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PHYS 342
Fall Semester 2011
Homework No. 2
Due: Sept. 12, 2011
1.
This problem is designed to lead you to the conclusion that
mathematically
speaking,
the superposition of two waves moving in
opposite
directions is exactly equivalent to a
standing wave.
The displacement of the nth mode of oscillation in the ydirection of a stretched string tethered
at both ends of length L with a mass per unit length
μ
subjected to a tensional force T can be
written as a
standing wave
given by
Note that each allowed mode of vibration is specified by the
integer
index n. The amplitude of
each mode of vibration is specified by A
n
. The angular frequency of each mode is specified by
n
.
a)
According to Euler’s formula, you can always write
It is important that you understand these identities, since we will make use of them
often during the semester. By multiplying these two expressions together, show that
you can rewrite y
n
(x,t) as
(,)
s
i
n
c
o
s
nn
n
n
nx
y xt
A
t
L
T
n
L
s
i
n
s
i
n
2
n
n
A
t
t
LL
11
sin
and
cos (
)
22
in x
in x
it
n
ee
t
e
e
Li
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Note the symmetry of this equation. One sine function has an argument that is given by
the difference of two quantities; the other sine function has an argument that is the sum
of the same two quantities.
b)
When discussing waves on a string, the
spatial
variation of the argument of a sine
wave is usually written as
where
λ
n
defines the wavelength of the n
th
mode.
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