# 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 , where is the speed of the wave and the wave function is .

n

L

f=v/(2L)

1

v

y(x,t)=Asin(κx)cos(ωt+δ)

1

1

1

1

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 and 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 .

y(x,t)=ΣAsin(κx)cos(ωt+δ)

n

n

n

n

n

κ=2π/λ

n

ω=2πfeAeδ

n

n

n

n

t=0

x=1/2L

x=1/2L

n

A=0

n=even