THIS LECTURE
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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.

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Waves on a string

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Waves on a string

THIS LECTURE

Waves on a string

  • Standing waves

  • Dispersive and non-dispersive waves


Waves on a string

Travelling waves

No boundaries

x

With boundaries

Standing waves

Two ends fixed

One end fixed


Waves on a string

Standing waves

Two ends fixed


Waves on a string

Standing waves

Two ends fixed


Waves on a string

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


Waves on a string

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


Waves on a string

Standing waves

One end fixed


Waves on a string

Superposition of standing waves


Waves on a string

Relative intensities of the harmonics

for different instruments


Waves on a string

Playing different instruments

x

x


Waves on a string

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


Waves on a string

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 VgVp

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


Waves on a string

Group velocity

If VpVg  dispersive wave

If Vp=Vg  non-dispersive wave

Group and phase velocity

Phase velocity

Relation between Vg and Vp


Waves on a string

Superposition

Wave-packet

Superposition of sinusoidal waves

Sinusoidal waves

w1, k1

w2, k2

w3, k3


Waves on a string

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


Waves on a string

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


Waves on a string

Dispersion relation

Group velocity

Phase velocity

Real string

Ideal string


Waves on a string

Problem

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


Waves on a string

The resulting wave is given by

Group velocity

2p/Dk

Phase velocity

2p/k

Superposition of sinusoidal waves


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