# A Comparison Study for Investigation of Diffracted Waves between Parallel Edges and Edges with Arbitrary Angle

Authors: Jiro Iwashige; Leonard Barolli; Motohiko Iwaida; Saki Kameyama
Journal: Mobile Information Systems
 (2015)

---

## Abstract

Estimation of the scattering of electromagnetic waves by buildings and other obstacles is
very important for wireless communications. In our previous work, we have examined the
diffracted fields by two horizontal edges which make an arbitrary angle. In this paper, we
compare the diffracted fields between parallel edges and edges with arbitrary angle. The numerical
calculation results show that when the angle between two edges increases, relatively strong
orthogonal polarized wave components appear, but the principal components are almost the
same as the incident wave.


---

## Body

## 1. Introduction

Electromagnetic waves can be scattered by edges of the buildings in urban areas or edges of the mountain (ridges) for long communication. Thus, the communication quality deteriorates. Therefore, it is important to estimate propagation path from the viewpoint of the effective utilization of the radio wave resources. However, the prediction of the propagation path is not easy.Until now, there are many research works dealing with estimation of the radio wave propagation path in the urban areas [1, 2]. These papers mainly analyzed and examined the propagation path of radio waves when in the urban area there are many buildings. For the case when the waves exceed building roof, Zhang analyzed propagating radio wave over a large number of buildings [3].In our previous works, we have examined the diffracted fields by two horizontal edges which make an arbitrary angle [4, 5]. But it is almost impossible to find diffraction points on these edges analytically [6], so we used a method which can find the diffraction points by repetition process [5]. However, it should be noted that edges are not always horizontal. Thus, it is necessary to examine the diffracted field by edges which change in both horizontal and vertical directions [7]. We also discussed the fields diffracted by wedges and showed that wedge shapes, materials, and incident polarization do not have significant effect on diffracted fields except near the wedge surfaces [8].In this paper, first we survey the double diffraction by two edges which make an arbitrary angle. Then, we refer to coordinate transformation between a total coordinate system and an edge-fixed one and explain the method for finding the diffraction points on two edges with an arbitrary angle which change in both horizontal and vertical directions. For the analysis of diffracted fields, we used GTD (geometrical theory of diffraction) [9, 10]. We make the analysis by including the slope diffraction considering Holm’s method [11, 12]. The numerical calculations are carried out on perfect conductor. The results show that when the angle between two edges increases, relatively strong orthogonal polarized wave components appear, but the principal components are the same as the incident wave.This paper is organized as follows. In Section2, we present diffraction by two edges containing coordinate transformation and the method of finding the diffraction points and introduce the diffraction by two edges. In Section 3, we present the numerical examples. Finally, some conclusions are given in Section 4.


## 2. Diffraction by Two Edges

### 2.1. Aspect of Double Diffraction

In Figure1(d), the aspect of double diffraction by edges 1 and 2 is shown. We express edge i (i
=
1,2) by e
i, which makes an arbitrary angle in horizontal and vertical planes. The wave from a source S
0
(
x
S
,
y
S
,
z
S
) is diffracted twice, at point Q
1 on e
1 and at point Q
2 on e
2, and it reaches an observation point P
0
(
x
P
,
y
P
,
z
P
). We use three coordinate systems #
0, #
1 and #
2, and the origins of #
1 and #
2 are, respectively, at (
a
1
,
b
1
,
c
1
) and (
a
2
,
b
2
,
c
2
) of coordinate system #
0. The edges e
1 and e
2 agree with y
1-axsis and y
2-axsis of coordinate systems #
1 and #
2. It should be noted that the subscript i in (
x
,
y
,
z
)
i means the coordinate system #
i, but the subscript 0 of #
0 is omitted. Figure 1(a) shows the front view of two wages. Figures 1(b) and 1(c) show the top view and side view of Figure 1(d). The wedges are supposed to rotated by angle ψ
i (i
=
1,2) counterclockwise in a horizontal plane as shown in Figure 1(b) and inclined by angle τ
i in a vertical plane in Figure 1(c) (in Figure 1, ψ
2 and τ
2 are drawn as ψ
2
<
0 and τ
2
<
0).Diffraction by two edges.
(a)
Front view of two wedges
(b)
Top view of two edgese
1 and e
2
(c)
Side view of two edgese
1 and e
2
(d)
Aspect of double diffraction by two wedges


### 2.2. Coordinate Transformation

It is difficult to analyze the diffracted field by edges with an arbitrary angle. But the analysis becomes easier by using coordinate transformation as described below.Let us consider Figure1 to explain the coordinate transformation. The origin of coordinate system #
i (i
=
1,2) is put at (
a
i
,
b
i
,
c
i
) of coordinate system #
0, and #
i is supposed to be rotated by ψ
i counterclockwise around z
i-axis (#
i
z-axis) as shown in Figure 1(b). Furthermore, the z
i-axis is inclined by τ
i around x
i-axis (Figure 1(c)). A point (
x
i
,
y
i
,
z
i
)
i on #
i is obtained from a point (
x
,
y
,
z
) on #
0 by the following equation:
(1)
x
i
y
i
z
i
=
T
i
0
x
-
a
i
y
-
b
i
z
-
c
i
,
T
i
0
=
1
0
0
0
cos
⁡
τ
i
sin
τ
i
0
-
sin
τ
i
cos
⁡
τ
i
cos
⁡
ψ
i
sin
ψ
i
0
-
sin
ψ
i
cos
⁡
ψ
i
0
0
0
1
.
On the other hand, a point (
x
,
y
,
z
) on #
0 is obtained from a point (
x
i
,
y
i
,
z
i
)
i on #
i as follows:
(2)
x
y
z
=
T
0
i
x
i
y
i
z
i
+
a
i
b
i
c
i
,
T
0
i
=
cos
⁡
ψ
i
-
sin
ψ
i
0
sin
ψ
i
cos
⁡
ψ
i
0
0
0
1
1
0
0
0
cos
⁡
τ
i
-
sin
τ
i
0
sin
τ
i
cos
⁡
τ
i
.
In addition, the relationship between a vector A
=
x
^
A
x
+
y
^
A
y
+
z
^
A
z on #
0 and a vector A
i
=
x
^
i
A
i
x
+
y
^
i
A
i
y
+
z
^
i
A
i
z on #
i is shown as follows:
(3)
A
i
=
T
i
0
A
,
A
=
T
0
i
A
i
.


### 2.3. Diffracted Field by Edgee
i

Firstly, we would consider a diffraction by an edgee
i (i
=
1, 2) of a wedge i. The aspect of diffraction at Q
i on an edge e
i on the coordinate system #
i is shown in Figure 2. The wedge which constitutes e
i has the external angle n
i
π and it consists of 0-plane and n-plane which are in the angle of θ
i
0, θ
i
n from the z
i
(
z
i
<
0
) axis. The source point S
0, the observation point P
0, and a unit vector of the current at S
0 are expressed as S
0
(
x
i
S
,
y
i
S
,
z
i
S
)
i, P
0
(
x
i
P
,
y
i
P
,
z
i
P
)
i, and i
^
i of coordinate system #
i. The electromagnetic wave emitted from the source S
0 is diffracted at point Q
i on e
i and reaches the observation point P
0. According to GTD, as the angle α
i
S between the incident path S
0
Q
i and e
i (y
i axis) is equal to the angle α
i
P between the path Q
i
P
0 after the diffraction and e
i [9], the point Q
i
(
0
,
y
Q
i
,
0
)
i is easily obtained as follows:
(4)
y
Q
i
=
ρ
P
0
i
y
i
S
+
ρ
S
0
i
y
i
P
ρ
P
0
i
+
ρ
S
0
i
,
where ρ
P
0
i and ρ
S
0
i are the distances from the e
i to the points P
0 and S
0, respectively. Thus, the distances, the angles, the unit vectors r
^
S
i and α
^
i
S, and so on in Figure 2 can be calculated.Figure 2
Aspect of diffraction by an edgee
i.Therefore, the single diffracted fieldE
i
d
1 at point Q
i on the e
i is obtained by the following equations [10–12]:
(5)
E
i
d
1
=
E
0
e
-
j
k
r
S
i
P
r
S
i
P
r
S
i
P
r
S
i
r
i
P
(
r
^
S
i
×
i
^
i
×
r
^
S
i
)
·
D
¯
S
i
P
,
where E
0 is a constant and k is the wave number. The r
S
i and r
i
P are the distances between S
0 and Q
i and between Q
i and P
0, respectively, and r
S
i
P
=
r
S
i
+
r
i
P. The r
^
S
i and i
^
i are unit vectors of r
S
i and the current at the source S
0. The D
¯
S
i
P is the dyadic diffraction coefficient of the diffracted wave from S
0 to Q
i until it reaches P
0. The diffracted field E
i
d
1 in (5) described by #
i is easily transformed to E
d
1 on #
0 using (3).


### 2.4. Diffraction PointsQ
1 and Q
2 on Two Edges

When edges make arbitrary angle in a plane or they are parallel in different heights, the diffraction points can be found easily [5]. But, when they make arbitrary angle in the space, it is impossible to find the diffraction points analytically. So, we use the following method to find the diffraction points [7] as shown in Figure 3.(1)
Let us assumeQ
1
(
0
,
y
Q
1
,
0
)
1, which is the diffraction point on e
1 as given in preceding subsection, to be a first diffraction point Q
1
(
1
) (subscript 1 means diffraction point on e
1 and superscript (1) means 1st step for finding the last diffraction point).
(2)
The temporary diffraction pointQ
2
(
1
) on the edge e
2 is given as follows (see Figure 3(a)).
(a)
A planeA
1 can be defined containing the point Q
1
(
1
) and e
2.
(b)
S
0 is rotated around e
1 with a constant distance of ρ
S
01 as shown in Figure 2; then S
0 reaches a point S
1
(
x
S
12
,
y
S
12
,
z
S
12
)
2 on the plane A
1.
(c)
Then using the distance fromS
1 to e
2 (ρ
S
12) and the distance from P
0 to e
2 (ρ
P
02), the first temporary diffraction point Q
2
(
1
)
(
0
,
y
Q
21
,
0
)
2 on e
2 is given by
(6)
y
Q
21
=
ρ
P
02
y
S
12
+
ρ
S
12
y
P
02
ρ
P
02
+
ρ
S
12
.
(3)
The 2nd temporary diffraction pointQ
1
(
2
) on the edge e
1 is given as follows.
(a)
A planeA
2 can be constructed using the point Q
2
(
1
) and e
1.
(b)
P
0 is rotated around C
P
2 with the length ρ
P
02 shown in Figure 3(a), and then P
0 reaches a point P
2
(
x
P
21
,
y
P
21
,
z
P
21
)
1 on the plane A
2 as shown in Figure 3(b).
(c)
Usingρ
S
01 and ρ
P
21, the 2nd temporary diffraction point Q
1
(
2
)
(
0
,
y
Q
12
,
0
)
1 on e
1 is given by
(7)
y
Q
12
=
ρ
P
21
y
S
01
+
ρ
S
01
y
P
21
ρ
P
21
+
ρ
S
01
.
(4)
After that, we repeat the processes (2) and (3) untilΔ
y
Q
i
n
=
|
y
Q
i
n
-
y
Q
i
n
-
1
|
(
i
=
1,2
;
n
=
2,3
,
…
) becomes small enough. Then, we can find the last diffraction points Q
i
(
≃
Q
i
(
n
)
).Diffraction points on edgese
2 and e
1.
(a)
PlaneA
1
(b)
PlaneA
2


### 2.5. Double Diffracted Fields

The double diffracted fieldE
d
2 by the point Q
1 on the edge e
1 and the point Q
2 on the edge e
2 is calculated by the following equation considering the slope diffraction [10–12]:(8)
E
d
2
=
E
0
e
-
j
k
r
S
12
P
r
S
12
P
r
S
12
P
r
S
1
r
12
r
2
P
r
^
S
1
×
i
^
1
×
r
^
S
1
·
D
¯
S
12
·
D
¯
12
P
+
j
k
r
12
∂
D
¯
S
12
∂
φ
12
·
∂
D
¯
12
P
∂
φ
21
=
E
(
1
)
d
2
+
E
(
2
)
d
2
,
where r
S
12
P
=
r
S
1
+
r
12
+
r
2
P and φ
i
j
(
i
,
j
=
1
,
2
) is the direction angle at the point Q
i toward the point Q
j measured from the 0-plane of the wedge i. The first term in the parenthesis of (8) is the first-order diffraction (
E
(
1
)
d
2
) and the second term is the second-order diffraction (E
(
2
)
d
2: the so-called slope diffraction). When we execute (8), the dyadic diffraction coefficients D
¯
S
12 and D
¯
12
P are calculated on #
1 and #
2, respectively. Then, they are transformed to #
0 by (3), and the double diffracted fields by e
1 and e
2 are obtained by performing scalar product in (8).


### 2.6. Total Field

The direct fieldE
inc from a source S to the observation point P is obtained as follows:
(9)
E
inc
=
E
0
e
-
j
k
r
S
P
r
S
P
(
r
^
S
P
×
i
^
×
r
^
S
P
)
,
where r
S
P is the distance between S and P, and the r
^
S
P and i
^ are unit vectors of r
S
P and the current at S for the coordinate system #
0.The single diffracted fieldE
i
d
1
(
i
=
1
,
2
) by an edge e
i and double diffracted field E
d
2 by e
1 and e
2 can be obtained by (5) and (8). Considering direct field E
inc from a source S and the field E
ref reflected by surfaces of the wedges, the total electric field E
t at an observation point P is calculated as the sum of these fields:
(10)
E
t
=
E
inc
+
E
ref
+
E
1
d
1
+
E
2
d
1
+
E
d
2
.


## 2.1. Aspect of Double Diffraction

In Figure1(d), the aspect of double diffraction by edges 1 and 2 is shown. We express edge i (i
=
1,2) by e
i, which makes an arbitrary angle in horizontal and vertical planes. The wave from a source S
0
(
x
S
,
y
S
,
z
S
) is diffracted twice, at point Q
1 on e
1 and at point Q
2 on e
2, and it reaches an observation point P
0
(
x
P
,
y
P
,
z
P
). We use three coordinate systems #
0, #
1 and #
2, and the origins of #
1 and #
2 are, respectively, at (
a
1
,
b
1
,
c
1
) and (
a
2
,
b
2
,
c
2
) of coordinate system #
0. The edges e
1 and e
2 agree with y
1-axsis and y
2-axsis of coordinate systems #
1 and #
2. It should be noted that the subscript i in (
x
,
y
,
z
)
i means the coordinate system #
i, but the subscript 0 of #
0 is omitted. Figure 1(a) shows the front view of two wages. Figures 1(b) and 1(c) show the top view and side view of Figure 1(d). The wedges are supposed to rotated by angle ψ
i (i
=
1,2) counterclockwise in a horizontal plane as shown in Figure 1(b) and inclined by angle τ
i in a vertical plane in Figure 1(c) (in Figure 1, ψ
2 and τ
2 are drawn as ψ
2
<
0 and τ
2
<
0).Diffraction by two edges.
(a)
Front view of two wedges
(b)
Top view of two edgese
1 and e
2
(c)
Side view of two edgese
1 and e
2
(d)
Aspect of double diffraction by two wedges


## 2.2. Coordinate Transformation

It is difficult to analyze the diffracted field by edges with an arbitrary angle. But the analysis becomes easier by using coordinate transformation as described below.Let us consider Figure1 to explain the coordinate transformation. The origin of coordinate system #
i (i
=
1,2) is put at (
a
i
,
b
i
,
c
i
) of coordinate system #
0, and #
i is supposed to be rotated by ψ
i counterclockwise around z
i-axis (#
i
z-axis) as shown in Figure 1(b). Furthermore, the z
i-axis is inclined by τ
i around x
i-axis (Figure 1(c)). A point (
x
i
,
y
i
,
z
i
)
i on #
i is obtained from a point (
x
,
y
,
z
) on #
0 by the following equation:
(1)
x
i
y
i
z
i
=
T
i
0
x
-
a
i
y
-
b
i
z
-
c
i
,
T
i
0
=
1
0
0
0
cos
⁡
τ
i
sin
τ
i
0
-
sin
τ
i
cos
⁡
τ
i
cos
⁡
ψ
i
sin
ψ
i
0
-
sin
ψ
i
cos
⁡
ψ
i
0
0
0
1
.
On the other hand, a point (
x
,
y
,
z
) on #
0 is obtained from a point (
x
i
,
y
i
,
z
i
)
i on #
i as follows:
(2)
x
y
z
=
T
0
i
x
i
y
i
z
i
+
a
i
b
i
c
i
,
T
0
i
=
cos
⁡
ψ
i
-
sin
ψ
i
0
sin
ψ
i
cos
⁡
ψ
i
0
0
0
1
1
0
0
0
cos
⁡
τ
i
-
sin
τ
i
0
sin
τ
i
cos
⁡
τ
i
.
In addition, the relationship between a vector A
=
x
^
A
x
+
y
^
A
y
+
z
^
A
z on #
0 and a vector A
i
=
x
^
i
A
i
x
+
y
^
i
A
i
y
+
z
^
i
A
i
z on #
i is shown as follows:
(3)
A
i
=
T
i
0
A
,
A
=
T
0
i
A
i
.


## 2.3. Diffracted Field by Edgee
i

Firstly, we would consider a diffraction by an edgee
i (i
=
1, 2) of a wedge i. The aspect of diffraction at Q
i on an edge e
i on the coordinate system #
i is shown in Figure 2. The wedge which constitutes e
i has the external angle n
i
π and it consists of 0-plane and n-plane which are in the angle of θ
i
0, θ
i
n from the z
i
(
z
i
<
0
) axis. The source point S
0, the observation point P
0, and a unit vector of the current at S
0 are expressed as S
0
(
x
i
S
,
y
i
S
,
z
i
S
)
i, P
0
(
x
i
P
,
y
i
P
,
z
i
P
)
i, and i
^
i of coordinate system #
i. The electromagnetic wave emitted from the source S
0 is diffracted at point Q
i on e
i and reaches the observation point P
0. According to GTD, as the angle α
i
S between the incident path S
0
Q
i and e
i (y
i axis) is equal to the angle α
i
P between the path Q
i
P
0 after the diffraction and e
i [9], the point Q
i
(
0
,
y
Q
i
,
0
)
i is easily obtained as follows:
(4)
y
Q
i
=
ρ
P
0
i
y
i
S
+
ρ
S
0
i
y
i
P
ρ
P
0
i
+
ρ
S
0
i
,
where ρ
P
0
i and ρ
S
0
i are the distances from the e
i to the points P
0 and S
0, respectively. Thus, the distances, the angles, the unit vectors r
^
S
i and α
^
i
S, and so on in Figure 2 can be calculated.Figure 2
Aspect of diffraction by an edgee
i.Therefore, the single diffracted fieldE
i
d
1 at point Q
i on the e
i is obtained by the following equations [10–12]:
(5)
E
i
d
1
=
E
0
e
-
j
k
r
S
i
P
r
S
i
P
r
S
i
P
r
S
i
r
i
P
(
r
^
S
i
×
i
^
i
×
r
^
S
i
)
·
D
¯
S
i
P
,
where E
0 is a constant and k is the wave number. The r
S
i and r
i
P are the distances between S
0 and Q
i and between Q
i and P
0, respectively, and r
S
i
P
=
r
S
i
+
r
i
P. The r
^
S
i and i
^
i are unit vectors of r
S
i and the current at the source S
0. The D
¯
S
i
P is the dyadic diffraction coefficient of the diffracted wave from S
0 to Q
i until it reaches P
0. The diffracted field E
i
d
1 in (5) described by #
i is easily transformed to E
d
1 on #
0 using (3).


## 2.4. Diffraction PointsQ
1 and Q
2 on Two Edges

When edges make arbitrary angle in a plane or they are parallel in different heights, the diffraction points can be found easily [5]. But, when they make arbitrary angle in the space, it is impossible to find the diffraction points analytically. So, we use the following method to find the diffraction points [7] as shown in Figure 3.(1)
Let us assumeQ
1
(
0
,
y
Q
1
,
0
)
1, which is the diffraction point on e
1 as given in preceding subsection, to be a first diffraction point Q
1
(
1
) (subscript 1 means diffraction point on e
1 and superscript (1) means 1st step for finding the last diffraction point).
(2)
The temporary diffraction pointQ
2
(
1
) on the edge e
2 is given as follows (see Figure 3(a)).
(a)
A planeA
1 can be defined containing the point Q
1
(
1
) and e
2.
(b)
S
0 is rotated around e
1 with a constant distance of ρ
S
01 as shown in Figure 2; then S
0 reaches a point S
1
(
x
S
12
,
y
S
12
,
z
S
12
)
2 on the plane A
1.
(c)
Then using the distance fromS
1 to e
2 (ρ
S
12) and the distance from P
0 to e
2 (ρ
P
02), the first temporary diffraction point Q
2
(
1
)
(
0
,
y
Q
21
,
0
)
2 on e
2 is given by
(6)
y
Q
21
=
ρ
P
02
y
S
12
+
ρ
S
12
y
P
02
ρ
P
02
+
ρ
S
12
.
(3)
The 2nd temporary diffraction pointQ
1
(
2
) on the edge e
1 is given as follows.
(a)
A planeA
2 can be constructed using the point Q
2
(
1
) and e
1.
(b)
P
0 is rotated around C
P
2 with the length ρ
P
02 shown in Figure 3(a), and then P
0 reaches a point P
2
(
x
P
21
,
y
P
21
,
z
P
21
)
1 on the plane A
2 as shown in Figure 3(b).
(c)
Usingρ
S
01 and ρ
P
21, the 2nd temporary diffraction point Q
1
(
2
)
(
0
,
y
Q
12
,
0
)
1 on e
1 is given by
(7)
y
Q
12
=
ρ
P
21
y
S
01
+
ρ
S
01
y
P
21
ρ
P
21
+
ρ
S
01
.
(4)
After that, we repeat the processes (2) and (3) untilΔ
y
Q
i
n
=
|
y
Q
i
n
-
y
Q
i
n
-
1
|
(
i
=
1,2
;
n
=
2,3
,
…
) becomes small enough. Then, we can find the last diffraction points Q
i
(
≃
Q
i
(
n
)
).Diffraction points on edgese
2 and e
1.
(a)
PlaneA
1
(b)
PlaneA
2


## 2.5. Double Diffracted Fields

The double diffracted fieldE
d
2 by the point Q
1 on the edge e
1 and the point Q
2 on the edge e
2 is calculated by the following equation considering the slope diffraction [10–12]:(8)
E
d
2
=
E
0
e
-
j
k
r
S
12
P
r
S
12
P
r
S
12
P
r
S
1
r
12
r
2
P
r
^
S
1
×
i
^
1
×
r
^
S
1
·
D
¯
S
12
·
D
¯
12
P
+
j
k
r
12
∂
D
¯
S
12
∂
φ
12
·
∂
D
¯
12
P
∂
φ
21
=
E
(
1
)
d
2
+
E
(
2
)
d
2
,
where r
S
12
P
=
r
S
1
+
r
12
+
r
2
P and φ
i
j
(
i
,
j
=
1
,
2
) is the direction angle at the point Q
i toward the point Q
j measured from the 0-plane of the wedge i. The first term in the parenthesis of (8) is the first-order diffraction (
E
(
1
)
d
2
) and the second term is the second-order diffraction (E
(
2
)
d
2: the so-called slope diffraction). When we execute (8), the dyadic diffraction coefficients D
¯
S
12 and D
¯
12
P are calculated on #
1 and #
2, respectively. Then, they are transformed to #
0 by (3), and the double diffracted fields by e
1 and e
2 are obtained by performing scalar product in (8).


## 2.6. Total Field

The direct fieldE
inc from a source S to the observation point P is obtained as follows:
(9)
E
inc
=
E
0
e
-
j
k
r
S
P
r
S
P
(
r
^
S
P
×
i
^
×
r
^
S
P
)
,
where r
S
P is the distance between S and P, and the r
^
S
P and i
^ are unit vectors of r
S
P and the current at S for the coordinate system #
0.The single diffracted fieldE
i
d
1
(
i
=
1
,
2
) by an edge e
i and double diffracted field E
d
2 by e
1 and e
2 can be obtained by (5) and (8). Considering direct field E
inc from a source S and the field E
ref reflected by surfaces of the wedges, the total electric field E
t at an observation point P is calculated as the sum of these fields:
(10)
E
t
=
E
inc
+
E
ref
+
E
1
d
1
+
E
2
d
1
+
E
d
2
.


## 3. Numerical Examples

In this section, we present the numerical calculation. The calculations were carried out for the case where the origin of coordinate system#
1 is located at (
a
1
,
b
1
,
c
1
)
=
(
-
50
m
,
0
,
h
1
) of coordinate system #
0 and for #
2 at (
a
2
,
b
2
,
c
2
)
=
(
0
,
0
,
h
2
), where h
1 and h
2 are the heights of each origin for the coordinate system #
0 as shown in Figure 1. The angles of edges with y-axes are ψ
1, ψ
2, τ
1, and τ
2 as shown in Figures 1(b) and 1(c), but we do not consider the field diffracted by an edge formed at an intersection of two wedges and the field reflected by the wedge surfaces and the ground. The source S
0 is at (
x
S
,
y
S
,
z
S
)
=
(
-
100
m
,
0
,
h
S
), the observation point P
0 is at (
x
P
,
y
P
,
z
P
)
=
(
x
P
,
0
,
1.5
m
), and E
0
=
1 in (5), (8), and (9).When the difference|
ψ
1
-
ψ
2
| or |
τ
1
-
τ
2
| becomes large, two edges cross each other near to x-axis or z-axis as shown in Figure 1, and the diffraction points Q
1 and/or Q
2 go out the calculable range. So, we firstly checked the limitation of ψ
1, ψ
2 and τ
1
,
τ
2 values. We found that we can nearly calculate diffracted fields when |
ψ
1
-
ψ
2
|
<
9
0
∘ or |
τ
1
-
τ
2
|
<
9
0
∘.In Table1 the aspect of converging the locations of the diffracted points on two edges e
1 and e
2 is shown. Let the heights of the source and the edges be h
S
=
40 m, h
1
=
10 m, and h
2
=
30 m and let the angles ψ
1 and ψ
2 be 3
0
∘ and -
3
0
∘, respectively. Table 1 shows the effect of converging locations of Q
i
(
0
,
y
Q
i
,
0
)
i
(
i
=
1,2
) and the diffraction angles α
i at x
P
=
50 m and 300 m. In this table, n is the number of repetitions, Δ
y
Q
i
n
=
|
y
Q
i
n
-
y
Q
i
(
n
-
1
)
|, and Δ
α
i
=
|
α
i
1
-
α
i
2
|. According to GTD, the angle α
i
1 between the incident ray and e
i is equal to the angle α
i
2 between e
i and the diffracted ray as shown in Figure 2. Table 1(a) is for horizontal edges (τ
1
=
τ
2
=
0
∘) and Table 1(b) is for τ
1
=
1
0
∘ and τ
2
=
-
1
0
∘. The number of repetitions n has a tendency to increase when ψ
i, τ
i, and/or x
p become large. Although the calculations were carried out for many cases of various angles ψ
i, τ
i and various positions of the #i origins, their results show that the number of repetitions n is relatively small.Table 1
Relation betweenΔ
y
Q
n
i, Δ
α
i  (
i
=
1
,
2
), and n. (h
1
=
10 m, h
2
=
30 m and  ψ
1
=
3
0
°
,
ψ
2
=
-
3
0
°).
(a)
τ
1
=
τ
2
=
0
°
x
P [m]
n
Δ
y
Q
n
1 [m]
Δ
y
Q
n
2 [m]
Δ
α
1 [deg]
Δ
α
2 [deg]
50
1
1.0
—
1.8
×
1
0
-
1
1.2
2
2.3
×
1
0
-
2
1.6
4.0
×
1
0
-
3
2.7
×
1
0
-
2
3
4.9
×
1
0
-
4
3.4
×
1
0
-
3
8.7
×
1
0
-
5
5.9
×
1
0
-
4
4
1.1
×
1
0
-
5
7.3
×
1
0
-
5
1.9
×
1
0
-
6
1.3
×
1
0
-
5
300
1
3.9
—
1.5
5.1
2
2.0
×
1
0
-
1
1.3
6.7
×
1
0
-
2
2.4
×
1
0
-
1
3
9.1
×
1
0
-
3
5.6
×
1
0
-
2
3.1
×
1
0
-
3
1.1
×
1
0
-
2
4
4.2
×
1
0
-
4
2.6
×
1
0
-
3
1.4
×
1
0
-
4
5.1
×
1
0
-
4
5
1.9
×
1
0
-
5
1.2
×
1
0
-
4
6.4
×
1
0
-
6
2.3
×
1
0
-
5
(b)
τ
1
=
1
0
°, τ
2
=
-
1
0
°
x
P [m]
n
Δ
y
Q
n
1 [m]
Δ
y
Q
n
2 [m]
Δ
α
1 [deg]
Δ
α
2 [deg]
50
1
5.6
—
3.3
9.4
2
7.5
×
1
0
-
1
2.4
3.8
×
1
0
-
1
1.1
3
8.8
×
1
0
-
2
2.6
×
1
0
-
1
4.4
×
1
0
-
2
1.3
×
1
0
-
1
4
1.0
×
1
0
-
2
3.0
×
1
0
-
2
5.1
×
1
0
-
3
1.5
×
1
0
-
2
5
1.2
×
1
0
-
3
3.5
×
1
0
-
3
5.8
×
1
0
-
4
1.7
×
1
0
-
3
6
1.4
×
1
0
-
4
4.0
×
1
0
-
4
6.8
×
1
0
-
5
2.0
×
1
0
-
4
7
1.6
×
1
0
-
5
4.6
×
1
0
-
5
7.8
×
1
0
-
6
2.3
×
1
0
-
5
300
1
5.8
—
7.8
11.6
2
1.6
5.4
1.9
2.8
3
3.9
×
1
0
-
1
1.2
4.4
×
1
0
-
1
6.8
×
1
0
-
1
4
9.3
×
1
0
-
2
2.7
×
1
0
-
1
1.1
×
1
0
-
1
1.6
×
1
0
-
1
5
2.2
×
1
0
-
2
6.4
×
1
0
-
2
2.5
×
1
0
-
2
3.9
×
1
0
-
2
6
5.3
×
1
0
-
3
1.5
×
1
0
-
2
6.0
×
1
0
-
3
9.3
×
1
0
-
3
7
1.3
×
1
0
-
3
3.6
×
1
0
-
3
1.4
×
1
0
-
3
2.2
×
1
0
-
3
8
3.0
×
1
0
-
4
8.7
×
1
0
-
4
3.4
×
1
0
-
4
5.3
×
1
0
-
4
9
7.2
×
1
0
-
5
2.1
×
1
0
-
4
8.2
×
1
0
-
5
1.3
×
1
0
-
4In [8], we mentioned that the diffracted fields by wedges are not affected too much by their shapes, material, and incident polarization except near the surfaces of wedges. So, we show the following calculation for the perfectly conducting wedges whose inner angles are zero (θ
i
0
=
θ
i
n
=
0
∘
;
i
=
1,2). Figures 4–6 show the change of the total electric field |
E
t
|
=
E
t and its incident and diffracted components with a distance x
P from a surface of the wedge 2; when the wave is incident to the perfectly conducting wedges whose inner angles are θ
i
0
=
θ
i
n
=
0
∘
(
i
=
1,2
), the heights of source and two edges are h
S
=
40 m, h
1
=
10 m, and h
2
=
30 m.Figure 4
Fields diffracted by two wedges and their components for three frequencies (800, 1600, and 3200 MHz) (i
^
=
y
^, h
1
=
h
2
=
30m, and ψ
1
=
ψ
2
=
τ
1
=
τ
2
=
0
∘).Fields diffracted by two wedges for horizontal polarized incidence.
(a)
ψ
1
=
ψ
2
=
τ
1
=
τ
2
=
0
∘
(b)
ψ
1
=
3
0
∘, ψ
2
=
-
3
0
∘, τ
1
=
τ
2
=
0
∘
(c)
ψ
1
=
ψ
2
=
0
∘, τ
1
=
3
0
∘, τ
2
=
-
3
0
∘Fields diffracted by two wedges for vertical polarized incidence.
(a)
ψ
1
=
ψ
2
=
τ
1
=
τ
2
=
0
∘
(b)
ψ
1
=
3
0
∘, ψ
2
=
-
3
0
∘, and τ
1
=
τ
2
=
0
∘
(c)
ψ
1
=
ψ
2
=
0
∘, τ
1
=
3
0
∘, and τ
2
=
-
3
0
∘Figure4 shows the frequency characteristics of the fields when the horizontal polarized wave (i
^
=
y
^) is incident and ϕ
1
=
ϕ
2
=
τ
1
=
τ
2
=
0
∘. In this figure, the fields are the total fields (E
y
t), single diffracted fields (E
2
,
y
d
1) by the e
2, double diffracted components (E
(
1
)
,
y
d
2), and the slope diffracted components (E
(
2
)
,
y
d
2), where the subscript y means y component. E
S
P is the direct field from S to P without wedges. At x
P
=
285m the direct field E
inc appears, so we called the point x
P
=
285.0m SB
i (shadow boundary for an incident wave) and the region near to x
P
=
285m is called a transition region. When the frequency is doubled, outside the transition region, E
2
,
y
d
,
1, E
(
1
)
,
y
d
,
2, and E
(
2
)
,
y
d
,
2 are deceased at 3 dB, 6 dB, and 12 dB, respectively [13]. For example, at x
P
=
100m E
2
,
y
d
,
1 is -
69.3dB, -
72.4dB, and -
75.4dB at 800 MHz, 1600 MHz, and 3200 MHz, and E
(
1
)
,
y
d
,
2 and E
(
2
)
,
y
d
,
2 are attenuated at 6 dB and 12 dB when frequency is doubled.Figure5 shows the fields when the horizontal polarized wave (i
^
=
y
^) is incident on the wedges. Figure 5(a) is the case when two edges e
1 and e
2 are parallel (ψ
1
=
ψ
2
=
τ
1
=
τ
2
=
0
∘). Only horizontal wave E
y is diffracted and the total field E
t
(
=
|
E
y
i
+
E
y
d
1
+
E
(
1
)
,
y
d
2
+
E
(
2
)
,
y
d
2
|
) increases according to distance x
P because in this case the direct field E
inc is added in the range x
P
≥
285.0m.Figure5(b) is the case when ψ
1
=
3
0
∘, ψ
2
=
-
3
0
∘, and τ
1
=
τ
2
=
0. As two edges are rotated in horizontal plane, the orthogonal components (E
x, E
z) are generated, and when the angle between two edges increases, relatively strong orthogonal polarized wave components appear to the incident wave. But the horizontal components (E
y
t, E
2
,
y
d
1, E
(
1
)
,
y
d
2, and E
(
2
)
,
y
d
2) change scarcely from Figure 5(a).Figure5(c) shows the case that two edges are not rotated in horizontal plane but in vertical plane (ψ
1
=
ψ
2
=
0
∘, τ
1
=
3
0
∘, and τ
2
=
-
3
0
∘). Comparing Figure 5(c) with Figure 5(b), the orthogonal (z, x) components are different from each other, but the principal components (E
t, E
y
i, and E
2
,
y
d
1) are almost the same as in Figures 5(a) and 5(b).Figure6 shows fields for vertically polarized incidence (i
^
=
z
^). There are fields diffracted in x and z components. So, the total field E
t is composed by x and z components. In Figure 6(a) the case when the edges e
1 and e
2 are parallel (ψ
1
=
ψ
2
=
τ
1
=
τ
2
=
0
∘) is shown. Comparing Figure 6(a) with Figure 5(a), in Figure 6(a), the x components of the diffracted fields E
2
,
x
d
1, E
(
1
)
,
x
d
2, and E
(
2
)
,
x
d
2 at x
P
=
0 are not zero because of the boundary condition for conductor. Then, the curve of E
t starts with finite value at x
P
=
0, but in Figure 5(a) the E
t is zero. But both E
t curves in Figures 5(a) and 6(a) almost agree in the region where x
P is large. Figure 6(b) is the fields when ψ
1
=
3
0
∘, ψ
2
=
-
3
0
∘, and τ
1
=
τ
2
=
0
∘. The orthogonal components E
2
,
y
d
1, E
(
1
)
,
y
d
2, and E
(
2
)
,
y
d
2 are generated, but the principal components (E
t, E
i, E
2
,
z
d
1, and E
2
,
x
d
1) change a little bit. Figure 6(c) shows the case for ψ
1
=
ψ
2
=
0
∘, τ
1
=
3
0
∘, and τ
2
=
-
3
0
∘. The y component of single diffracted wave E
2
,
y
d
1 is zero, and the double diffracted components (E
(
1
)
,
z
d
2, E
(
1
)
,
x
d
2, etc.) are different from those of Figure 6(b), but the principal components are almost the same as in Figures 6(a) and 6(b).


## 4. Conclusions

In this paper, we considered the diffracted fields by two edges, parallel edges and edges with arbitrary angle. We used three coordinate systems#
0, #
1, and #
2 and considered coordinate transformation. Next, we explained the method to find the diffraction points on two edges. We analyzed the diffracted fields by two edges using the GTD. The numerical calculations were carried out for two wedges of perfect conductor for horizontal and vertical polarized incidence.From the numerical results, we conclude as follows.(i)
The diffraction points on two edges with arbitrary angle in horizontal and vertical planes can be confirmed by few repetitions of our method.
(ii)
The number of repetitions is increased when the angles and/or distance to the observation point are increased.
(iii)
When the frequency is doubled, outside the transition region, the single-, double-, and slope-diffracted fields are deceased 3 dB, 6 dB, and 12 dB, respectively.
(iv)
When the wave is incident perpendicularly to parallel edges, the diffracted fields have the same component as incident polarization, but the orthogonal component wave to incident wave is generated by edges with arbitrary angle.
(v)
When the angle between two edges increases horizontally and/or vertically, relatively strong orthogonal polarized wave components appear, but the principal components are the same as the incident wave.In the future work, we would like to confirm numerical results by experiments.



---

*Source: 268989-2015-02-19.xml*