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

.