252 Chapter 10 Diffraction Interaction of Neighboring Edges on a Ruled Surface
with N = α/π and
F(x) = 1 + i2xe−ix
2
·
sgn(x)∞
2
x
eit dt.
(10.26)
For large arguments |x|
1,
and
w(s, ϕ) ≈ −
F(x) ≈
i
2x2
sin
π
sin
ϕ
ei(s+π/4)
2
N N
%cos
π
ϕ
&2
√
(2s)3/2
.
N2
− cos
π
N
N
(10.27)
(10.28)
One can show that the field u1s(R1, ϕ1) + us(t) and its normal derivative are con-
tinuous at the
shadow boundary (ϕ
1 =
0, ϕ
=
π ). Away from this boundary (when
ϕ
√
cos 2
1), the function (10.21) reduces to the ray asymptotic (10.20).
2kχ sin γ0
Here it is pertinent
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:
(t)
(t)
1
inc
eikR+iπ/4
Eϕ
= −Z0Hϑ
=
Z0Ht
(ζst )g(ϕ, 0, α)
sin2 γ0
√
,
2
2π kR(1 + R/ρ)
(t)
(t)
eq
∂f (ϕ, 0, α)
eikR+iπ/4
Eϑ
= Z0Hϕ
= −Et
(ζst )
sin2 γ0
√
,
∂ϕ0
2π kR(1 + R/ρ)
where
inc
= u01g(0, ϕ01
, α1)
√
eikR10+iπ/4
,
Ht (ζst )
sin γ01
2π kR10(1 + R10/ρ1)
∂Einc(R1
=
=
1
, ϕ1)
R1
R10, ϕ1
0
,
Eteq(ζst ) = − i2kR10 sin γ01 sin γ0
t
∂ϕ1
(10.29)
(10.30)
(10.31)
(10.32)
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10.3 Diffraction of Electromagnetic Waves 253
and
inc
= u01f (ϕ1, ϕ01, α1)
√
eikR10+iπ/4
Et
sin γ01
.
(10.33)
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)
=
1
inc
eikR+iπ/4
Ht
Ht
(ζst )g(ϕ, 0, α)
√
,
(10.34)
2
and
sin γ0
2π kR(1
+ R/ρ)
(t)
eq
∂f (ϕ, 0, α)
eikR+iπ/4
Et
= Et
(ζst )
sin γ0
√
.
(10.35)
∂ϕ0
2π kR(1 + R/ρ)
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,
(10.36)
and
∂usinc
∂Etinc
us = Et ,
if
=
,
(10.37)
∂n
∂n
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)
eikR
sin γ0
√
(10.38)
R(1 + R/ρ)
and
1
∂Einc(ζ
st
)
eikR
Eϑ(t) = Z0Hϕ(t) =
t
Ds(χ , ϕ, γ0)
√
.
(10.39)
ik sin γ0
∂n
R(1 + R/ρ)
TEAM LinG
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).
TEAM LinG
(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
uinc = u0eikφi
(11.1)
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
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
f (ϕ, ϕ0
, α)
ikR
uhpr
= u0eikφi · a g(ϕ, ϕ0
, α)
e
,
(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)
(P)
=
1
g(ψ , 0, α)
eikR
u(m)
(ζ )dζ
=
a
g(ψ , 0, α)u(m)
eikR
(11.3)
h
4π
R
L ¯h
2
¯h R
¯ (m)
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
11.2 Multiple Hard Diffraction 257
waves of the (m + 1) order. More precisely, u¯h(m) is the field of the m-order wave at the diffraction point ζ . One can show that
(1)
pr
1
ikφi
ei(2ka−π/4)
u¯h
≡ uh
=
u0e
g(α, ϕ0, α)
√
(11.4)
2
π ka
and
¯ (m) = ¯ (m−1)
uh uh
These relationships lead to
ei(2ka−π/4)
g(0, 0, α) √ , m = 2, 3, 4, . . . . (11.5) 4 π ka
ei(2ka−π/4)
m
(m)
= u0eik
i
m
−
1
u¯h
φ 2g(α, ϕ0, α) g(0, 0, α)
4√
,
m = 1, 2, 3, . . . .
π ka
-
.
(11.6)
Therefore, the total field of all edge waves on the focal line equals
∞
uhew(P) = uhpr (P) + uh(m)(P)
=
·
m=2
+
4√π ka
R
u0elkφi
a
eikR
g(ϕ, ϕ0, α)
g(ψ , 0, α)g(α,
ϕ0, α)
ei(2ka−π/4)
+
m 3 [
]
4√π ka
m−1
e
−
g(ψ , 0, α)g(α, ϕ0, α)
∞
g(0, 0, α) m−2
. (11.7)
=
Here, the series is the geometric progression that can be converted to its sum. The physical meaning of Equation (11.7) is clear. The first term in the braces relates to the primary edge waves, the second to the secondary waves, and the third term represents the sum of all multiple edge waves of order 3 and higher.
The total scattered field on the focal line also includes the reflected rays (6.187) in front of the object (z < 0) and the shadow radiation (6.228) behind the object (z > 0). This approximation for the scattered field actually represents the incomplete asymptotic expansion, because it includes only the first term in the individual asymptotic expansion for each multiple edge wave. Also, Equation (6.187) is only the first term in the asymptotic expansion for the reflected field.
Expression (11.7) can be used to calculate the total scattering cross-section. In the case of the incident plane wave, uinc = eikz, this quantity is defined as
σh,s =
4π
Im(uh,stot · Re−ikR),
(11.8)
k
TEAM LinG
8ka π ka
258 Chapter 11 Diffracted at a Convex Body of Revolution with a Flat Base
where uh,stot is the total field scattered in the forward direction (ψ = π/2). This field consists of the following components:
•The shadow radiation, which is equivalent to the PO field (for the forward direction) and determined by Equation (6.228), where one should set u0 = 1;
•The primary edge waves generated by the nonuniform/fringe scattering sources
jh,s(1) and determined by Equations (6.41), (6.42) or (11.2), where one should set u0 = 1 and u0 exp(ikφi) = 1, respectively;
•The sum of all multiple edge waves of order 2 and higher.
The substitution of this total field into Equation (11.8) results in the following asymptotic expression:
2
π
σh = 2π a2 +1 +
g %
, 0, α& g(α, ϕ0, α)
ka
2
sin m(2ka − π/4)]
∞
g(0, 0, α)
m
1
,
(11.9)
×
[
]
−
[4m(π ka)m/2
m=1
which is incomplete in the sense mentioned above. Notice that the series in Equation (11.9) equals zero, when 2ka − π/4 = lπ (l = 1, 2, 3, . . .). In this case, all corrections to the first term in Equation (11.9) are determined by the higher-order terms in the individual asymptotic expansions for each multiple edge wave.
11.3MULTIPLE SOFT DIFFRACTION
The primary edge wave excited by the incident wave (11.1) is determined by Equation (11.2). The higher-order edge waves arise due to the slope diffraction of waves running along the flat base of the scattering object. These higher-order edge waves are calculated on the basis of Equation (10.19). For the (m + 1)-order edge wave arriving at point P on the focal line, it can be written in the form
u(m+1)(P)
=
1
∂f (ψ , 0, α) eikR
u(m)dζ
au(m)
∂f (ψ , 0, α)
eikR
. (11.10)
s
2π
∂ϕ0
R
L ¯s
= ¯s
∂ϕ0
R
¯ (m)
Here, us is the amplitude factor of the wave, which is equivalent to the m-order edge
ζ
ζ
−
π
a. This quantity
wave coming to the edge point ζ from its opposite point ¯ =
is calculated with application of Equations (10.18) and (10.20), where one should set γ0 = γ01 = π/2, R = 2a, and ρ = −a. These calculations result in
u¯s(1)
u¯s(m)
= −u0e
ikφi ∂f (α, ϕ0, α) ei(2ka+π/4)
8ka√
,
∂ϕ
π ka
u(m−1) ∂2f (0, 0, α) ei(2ka+π/4) ,
= ¯s
∂ϕ∂ϕ0
√
(11.11)
(11.12)
TEAM LinG
11.3 Multiple Soft Diffraction 259
or
ikφi ∂f (α, ϕ0, α)
m−1
ei(2ka+π/4)
m
(m)
= −u0e
∂2f (0, 0, α)
.
u¯s
8ka√
(11.13)
∂ϕ
∂ϕ∂ϕ0
π ka
After the substitution of (11.13) into Equation (11.10) we obtain
u(m+1)(P)
= −
u
eikφ
i
·
a
eikR
R
s
0
×
∂f (ψ , 0, α) ∂f (α, ϕ0, α)
∂2f (0, 0, α)
m−1
ei(2ka+π/4)
8ka√
∂ϕ0
∂ϕ
∂ϕ ∂ϕ0
π ka
The total field of all edge waves on the focal lines equals
m
.
(11.14)
∞
usew(P) = uspr (P) + us(m)(P)
=
·
m=1
−
R
∂ϕ0
∂ϕ
u0eikφi
a
eikR
f (1)
(ϕ, ϕ0, α)
∂f (ψ , 0,
α) ∂f
(α, ϕ0
, α)
2
m 2
i(2ka π/4) m−1
.
∂ f (0, 0, α)
−
e
+
∞
(11.15)
× m
2
∂ϕ ∂ϕ0
8ka√π ka
=
The series in Equation (11.15) is a geometrical progression.
Now we apply Equation (11.15) for calculation of the total scattering crosssection. In the case of the incident plane wave uinc = eikz, it is defined by
Equation (11.8), where
∞
ustot = ush + us(1) +
us(m).
(11.16)
m=2
Here, ush is the shadow radiation (6.228) (where one should set u0 = 1), the quantity
us(1) = a f (1)(ϕ, ϕ0, α)
eikR
(11.17)
R
is the primary edge wave generated by the nonuniform scattering sources js(1), and the series represents the sum of all edge waves of order 2 and higher. Thus, the total
scattering cross-section of the acoustically soft object equals
+
σs = 2π a2
1 −
2 ∂f (π/2, 0, α) ∂f (α, ϕ0, α)
ka
∂ϕ0
∂ϕ
∂2f (0, 0, α)
m−1
sin[m(2ka + π/4)]
∞
.
(11.18)
× m
=
1
∂ϕ ∂ϕ0
(8ka)m(π ka)m/2
TEAM LinG
260 Chapter 11 Diffracted at a Convex Body of Revolution with a Flat Base
We emphasize again that approximations (11.15) and (11.18) are the incomplete asymptotic expansions in the sense discussed above in Section 11.2. The series in Equation (11.18) equals zero when 2ka + π/4 = lπ (l = 1, 2, 3, . . .). In this case, all corrections to the first term in (11.18) are determined by the higher-order terms in the individual asymptotic expansions for each multiple edge wave.
PROBLEMS
11.1Prove Equations (11.4) to (11.6) for the primary and multiple acoustic edge waves on a hard scattering object. Explain all of the details, including caustic parameters, phase shifts, directivity factors, and fractional coefficients.
11.2Prove Equations (11.11) to (11.13) for the primary and multiple acoustic edge waves on a soft scattering object. Explain all of the details, including caustic parameters, phase shifts, directivity factors, and fractional coefficients.
TEAM LinG
Chapter 12
Focusing of Multiple
Edge Waves Diffracted at
a Disk
The theory presented in this chapter is based on the papers by Ufimtsev (1989, 1991). It represents the extension of the previous chapter to the disk diffraction problem, where it is necessary to take into account the edge waves propagating along both faces of the disk (Fig. 12.1). This problem is complicated by the fact that the wave traveling along one face of the disk generates (due to diffraction at the edge) the higher–order waves not only on the same face but also on the other face. However, its solution can be lightened if we utilize the symmetry of the scattered field. Let us consider the scattering at an arbitrary plate located in the plane z = 0. It follows from Equation (1.10) that
ussc(−z) = ussc(z),
uhsc(−z) = −uhsc(z).
(12.1)
Here, the first equality is obvious and the second is caused by the factor
∂ eikr
=
eikr
n
eikr
n
d
eikr
(
r
n)
d
eikr
(r
n), (12.2)
∂n r
r
r
= − dr r
· ˆ = −
· ˆ = − dr r
· ˆ
ˆ
· ˆ
where rˆ(−z) · nˆ = −rˆ(z) · nˆ.
The geometry of the problem is shown in Figure 12.1. The incident wave is given by uinc = eikz. The scattered field is investigated at the points P on the z-axis, which is the focal line of the edge-diffracted waves.
12.1MULTIPLE HARD DIFFRACTION
The primary edge waves excited by the incident wave directly are given by Equation (11.2), where one should set u0 exp(ikφi) = 1. The (m + 1)-order waves are
Fundamentals of the Physical Theory of Diffraction. By Pyotr Ya. Ufimtsev
