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Waves on a string

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THIS LECTURE

Waves on a string

- Standing waves

- Dispersive and non-dispersive waves

Travelling waves

No boundaries

x

With boundaries

Standing waves

Two ends fixed

One end fixed

Standing waves

Two ends fixed

Standing waves

Two ends fixed

x

x

Travelling waves

Standing waves

Boundaries

No boundaries

2

2

Each section of the string vibrates with same frequency w

Each section of the string vibrates

with same amplitude A

Each section of the string vibrates

with different phase f = kx

x

x

Travelling waves

Standing waves

Boundaries

No boundaries

2

2

Each section of the string vibrates with same frequency w

Each section of the string vibrates with same frequency w

Each section of the string vibrates

with same amplitude A

Each section of the string vibrates

with different amplitude 2Asin(knx)

Each section of the string vibrates

with different phase f = kx

Each section of the string vibrates

with phase 0 or out of phase by p

Standing waves

One end fixed

Superposition of standing waves

Relative intensities of the harmonics

for different instruments

Playing different instruments

x

x

Dispersive wave: it changes shape

t = 0

t > 0

Dispersive and non-dispersive waves

Non-dispersive wave: it does not change shape

t = 0

t > 0

Two velocities to describe the wave

Group velocity, Vg

Velocity at which the envelope

of wave peaks moves

Phase velocity, Vp

Velocity at which successive

peaks move

For non-dispersive waves Vg =Vp

For dispersive waves VgVp

http://www.isvr.soton.ac.uk/SPCG/Tutorial/Tutorial/Tutorial_files/Web-further-dispersive.htm

Group velocity

If VpVg dispersive wave

If Vp=Vg non-dispersive wave

Group and phase velocity

Phase velocity

Relation between Vg and Vp

Superposition

Wave-packet

Superposition of sinusoidal waves

Sinusoidal waves

w1, k1

w2, k2

w3, k3

Dispersive wave

Sinusoidal waves have different speed

w1/ k1= c1

w2/ k2= c2

w3/ k3= c3

Non-dispersive wave

Sinusoidal waves have the same speed

Wave propagates

with speed c

maintaining its shape

w1/ k1= c

w2/ k2= c

t = 0

w3/ k3= c

t > 0

Wave

changes its shape

t = 0

t > 0

Dispersion relation

c= slope

Real string (e.g. a piano string)

c2

c1

Waves on a string

Ideal string

Vp=w/k=c does not

depend on k

Non-dispersive wave

Vp=w/k=c

depends on k

Dispersive wave

Dispersion relation

Group velocity

Phase velocity

Real string

Ideal string

Problem

Determine phase and group velocity for waves whose dispersion relation is described by :

The resulting wave is given by

Group velocity

2p/Dk

Phase velocity

2p/k

Superposition of sinusoidal waves