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Physics 2214 Problem Set #6
(Due 11:15 am, Thursday 10/16/08)
1.
Consider the superposition of two harmonic sound waves with equal amplitudes but
slightly
different
k
’s and
ω
’s:
11
2 2
()(
)
(,)
ikx
t
t
sxt
Ae
+
+
=+
where
k
1
>
k
2
and
1
>
2
. The average wavenumber and angular frequency are
( )
( )
12
1
2
22
and
kk
k
ωω
=
+
(a) Show that
s
(
x,t
) can be written in the form
(
)
t
Axte
+
=
and find
A
(
x
,
t
).
Interpretation:
(
)
t
e
+
represents a carrier wave with the average
wavenumber and frequency and
A
(
x
,
t
)—the
envelope
—is an amplitude that depends on
position and time.
Express
A
(
x
,
t
) in terms of
Δ
k
=
k
1
–
k
2
and
Δ
=
1
–
2
, where
Δ
k
ا
k
and
Δ
ا
.
(b) At what speed does a point of constant phase—e.g. a peak of the carrier wave—travel?
This
is called the phase speed
v
ph
.
(c) At what speed does the
envelope
travel? This is called the group speed
v
gr
.
(d) Show that in a dispersionless medium (which means
v
ph
is independent of
k
),
v
ph
=
v
gr
. To see
what happens in a dispersive medium, use the applet at
http://galileo.phys.virginia.edu/classes/109N/more_stuff/Applets/sines/GroupVelocity.html
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 Fall '07
 GIAMBATTISTA,A
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