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Physics 53
Wave Motion 3
Q: What's the difference between a violin and a viola?
A: The viola burns longer.
— Old Musician’s Joke
Standing waves in a string
We consider again two harmonic waves with the same amplitude, wavelength and
frequency, but now moving in
opposite
directions.
We can get the formula for the sum of these waves from the formula in Eq (1) of Wave
Motion 2, by the simple trick of replacing
k
by –
k
in the second wave. The result is
y
(
x
,
t
)
=
2
A
⋅
cos(
kx
−
φ
/2)
⋅
cos(
ω
t
−
/2)
.
Note that the dependences on
x
and on
t
are now in separate factors. This kind of
disturbance does
not
result in a net transport of energy in either direction, although
there is energy
in
it. It is called a
standing wave
. The particles execute SHM with an
amplitude and energy that varies from place to place.
One can produce standing waves by reFection from a boundary between two media,
such as where a string is attached to a wall. The standing waves arise from the
superposition of the incident and reFected waves.
To get equal amplitudes in the two waves, the reFection coef±cient must be 1, which we will assume as an
approximation. As was noted earlier, for a string attached to a wall,
R
is very nearly equal to 1.
We begin with waves in a string ±xed to walls at both ends. Let the string have length
L
,
±xed at walls located at
x
= 0 and
x
=
L
. A ±xed end cannot move, so we must have
y
(0,
t
) = 0 at all times, which by Eq (1) gives
φ
=
π
. The wavefunction can thus be written
y
(
x
,
t
)
=
2
A
⋅
sin(
kx
)
⋅
sin(
t
)
.
We must also have
y
(
L
,
t
)
=
0
at all times, so
sin(
kL
)
=
0
. This can hold only for those
values of
k
that satisfy
kL
=
π
, 2
,3
,
…
Zero or negative values of
kL
do not exist, of course, so only positive multiples of
π
appear.
This condition restricts the values of
k
, and thus of the wavelengths and frequencies of
the standing waves, to the following values:
PHY 53
1
Wave Motion 3
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View Full DocumentString fxed at both ends (or pipe open
at both ends)
λ
n
=
2
L
n
f
n
=
n
⋅
v
2
L
⎫
⎬
⎪
⎪
⎭
⎪
⎪
where
n
=
1,2,3,.
..
Only standing waves obeying these restrictions can exist in the string fxed at both ends.
The values oF
n
give the various “modes” oF oscillation oF the string. They are usually
called
harmonics
. The case
n
= 2, For example, is the 2
nd
harmonic. The 1
st
harmonic is
also called the
fundamental
.
Shown is a string oF length vibrating in the 5th harmonic, at the times when
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 Spring '07
 Mueller
 Physics

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