos γ ), |
0 < γ < π/2. |
(14.1) |
The scattered field is evaluated in the plane y0z (ϕ = π/2 and ϕ = 3π/2). It is convenient to indicate the scattering direction by the angle (0 ≤ ≤ 2π ),
ϑ , |
|
ϑ , |
if ϕ |
= |
π/2 |
(14.2) |
= 2π |
− |
if ϕ |
3π/2, |
|||
|
|
|
= |
|
|
where ϑ (0 ≤ ϑ ≤ π ) is the ordinary spherical coordinate of the field point (R, ϑ , ϕ). One should not confuse this angle with the local angle θ used for description of an EEW diverging from the diffraction point ζ at the edge. The relevant local coordinates r, θ , φ were introduced above in Section 13.2.1.
14.1.1PO Approximation
The incident wave (14.1) generates the uniform component js,h(0) of the scattering sources only on the left base (disk) and on the lower lateral part (π ≤ ψ ≤ 2π ) of the cylindrical surface. The scattered field is determined by Equation (1.32). We omit all intermediate calculations and obtain the final expressions for this field
us,h(0) = u0 s,h(0)( , γ ) |
eikR |
, |
(14.3) |
R |
Fundamentals of the Physical Theory of Diffraction. By Pyotr Ya. Ufimtsev
Copyright © 2007 John Wiley & Sons, Inc.
287
TEAM LinG
288 Chapter 14 Bistatic Scattering at a Finite-Length Cylinder
Figure 14.1 Cross-section of the cylinder by the y0z-plane. Dots 1, 2, and 3 are the stationary-phase points.
where
(0) |
|
2 |
|
J1( p) |
|
iq |
|
ikal |
|
sin q |
2π |
|
ip sin ψ |
|
|
||||||||
= ika |
cos γ |
e |
− |
sin γ |
|
e |
sin ψ dψ , |
(14.4) |
|||||||||||||||
s |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||
|
|
p |
|
|
|
π |
|
q |
π |
|
|
||||||||||||
(0) |
|
2 |
|
|
J1( p) |
|
iq |
|
ikal |
|
|
sin q |
2π |
|
|
ip sin ψ |
|
|
|||||
= ika |
cos |
|
− |
|
|
|
e |
sin ψ dψ , |
(14.5) |
||||||||||||||
h |
|
|
e |
|
|
|
sin |
|
|
|
|||||||||||||
|
p |
|
|
π |
q |
π |
|
||||||||||||||||
and |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
p = ka(sin γ − sin ), |
|
|
q = kl(cos − cos γ ). |
|
(14.6) |
The first terms in functions (s,h0), which contain the Bessel function J1( p), relate to the field scattered by the disk, and the second terms describe the scattering at the lateral (cylindrical) surface. These terms were evaluated numerically and their magnitudes are plotted in Figures 14.2 and 14.4.
It follows from the above field expressions that the total scattering cross-section (1.53) is given by
σs,htot = 2A, |
(14.7) |
where |
|
A = π a2 cos γ + 4al sin γ |
(14.8) |
is the area of the shadow beam cross-section.
According to Equations (14.4) and (14.5) the normalized scattering cross-section (13.14) is defined as
2 |
|
|
2 |
|
|
|
|
||||
σnorm(0) ( , γ ) = |
|
s,h(0) |
( , γ ) . |
(14.9) |
|
ka2 |
|||||
|
|
|
|
|
|
This quantity was calculated numerically and the results are presented in Figures 14.2 to 14.5 for the incident wave direction γ = 45◦. Figures 14.2 and 14.4 show the scattering at the individual parts of the cylinder and demonstrate how the total scattered field is formed.
TEAM LinG
14.1 Acoustic Waves 289
Figure 14.2 Scattering at the individual parts of a soft cylinder. According to Equation (14.62), this figure also demonstrates the PO approximation for electromagnetic waves (with Ex -polarization) scattered from the parts of a perfectly conducting cylinder.
14.1.2 Shadow Radiation as a Part of the Physical Optics Field
In Section 1.3.4 it was shown that the shadow radiation is the constituent part of the PO field. It was noted there that this field concentrates in the vicinity of the shadow region. Now we can verify this property by numerical investigation of the shadow radiation generated by the finite-length cylinder. The most appropriate procedure for doing this work would be the direct application of the shadow contour theorem demonstrated in Section 1.3.5. However, we can facilitate our work by utilizing the relationship given in Equation (1.73),
ush = 21 [us(0) + uh(0)], |
(14.10) |
and the numerical results obtained in the previous section for the PO fields us(0) and uh(0). Figure 14.6 shows the spatial distribution of the shadow radiation found in this way. The normalized scattering cross-section (14.9) is plotted here in a decibel
TEAM LinG
290 Chapter 14 Bistatic Scattering at a Finite-Length Cylinder
Figure 14.3 Scattering at a soft cylinder. According to Equation (14.62), this figure also demonstrates the PO approximation for electromagnetic waves (with Ex -polarization) scattered from a perfectly conducting cylinder.
scale. It is clearly seen that the shadow radiation really represents the nature of the scattered field in the forward sector (0◦ ≤ ≤ 90◦). According to Equations (14.62) and (14.75), this figure also demonstrates the shadow radiation and PO field scattered from a perfectly conducting cylinder.
14.1.3PTD for Bistatic Scattering at a Hard Cylinder
Here we consider the scattering only at a hard cylinder. This problem is more important from a practical point of view. According to PTD the scattered field is generated by the uniform ( jh(0)) and nonuniform ( jh(1)) scattering sources induced by the incident wave on the cylinder. The field created by jh(0) represents the PO field investigated in the previous sections. Now we calculate the field generated by that part of jh(1) that concentrates near the circular edges of the cylinder and which is also called the fringe source. Then we will combine both components of the scattered field and provide the results of numerical calculation.
TEAM LinG
14.1 Acoustic Waves 291
Figure 14.4 Scattering at the individual parts of a hard cylinder. According to Equation (14.75), this figure also demonstrates the PO approximation for electromagnetic waves (with Hx -polarization) scattered from the parts of a perfectly conducting cylinder.
The field generated by jh(1) is found by integrating the elementary edge waves introduced in Chapter 7. This field can be written in the following form:
where |
|
|
|
|
|
uh(1) = uh(1)left + uh(1)right |
|
|
|
|
(14.11) |
||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||
uh(1)left = u0 |
a |
|
eikR |
|
|
|
2π |
Fh(1)left (ψ , θ , φ)eip sin ψ dψ |
|
||||||||||||
|
|
|
|
|
|
|
eikl(cos −cos γ ) |
|
(14.12) |
||||||||||||
2π |
|
R |
|
0 |
|||||||||||||||||
and |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||
uh(1)right = u0 |
a eikR |
e− ikl(cos −cos γ ) |
2π |
Fh(1)right (ψ , θ , φ)eip sin ψ dψ . |
(14.13) |
||||||||||||||||
2π |
|
|
R |
|
π |
||||||||||||||||
Here, p = ka(sin γ − sin ) |
and |
the |
angle |
(0 |
≤ |
|
≤ |
2π ) is defined by |
|||||||||||||
(1)left |
|
|
(1)right |
|
|
|
|
||||||||||||||
Equation (14.2). The summands uh |
|
and uh |
|
represent the components related |
TEAM LinG