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ECE 450 Fall 2009
Homework 10
Solutions
Due: Tuesday, Nov 3
rd
, 2009
1.
The medium is dispersive, i.e., different frequency components will travel at different speeds,
which is determined by the dispersion relation given as
gk
ω
=
.
a.
Given the relationships for
v
p
and
v
g
, express both them as a function of frequency
ω
what the dispersion relation indicates. There are a couple of ways to approach the
problem, all of which lead to the same results.
i.
Given
gk
=
,
2
2
p
gk
g
k
gg
v
kk
ωω
=
=⇒=
=
=
♠
()
2
2
2
2
g
gk
gk
gk
g
g
v
k
=→
=
∂∂
=⇒
=
∂
==
♠
∂
ii.
Æ
Given
gk
=
and the fact that
p
k
v
=
,
2
p
p
p
g
v
g
gv
v
=
=♠
Æ
Proceed with using
k
∂
∂
directly on the dispersion relation to determine
v
g
.
1
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1
1
g
2
2
·
2
2
Using
,
2
g
gk
gk
g
kk
gk
k
g
g
v
k
ω
−
∂∂
TT
==
=
=
∂
♠
∂
** Relationship:
v
g
= 1/2
v
p
and as a side note, notice how both
decreases
as
frequency increases.
b.
So the rock falling into the calm, deep ocean creates a spectrum waves with periods
longer than
T
min
and shorter than
T
max
, i.e.,
im
T
in
ax
m
≤
≤
, (
T
i
exemplifies the period of
an individual frequency component). Given the fact that the period
T
is inverse
proportional to frequency (i.e.,
1
T
∝
), you can equate
x
min
ma
T
→
and
n
max
mi
T
→
ax

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 Spring '08
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 Frequency

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