Homework 8 solutions
8.1 Omega beta diagram for the PPP
n
1
3
..
:=
a
0.01
:=
c
3 10
8
⋅
:=
omegacutoff
n
π
n
⋅
a
c
⋅
:=
omegacutoff
2
π
⋅
0
1.5
10
10
×
3
10
10
×
4.5
10
10
×
⎛
⎜
⎜
⎜
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎟
⎟
⎠
=
i
1
1000
..
:=
f
i
100 10
9
⋅
i
1000
⋅
:=
omega
i
f
i
2
⋅
π
⋅
:=
k0
i
omega
i
c
:=
kz
i n
,
omega
i
c
1
omegacutoff
n
omega
i
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
2
−
⋅
:=
kz
i n
,
Re kz
i n
,
(
)
i
Im kz
i n
,
(
)
⋅
−
:=
making sure that
k is beta  j alpha
0
1
.
10
11
2
.
10
11
3
.
10
11
4
.
10
11
5
.
10
11
6
.
10
11
7
.
10
11
1000
500
0
500
1000
1500
2000
2500
Re kz
i 1
,
(
)
Re kz
i 2
,
(
)
Re kz
i 3
,
(
)
Im kz
i 1
,
(
)
Im kz
i 2
,
(
)
Im kz
i 3
,
(
)
k0
i
95 10
9
⋅
191 10
9
⋅
omega
i
I have highlighted the first two cutoff omegas where the propagation constant comes out of being
j alpha to become purely real.
You also see how as frequency goes up kz approaches k0
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only plot real angles
slope
n
kx
n
Re knz
n
(
)
0.0000001
+
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
:=
Then the ray representing the wave travels with the slope:
z
p
p
a
1000
⋅
:=
p
0
1000
..
:=
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 Spring '09
 diaz
 Slope, Frequency

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