Lecture 9Interference for sound waves.

Beats. Doppler effect.

Interference

Let S1, S2 be two sources that emit spherical sound waves in phase.

S1

S2

P

d1

d2

At point P:

1 1max 1( , ) cos( )s r t s kd t

2 2max 2( , ) cos( )s r t s kd t

2 1 oddk d d n Destructive interference 2 1 odd 2

d d n

If both waves have the same amplitude (equal distance to sources), these points can even have zero intensity!

2 1Phase diff erence k d d

This is what matters…

+ amplitude

- amplitude

Interference in real life?

Your stereo equipment does not seem to produce these minimum intensity spots…

• many frequencies at the same time

• multiple reflections on walls, ceiling, furniture…

DEMO: Interference

Beats

Consider two harmonic waves meeting at x = 0. Same amplitudes, but 2 = 1.15 1.

The displacement versus time for each is shown below:

C(t) = A(t) + B(t)Constructiveinterference

Destructiveinterference

1cos( )A t

2cos( )A t

Beats (math)

1 2 1 2

1 1where and

2 2L H

cos(Lt)

1 2cos( ) cos( ) 2 cos cosL HA t A t A t t

DEMO: Beats

Video

Note: What you actually hear (beats) has frequency

beat 1 22Lfff f

Beat 1 Beat 2 Beat 3

Doppler Shift

Even better: http://www.lon-capa.org/~mmp/applist/doppler/d.htm

DEMO: Whistle

Doppler math: moving source

t = 0vS (source)

s' vT

s

'

vv vff f

s

'v

ffv v

L Ss

vff

v v

Source moving with vS (vS>0 from listener to

source) Stationary listener

• Speed of sound v is constant.

vST

Front of wave emitted at t = 0

t = T

• Listener (ear) perceives ’

’

• Source emits

Doppler math: moving listener

v (sound)

t = 0

v L (listener)

vS L' 'vT

L

' '

vv vff f

L'v v

ffv

vLT’

t = T’

’

LL S

v vff

v

Stationary source. Source moving with vL

(vL>0 from listener to source)

Moving source and moving listener

LL S

S

v vff

v v

vL, vS > 0 in direction from listener to source

(v > 0 always)

To get signs correct1) sketch the situation, including a few wavefronts2) decide whether observed wavelength or period will be

shorter or longer3) use this to guide whether frequency increases, decreases 4) keep in mind speed of sound does not depend on what the

source or observer is doing

ACT: Doppler

1. Heard frequency is greater than 500 Hz and increases as train slows down

2. Heard frequency is greater than 500 Hz and decreases as train slows down

3. Heard frequency is less than 500 Hz and increases as train slows down

4. Heard frequency is less than 500 Hz and decreases as train slows down

A train is approaching you as you stand on a platform at a railway station. As the train approaches, it slows down. All the while, the engineer is sounding the horn at a constant frequency of 500 Hz.

Source approaching listener: wavefronts are squeezed together:

f

Effect must be getting smaller (back to source frequency): f decreases

In-class example: Doppler

A source of sound has a characteristic frequency f. The speed of sound is v. Consider the following four scenarios:

1. Static source, vobserver = v/2 toward source

2. Static source, vobserver = v/2 away from source

3. Static observer, vsource = v/2 toward observer

4. Static observer, vsource = v/2 away from observer

Order f1, f2, f3, f4 from lowest to highest.A. f1 = f2 = f3 = f4

B. f2 = f4 , f1 = f3

C. f1 , f2 , f3 , f4

D. f2 , f4 , f1 , f3

E. f4 , f3 , f2 , f1

12 1.5

vv

ff fv

2

2 0.5

vv

ff fv

A source of sound has a characteristic frequency f. The speed of sound is v. Consider the following four scenarios:

1. Static source, vobserver = v/2 toward source

2. Static source, vobserver = v/2 away from source

3. Static observer, vsource = v/2 toward observer

4. Static observer, vsource = v/2 away from observer

Order f1, f2, f3, f4 from lowest to highest.

3 2

2

vff f

vv

3 0.67

2

vff f

vv

A. f1 = f2 = f3 = f4

B. f2 = f4 , f1 = f3

C. f1 , f2 , f3 , f4

D. f2 , f4 , f1 , f3

E. f4 , f3 , f2 , f1

It is NOT option B: 2 and 4 (or 1 and 3) are not equivalent. You need to think about the motion relative to air, too.

Shock waves

What if the source (a plane, for instance) is moving almost at the speed of sound?

http://www.lon-capa.org/~mmp/applist/doppler/d.htm

vsource v

Air piles up here.High pressure and density in front of plane

Large aerodynamic drag (plane pushes on air, air pushes back)

“Sound barrier”

Supersonic speeds

And what if vsource > v ? http://www.lon-capa.org/~mmp/applist/doppler/d.htm

vsource > v

Points of constructive interference are along the red lines (sides of a cone)

BIG amplitude there!

…a person on the yellow line hears a very loud sound (sonic boom)

When this cone touches the ground…

Mach number

vsource > vvsourcet

source source

sinvt v

v t v s Mach number

v

v

vt

2 ~130 Mach ~ 1.1