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Superposition of Standing Waves on a String

​
t
0
n
1
3
5
7
9
11
13
15
The shapes of the first
n
harmonics are shown for a vibrating string of length
L
fixed at each end. The frequency of the fundamental mode of vibration is
f
1
=v/(2L)
, where
v
is the speed of the wave and the wave function is
y
1
(x,t)=
A
1
sin(
κ
1
x)cos(
ω
1
t+
δ
1
)
.
In general, a vibrating string does not vibrate in a single harmonic mode. The motion is a superposition of several harmonics. The wave function is a linear combination of harmonic functions
y(x,t)=
Σ
n
A
n
sin(
κ
n
x)cos(
ω
n
t+
δ
n
)
, where
κ
n
=2π/λ
and
ω
n
=2π
f
n
e
A
n
e
δ
n
are constants determined by initial conditions of the problem. The initial shape of the string is shown in the lower plot when
t=0
. It is symmetric about the point
x=1/2L
and initial velocity zero throughout the string. The movement of the string after being released is still symmetric with respect to
x=1/2L
. Only the odd harmonics (
n
odd) are excited. The even harmonics are null with
A
n=even
=0
.
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