# Superposition of Standing Waves on a String

Superposition of Standing Waves on a String

The shapes of the first harmonics are shown for a vibrating string of length fixed at each end. The frequency of the fundamental mode of vibration is =v/(2L), where is the speed of the wave and the wave function is (x,t)=sin(x)cos(t+).

n

L

f

1

v

y

1

A

1

κ

1

ω

1

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 , where =2π/λ and =2πee are constants determined by initial conditions of the problem. The initial shape of the string is shown in the lower plot when . It is symmetric about the point and initial velocity zero throughout the string. The movement of the string after being released is still symmetric with respect to . Only the odd harmonics ( odd) are excited. The even harmonics are null with =0.

y(x,t)=sin(x)cos(t+)

Σ

n

A

n

κ

n

ω

n

δ

n

κ

n

ω

n

f

n

A

n

δ

n

t=0

x=1/2L

x=1/2L

n

A

n=even