Lecture 09 interference for sound waves. beats. doppler effect

1. Lecture 9
Interference for sound waves.
Beats. Doppler effect.

2. Interference
Let S1, S2 be two sources that emit spherical sound waves in phase.
S1 
P
d1
d2
At point P:
r
s1 (r ,t ) = s1 max cos(kd1 − ωt )
r
s2 (r ,t ) = s2 max cos(kd2 − ωt )
(
Phase difference = k d2 − d1
S2 
Destructive
interference
)
This is what matters…
k ( d2 − d1 ) = noddπ
d2 − d1 = nodd
If both waves have the same amplitude (equal distance to
sources), these points can even have zero intensity!
λ
2

3. + amplitude
- amplitude

4. 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

5. Beats
Consider two harmonic waves meeting at x = 0. Same amplitudes, but
ω2 = 1.15 ω1.
The displacement versus time for each is shown below:
A cos(ω1t )
A cos(ω2t )
C(t) = A(t) + B(t)
Constructive
interference
Destructive
interference

6. DEMO:
Beats
Beats (math)
Video
A cos(ω1t ) + A cos(ω2t ) = 2A cos ( ωLt ) cos ( ωHt )
where
ωL =
1
( ω1 − ω2 )
2
and
ωH =
1
( ω1 + ω2 )
2
cos(ωLt)
Beat 1
Beat 3
Beat 2
Note: What you actually hear (beats) has frequency
fbeat
fL
= = f1 − f2
2

7. Doppler Shift
DEMO:
Whistle
Even better: http://www.lon-capa.org/~mmp/applist/doppler/d.htm

8. Doppler math: moving source
• Speed of sound v is constant.
t=0
vS (source)
• Source emits
λ
• Listener (ear) perceives
λ’
λ ' = λ − v sT
t=T
vST
λ’
v
v vs
= −
f' f f
v
f '=f
v −v s
λ
Front of wave
emitted at t = 0
Source moving with vS
(vS>0 from listener to source)
Stationary listener
v
fL = fS
v +vs

9. Doppler math: moving listener
v (sound)
t=0
v L (listener)
λ
vS
λ ' = λ − v LT '
v
v v
= − L
f' f f'
t = T’
λ’ vLT’
v + vL
f '=f
v
Stationary source.
Source moving with vL
(vL>0 from listener to source)
v + vL
fL = fS
v

10. Moving source and moving listener
v +vL
fL = fS
v + vS
vL, vS > 0 in direction from listener to source
(v > 0 always)
To get signs correct
1) sketch the situation, including a few wavefronts
2) decide whether observed wavelength or period will be shorter or
longer
3) 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

11. ACT: Doppler
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.
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
Source approaching listener: wavefronts are squeezed together: λ↓
Effect must be getting smaller (back to source frequency): f decreases
f↑

12. 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.
B.
C.
D.
E.
f 1 = f2 = f3 = f4
f 2 = f4 , f1 = f3
f 1 , f2 , f3 , f4
f 2 , f4 , f1 , f3
f 4 , f3 , f2 , f1

13. 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.
B.
C.
D.
E.
f 1 = f2 = f3 = f4
f 2 = f4 , f1 = f3
f 1 , f2 , f3 , f4
f 2 , f4 , f1 , f3
f 4 , f3 , f2 , f1
v
v+
2 = 1.5f
f1 = f
v
f3 = f
v
v
v−
2
= 2f
f2 = f
f3 = f
v−
v
v
v+
v
2 = 0.5f
v
2
= 0.67f
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.

14. 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
Air piles up here.
vsource v
High pressure
and density in
front of plane
Large aerodynamic drag
(plane pushes on air, air
pushes back)
“Sound barrier”

15. Supersonic speeds
And what if vsource > v ?
http://www.lon-capa.org/~mmp/applist/doppler/d.htm
Points of constructive
interference are along the red
lines (sides of a cone)
BIG amplitude there!
When this cone
touches the ground…
vsource > v
…a person on the yellow line hears
a very loud sound (sonic boom)

16. Mach number
vsource > v
vsourcet
α
vt
sin α =
vt
v sourcet
=
v
v source
vs
= Mach number
v

17. 2α ~ 130°
Mach ~ 1.1