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13.1 Acoustic Waves 277

expression (13.29) for the hard cylinder, it is valid (strictly speaking) for directions π/2 + 0 ≤ ϑ ≤ π − 0 , when point 4 is invisible. However, the contribution of point 4 for the hard cylinder is ka (or kl) times less than the PO field in the direction ϑ = π (or ϑ = π/2) and it can be neglected in the case of large cylinders.

A quantitative influence of the field us,h(1) on the backscattering is illustrated graphically in the next section.

13.1.3 Total Backscattered Field

The total field is the sum

us,h(t) = us,h(0) + us,h(1),

(13.41)

where us(0) = −uh(0) and the terms uh(0), us,h(1) are determined by Equations (13.12), (13.28), and (13.29). Utilizing these approximations, we have calculated the normal-

ized scattering cross-section (13.14) and demonstrated the individual contribution of each term in Equation (13.41). Recall that the field us,h(0) is produced by the uniform scattering source js,h(0) and represents the PO approximation. The field us,h(1) is produced

by the nonuniform source js,h(1) concentrated near the edges and is denoted below as the fringe component of the backscattering. The numerical results are presented in the following for two sets of geometrical parameters of the cylinder: (a) d = 2a = λ, L = 2l = 3λ; and (b) d = 3λ, L = 9λ. Here, d is the diameter and L is the length of the cylinder.

Figure 13.4 Backscattering at a soft cylinder. According to Equation (13.46), the PO curve here also displays the backscattering of electromagnetic waves (with Ex -polarization) from a perfectly conducting cylinder.

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278 Chapter 13 Backscattering at a Finite-Length Cylinder

Figure 13.5 Backscattering at a hard cylinder. According to Equation (13.70), the PO curve here also displays the backscattering of electromagnetic waves (with Hx -polarization) from a perfectly conducting cylinder.

An interesting observation follows from Figures 13.4 to 13.7. Most of the maximums in the soft fringe field are located in the vicinity of the angular positions of the minimums of the PO field. The opposite situation is observed for the hard fringe field: its maximums are positioned near the maximums of the PO field. This observation explains why the minimums of the field scattered by soft cylinders are not as deep as those for the case of hard cylinders.

Figure 13.6 Backscattering at a soft cylinder. According to Equation (13.70), the PO curve here also displays the backscattering of electromagnetic waves (with Ex -polarization) from a perfectly conducting cylinder.

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13.2 Electromagnetic Waves 279

Figure 13.7 Backscattering at a hard cylinder. According to Equation (13.46), the PO curve here also displays the backscattering of electromagnetic waves (with Hx -polarization) from a perfectly conducting cylinder.

13.2ELECTROMAGNETIC WAVES

The original PTD of electromagnetic waves scattered from a finite perfectly conducting cylinder was published in the work of Ufimtsev (1958a, 1962). Below, we present in brief a revised version based on the concept of EEWs.

13.2.1 E-Polarization

The incident wave is defined as

Exinc = E0x eik(z cos γ +y sin γ ),

Eyinc = Ezinc = Hxinc = 0.

(13.42)

The uniform component (1.97) of the induced surface current is determined by

jx(0)disk = 2Y0E0x cos γ e−ikl cos γ eikρ sin γ sin ψ

 

and

 

 

jy(0)disk = jz(0)disk = 0

(13.43)

on the left base of the cylinder (Fig. 13.1), and by

 

jx(0)cyl = −2Y0E0x sin γ sin ψ eik(z cos γ +a sin γ sin ψ ), jy(0)cyl = 2Y0E0x sin γ cos ψ eik(z cos γ +a sin γ sin ψ ),

and

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280 Chapter 13 Backscattering at a Finite-Length Cylinder

jz(0)cyl = 2Y0E0x cos γ cos ψ eik(z cos γ +a sin γ sin ψ )

(13.44)

on the cylindrical part of the surface (−l ≤ z ≤ l , π ≤ ψ ≤ 2π ). Here, Y0 = 1/Z0 is the admittance of free space (vacuum).

(0) (0)

The field Ex generated by the current j is found with the help of Equa-

m m =

tions (1.92) and (1.93), where one should drop off the terms Aϕ,ϑ , because j

−[ˆ ×

E

] =

 

(1)

 

 

 

 

 

n

 

0 due to the boundary condition on a perfectly conducting surface. The

 

 

 

(1)

 

 

= −

) and right (z

=

l)

noununiform (fringe) currents j concentrate near the left (z

 

 

edges. The field Ex

radiated by these currents is calculated in accordance with the

theory developed in Section 7.8. The total scattered field is the sum

 

 

 

 

 

Ex = Ex(0)disk + Ex(0)cyl + Ex(1)left + Ex(1)right .

(13.45)

One can show that

 

 

 

 

 

 

 

 

 

 

Ex(0)disk = us(0)disk ,

Ex(0)cyl = us(0)cyl.

(13.46)

The quantities us(0)disk and us(0)cyl are defined in Section 13.1.1, where one should set u0 = E0x . Therefore, the PO curves in Figures 13.4 and 13.6 for the backscattering of acoustic waves from a soft cylinder also display the backscattering of electromagnetic

waves from a perfectly conducting cylinder.

The fields Ex(1)left and Ex(1)right are calculated by the integration of EEWs, which are the functions of the local spherical coordinates with the origin at an edge point x = a cos ψ , y = a sin ψ . To avoid the possible confusion with the basic coordinates R, ϑ , ϕ of the observation point, we re-denote the local coordinates as r, θ , φ. The necessary preliminary work is to define the local coordinates in terms of the basic coordinates ϑ , ψ .

First, notice that one can use the following approximations

rleft = R + a sin ϑ sin ψ + l cos ϑ ,

rright = R + a sin ϑ sin ψ − l cos ϑ ,

 

 

 

Rˆ

 

 

(13.47)

rleft

rright

≈

y sin ϑ

z cos ϑ

(13.48)

ˆ

≈ ˆ

 

= −ˆ

+ ˆ

 

for the observation point (x = 0, y = −R sin ϑ , z = R cos ϑ ) in the far zone (R

ka2,

Rkl2). Then, we introduce the unit vectors

 

 

 

 

 

θ

= ˆ

 

+ ˆ

 

 

+ ˆ

 

,

 

φ

= ˆ

φ

+ ˆ

φ

 

+ ˆ

φ

z,

 

(13.49)

 

 

 

 

 

 

x θ

y θ

y

z θ

z

 

 

x

y

y

z

 

 

 

 

 

 

ˆ

 

 

x

 

 

 

 

 

 

ˆ

 

 

x

 

 

 

 

 

and find their components from the equations

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Rˆ

·

ˆ

=

 

ˆ

·

[ ˆ

× ˆ] =

 

 

ˆ

·

 

ˆ

= −

(

 

 

 

 

 

 

 

 

 

ˆ

=

(13.50)

 

 

 

 

 

−

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

sin2 ϑ cos2 ψ ,

 

ˆ

× ˆ

 

 

θ

 

0,

θ

 

R

t

 

0,

t

 

 

θ

 

1

 

 

φ

 

R

θ ,

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13.2 Electromagnetic Waves 281

where ˆt = xˆ sin ψ − yˆ cos ψ is the tangent to the edge. According to these equations,

θx = −

 

 

 

 

sin ψ

θy =

 

 

cos ψ cos2 ϑ

 

 

 

 

 

 

 

 

,

 

 

 

 

,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 − sin2 ϑ cos2 ψ

1 − sin2 ϑ cos2 ψ

 

 

 

 

cos ψ sin ϑ cos ϑ

 

'

 

 

 

 

 

 

θz =

 

'

 

,

 

 

 

 

 

 

(13.51)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

'

1

 

 

sin2 ϑ cos2 ψ

 

 

 

 

 

 

 

 

 

 

 

− cos ψ cos ϑ

 

 

 

 

 

sin ψ cos ϑ

 

 

φx = −

 

 

 

 

 

,

φy = −

 

 

 

 

,

'

 

 

 

 

 

1 − sin2 ϑ cos2 ψ

1 − sin2 ϑ cos2

ψ

φz = −

sin ψ sin ϑ

 

 

 

'

 

 

 

 

 

 

 

 

.

 

 

 

 

 

 

 

(13.52)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 − sin2 ϑ cos2 ψ

 

 

 

 

 

 

 

The angle θ is

defined by the equation

 

 

 

 

 

 

 

 

 

 

 

'

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Rˆ · ˆt = cos θ = sin ϑ cos ψ .

(13.53)

In order to define the angles φ and φ0, one should note that they are measured from the illuminated face of the edge in the plane perpendicular to the tangent ˆt to the edge.

By projecting the vectors R

y sin ϑ

 

z cos ϑ and Q

 

ki

 

 

y sin γ

z cos γ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ˆ = −ˆ

 

 

 

 

 

+ ˆ

 

 

 

 

 

 

 

ˆ

= −ˆ

= −ˆ

− ˆ

on this plane, one obtains

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

sin φ = −

 

 

 

 

 

 

 

 

cos ϑ

 

 

 

,

 

 

 

cos φ =

 

 

 

 

 

 

 

 

sin ϑ sin ψ

(13.54)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

'

 

 

 

 

 

 

 

'

 

 

 

 

and

1 − sin2 ϑ cos2 ψ

 

 

 

1 − sin2 ϑ cos2 ψ

 

sin φ0 =

 

 

 

 

 

 

 

 

cos γ

 

 

,

 

 

 

cos φ0 =

 

 

 

 

 

 

 

 

sin γ sin ψ

(13.55)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

−

sin2 ϑ cos2 ψ

 

 

1

−

sin2 ϑ cos2 ψ

for the left edge'(z

 

 

l), and

 

 

 

 

 

 

 

 

 

 

 

'

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

= −

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

cos ϑ

 

 

 

 

 

 

and sin φ = −

 

 

 

 

 

 

sin ϑ sin ψ

 

 

,

 

 

 

cos φ = −

 

 

 

 

 

 

 

 

 

 

 

(13.56)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

'

 

 

 

 

 

 

'

 

 

1 − sin2 ϑ cos2 ψ

 

 

1 − sin2 ϑ cos2 ψ

 

sin φ0 = −

 

 

 

 

 

 

sin γ sin ψ

 

 

 

 

,

 

 

 

cos φ0 =

 

 

 

 

 

 

 

 

 

cos γ

 

 

 

 

 

(13.57)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

'

 

 

 

 

 

 

1 − sin2 ϑ cos2 ψ

 

 

 

1 − sin2 ϑ cos2 ψ

for the right

edge (z l).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

'

 

 

=

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Now, according to Section 7.8, one obtains

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a

 

 

eikR

 

2π

0sin ψ Fθ(1)(ψ , θ , φ) · θx

 

 

 

 

 

 

 

 

Ex(1)left = E0x

 

 

 

 

ei2kl cos ϑ

 

 

 

 

 

 

 

 

 

 

 

2π

 

R

0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(1)

(ψ , θ , φ)

·

θ

 

G(1)(ψ , θ , φ)

·

φ

 

ei2ka sin ϑ sin ψ dψ ,

 

 

+ cos

ϑ cos ψ [Gθ

 

 

 

 

 

 

 

 

x +

φ

 

 

 

 

 

 

 

 

 

 

 

x

]1

 

 

 

 

(13.58)

Ey(1,z)left = Hx(1)left

= 0,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(13.59)

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