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1. Consider the following standing wave, with period
T
= 2
π/ω
and wave
length
λ
= 2
π/k
,
y
1
=
A
cos
kx
sin
ωt
a) Suppose this solution describes a standing wave on a string that
has one end at
x
= 0 and the other at
x
=
L
. The end at
x
=
L
is tied down so that
y
= 0 at that point for all times. Find the
relation between
λ
and
L
for the three lowestfrequency standing
waves (i.e. the ﬁrst three normal modes of vibration).
b) Plot the 1st and 2nd normal modes from a) at time
t
=
T/
4 on
the axis provided below, labeling each clearly.
c) Now consider a second standing wave solution,
y
2
=
A
sin
kx
cos
ωt
.
Is the superposition,
y
=
y
1
+
y
2
a standing wave or a traveling
wave? You will ﬁnd it useful to rewrite
y
using one of the trig.
identities below and argue by inspection.
sin(
x
±
y
) = sin
x
cos
y
±
cos
x
sin
y
cos(
x
±
y
) = cos
x
cos
y
∓
sin
x
sin
y
Solution:
a) We must have:
y
(
L,t
) =
A
cos
kL
sin
ωt
= 0.
A
cannot be zero because
there would then be no standing wave. sin
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 Spring '08
 COHN
 Physics

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