PHY 102: Waves & Quanta

Topic 5

Sound

John Cockburn (j.cockburn@... Room E15)

•Sound Waves

•Speed of sound in fluids in fluids, gases and solids

•Doppler Effect

•Beats

Sound waves

• Unlike waves on a string (transverse waves), sound waves are longitudinal waves

•Particles undergo SHM about equilibrium positions PARALLEL to direction of wave propagation

•Compression/rarefaction of the medium through which the sound wave travels………………………………………………

•Wave travelling from left to right/right to left still described by:

)/cos(),( tkxAtxy

But this time, y is in same direction as x……

Sound waves

(Ear, for exampleis sensitive to pressure, not dislacement Fluctuations)

Speed of sound

In a fluid:

B

v Bulk modulus

density

In a solid rod:

Y

v Young’s modulus

Speed of sound

The Doppler Effect

• If we stand still near a source of sound, and the source is also not moving we hear sound of a certain frequency f0

•If either, or both, the listener and the source are moving relative to one another, then a different frequency fL is heard by the listener.

•In general, if the listener and the source are moving towards each other, fL > f0, ie a higher pitch is heard.

•If the listener and the source are moving away from each other, fL<f0, ie a lower pitch is heard.

•This is known as the Doppler Effect

•Ambulance sirens, car horns, etc. etc.

Explanation of the Doppler Effect

fv Listener moving towards stationary source with velocity vL

Explanation of the Doppler Effect

fvvfv L )(

Effective velocity of sound heard by listener = vL+v:

0f

v

)( Lvv

f

v

vvff L )(0

Moving listener, stationary source

Moving Source

It takes a time T for one complete cycle of the wave to be emitted.(T = 1/f0)

During this time, the wavefront has moved a distance vT, and the source has moved a distance vsT

The wavelength is:

0

)()(

f

vvvvT ss

0

)()(

f

vvvvT ss

In front of source

Behind source

Moving listener approaching moving source

)( Lvv

f

(Stationary source, earlier slide)

)(

)(0

s

L

vv

vvff

(Modified λ, due to moving source)

And now the good news……………………

)(

)(0

s

L

vv

vvff

Works for ALL relative motions of source and listener in acoustic Doppler shift calculations, PROVIDED you DEFINE the positive direction as the listener to source direction, and set the signs of vL and vs accordingly……………………………………………………………………………………

Example calculations

Two train whistles A and B each have a frequency of 392 Hz. A is stationary, while B is moving to the right, away from A, at a speed of 35ms-1. A listener is between the 2 trains and is also moving to the right, with a speed of 15ms-1. What frequencies does the listener hear from whistles A and B?

(speed of sound in air = 340ms-1)

Whistle APositive direction

)(

)(0

s

L

vv

vvff

)0(

)(0

v

vvff L

Whistle BPositive direction

)(

)(0

s

L

vv

vvff

)35340(

)15340(0

ff

A police car (whose driver has clearly been watching too much Starsky and Hutch), with a 300Hz siren is approaching a warehouse at 30ms-1 with the intention of crashing through the door. What frequency does the driver hear reflected from the warehouse?

(speed of sound in air = 340ms-1)

Two part question………………………..

In the first part, treat the warehouse door as the “listener”…..

)(

)(0

s

L

vv

vvff

Sound at this frequency is reflected back towards the police car. This becomes the “new” f0 for the 2nd part of the question…………………..

Now, the warehouse door is the stationary source of the reflected sound……..

0)(

)0(0 v

vv

vv

vff L

s)(

)(0

s

L

vv

vvff

Beats

These occur from the superposition of 2 waves of close, but different frequency:

babeat fff