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1.5 Electromagnetic Waves 29

Equations (4.1.12) and (4.1.13) of Uﬁmtsev (2003) show that in the PO approximation, the ﬁeld backscattered by convex perfectly conducting objects does not depend on the polarization of the incident wave.

Consider another important consequence of the PO approximations (4.1.12) and (4.1.13) of Uﬁmtsev (2003). These equations were derived under the following conditions:

• The incident wave is a plane wave propagating in the direction

ˆ	= ˆ	+ ˆ
ki	y sin γ	z cos γ .
		ˆ = −
• The observation point is in the backscattering direction m			ki (in the plane
			ˆ

yoz (ϕ = −π/2)).

•Equation (4.1.12) is valid for the incident wave with E-polarization, Exinc =

E0x eik( y sin γ +z cos γ ).

•Equation (4.1.13) is valid for the incident wave with H-polarization, Hxinc =

H0x eik( y sin γ +z cos γ ).

In view of these comments, the PO approximations (4.1.12) and (4.1.13) in Uﬁmtsev (2003) can be written as

E(0)

ik eikR

Eince−ikr

cos (ki

n)ds,

(1.98)

= − 2π R

Sil

· ˆ

and

H(0)

ik eikR

Hince−ikr

cos (ki

n)ds.

(1.99)

2π R

Sil

· ˆ

Comparison of these equations with Equation (1.37) reveals the following fundamental relationships, which exist between the PO approximations for backscattered acoustic and electromagnetic waves:

Ex(0) = us(0)	if	Exinc = uinc	(1.100)
and
Hx(0) = uh(0)	if	Hxinc = uinc.	(1.101)

Utilizing the vector equivalency theorems (Uﬁmtsev, 2003) and the idea of Section 1.3.4, one can represent the PO ﬁeld in a form similar to Equation (1.70):

EPO ≡ E(0) = Ereﬂ + Esh, HPO ≡ H(0) = Hreﬂ + Hsh.

(1.102)

TEAM LinG

30 Chapter 1 Basic Notions in Acoustic and Electromagnetic Diffraction Problems

reﬂ reﬂ sh sh

Here, E , H and E , H are the reﬂected ﬁeld and the shadow radiation, respectively. Their far-ﬁeld approximations are

Ereﬂ

ik eikR

Sil {

Hinc

Einc

e−ikr cos ds,

4π R

] · ˆ

+ [ ˆ ×

] · ˆ}

Ereﬂ

ik eikR

Sil {

Hinc

Einc

e−ikr cos ds,

4π R

] · ˆ − [ˆ ×

] · ˆ }

Esh

ik eikR

Sil {

Hinc

Einc

e−ikr cos ds,

4π R

] · ˆ

− [ ˆ ×

] · ˆ}

Esh

ik eikR

Sil {

Hinc

Einc

e−ikr cos ds,

4π R

] · ˆ + [ˆ ×

] · ˆ }

Hreﬂ

= [ R × Ereﬂ]/Z0,

Hsh = [ R × Esh]/Z0.
ϑ (ϕ)	ϑ (ϕ)	increase.
Here, ˆ ˆ is the unit vector in the direction of the angle		increase.
Suppose that the incident wave is given as
Exinc = Z0Hyinc = E0x eikz.

Then, one can derive the following relationships:

(1.103)

(1.104)

(1.105)

(1.106)

(1.107)

(1.108)

Exsh =	ik		eikR			Exreﬂ = 0	(ϑ = 0)
		E0x A			,			(1.109)
	2π			R
for the forward direction, and
			Exsh = 0			(ϑ = π )		(1.110)

for the backscattering direction. Here, A is the area of the shadow region cross-section (Fig. 1.4). Thus,

ExPO ≡ Exsh

= E0x

eikz

(ϑ = 0)

(1.111)

2π

for the forward direction, and

eikR

· zˆ)e

i2kz

= −

E0x

Sil (nˆ

(ϑ = π )

(1.112)

2π

for the backscattering direction.

The Shadow Contour Theorem established in Section 1.3.5 for acoustic waves is also valid for electromagnetic waves.

TEAM LinG

Problems 31

Shadow radiation can be interpreted as the ﬁeld scattered by black bodies. Chapter 1 of Uﬁmtsev (2003) presents explicit expressions and the results of numerical calculation for the ﬁeld scattered by arbitrary 2-D black cylinders and black bodies of revolution.

The deﬁnition of the nonuniform component of the surface sources introduced in Section 1.4 is also applicable for electric surface currents. Its modiﬁcation is presented below in Sections 7.9.1 and 7.9.2.

PROBLEMS

1.1The incident wave uinc = u0 exp[−ik(x cos ϕ0 + y sin ϕ0)] excites the scattering sources js = 2∂uinc/∂y on the illuminated side ( y = +0) of a soft inﬁnite plane. y = 0 (Fig. P1.1). Start with the integral (1.10) and calculate the scattered ﬁeld ussc generated by these sources above and below the plane.

(a) Express the integral over the variable ζ through the Hankel function (3.7). Use the integral representation (3.8) of this function and obtain the Fourier integral for a plane wave.

(b) Consider the total ﬁeld ust = usinc + ussc in the region y < 0 and realize the blocking role of the scattering sources js.

1.2Solve the scattering problem similar to Problem 1.1, but for a hard reﬂecting plane.

1.3	The incident wave Ezinc = E0z exp[−ik(x cos ϕ0 + y sin ϕ0)] excites the surface current
	j	=	2	[ ˆ ×	Hinc	]	on the illuminated side (y	= +		sc
				y					0) of a perfectly conducting inﬁnite plane
	(Fig. P1. 1). Start with Equations (1.87) and (1.89) and calculate the scattered ﬁeld Ez
	generated by these currents above and below the plane.
	(a)		Express the integral over the variable ζ through the Hankel function (3.7). Use the
			integral representation (3.8) of this function and obtain the Fourier integral for a plane
			wave.
	(b)		Consider the total ﬁeld Ezt = Ezinc + Ezsc in the region y < 0 and realize the blocking
			role of the surface currents jz.
1.4	Solve the				problem analogous to Problem 1.3 but with the incident wave					Hzinc =
	H0z exp[−ik(x cos ϕ0 + y sin ϕ0)]. Start with Equation (1.88).

1.5Suppose that the incident wave uinc = u0 exp[ik(x cos φ0 + y sin φ0)] hits a soft strip as shown in Figure 5.1. Use Equation (1.71) and calculate the reﬂected part of the PO ﬁeld scattered by this strip.

(a) Express the integral over the variable ζ through the Hankel function (3.7), apply its asymptotic approximation (2.29), and express the far ﬁeld (r ka2) in closed form.

(b) Estimate the ﬁeld in the directions φ = φ0, φ = π − φ0, φ = π + φ0, and φ = −φ0.

Figure P1.1 Excitation of an inﬁnite plane by an incident wave.

TEAM LinG

32 Chapter 1 Basic Notions in Acoustic and Electromagnetic Diffraction Problems

(c) Apply the optical theorem (5.16) to the ﬁeld in the direction φ = π − φ0 and provide the geometrical interpretation of the total reﬂecting cross-section.

(d) Compute and plot the directivity pattern of the reﬂected ﬁeld, setting a = 2λ, φ0 = 45◦. What are the interesting properties of this ﬁeld?

1.6Solve the problem similar to Problem 1.5, but for a hard strip.

1.7Suppose that the incident wave uinc = u0 exp[ik(x cos φ0 + y sin φ0)] hits a soft strip as shown in Figure 5.1. Use Equation (1.72) and calculate the shadow radiation part of the PO ﬁeld scattered by this strip.

(a) Express the integral over the variable ζ through the Hankel function (3.7), apply its asymptotic approximation (2.29), and calculate the far ﬁeld (r ka2) in closed form.

(b) Estimate the ﬁeld in the directions φ = φ0, φ = π − φ0, and φ = π + φ0.

(c) Apply the optical theorem (5.16) to the ﬁeld in the direction φ = φ0 and give the geometrical interpretation of the total power of the shadow radiation.

(d) Compute and plot the directivity pattern of the shadow radiation, setting a = 2λ, φ0 = 45◦. What are the interesting properties of this ﬁeld?

1.8Is the difference between the reﬂected parts of the PO ﬁeld scattered by soft and hard objects (of the same shape and size) illuminated by the same incident wave.

1.9Is any difference between the shadow parts of the PO ﬁeld scattered by soft and hard objects (of the same shape and size) illuminated by the same incident wave.

1.10The incident wave Ezinc = E0z exp[ik(x cos φ0 + y sin φ0)] hits a perfectly conducting strip as shown in Figure 5.1. Calculate the reﬂected part of the PO scattered ﬁeld.

(a) Start with Equations (1.87), (1.88) and (1.89). Apply je,reﬂ		n	Hinc, jm,reﬂ	=
ˆ ×	inc	= ˆ ×		=
ˆ ×	E . Prepare the integral expression for the reﬂected ﬁeld.
n	E . Prepare the integral expression for the reﬂected ﬁeld.

(b) Express the integral over the variable ζ through the Hankel function (3.7), apply its asymptotic approximation (2.29), and express the far ﬁeld (r ka2) in closed form.

(c) Estimate the ﬁeld in the directions φ = φ0, φ = π − φ0, φ = π + φ0, and φ = −φ0. (d) Apply the optical theorem (5.16) to the ﬁeld in the direction φ = π − φ0 and provide

the geometrical interpretation of the total reﬂecting cross-section.

(e) Compute and plot the directivity pattern of the reﬂected ﬁeld, setting a = 2λ, φ0 = 45◦. What are the interesting properties of this ﬁeld?

1.11The incident wave Ezinc = E0z exp[ik(x cos φ0 + y sin φ0)] hits a perfectly conducting strip as shown in Figure 5.1. Calculate the shadow radiation part of the PO scattered ﬁeld.

(a) Start with Equations (1.87), (1.88) and (1.89). Apply

	je,sh	n	Hinc, jm,sh	n	Einc.
		= ˆ ×		= − ˆ ×
	Prepare the integral expression for the shadow radiation.
(b)	Express the integral over the variable ζ through the Hankel function (3.7), apply its
	asymptotic approximation (2.29), and calculate the far ﬁeld (r					ka2) in closed form.
(c)	Estimate the ﬁeld in the directions φ = φ0, φ = π − φ0, and φ = π + φ0.

(d)	Apply the optical theorem (5.16) to the ﬁeld in the direction φ = φ0 and provide the
	geometrical interpretation of the total power of the shadow radiation.
(e)	Compute and plot the directivity pattern of the reﬂected ﬁeld, setting a = 2λ,
	φ0 = 45◦. What are the interesting properties of this ﬁeld?

TEAM LinG

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