16.107 L01 Jan 25/02
Beats
Consider the pressure variations ∆p of two sound
waves when detected separately and when detected
simultaneously - can you hear the difference?
Resultant has the average frequency and the sound intensity
varies at a frequency equal to the difference of the two waves
Example
• We wish to tune a piano note to f = 220 Hz
• We have a tuning fork with a frequency of 440Hz
• How do we proceed?
• Frequency difference = 220Hz ,which is too large to
produce beats => need small difference!
• second harmonic of the note is 440 Hz
• if piano is not in tune, say 223 Hz, then 2nd
harmonic and the fork produce beats of 6Hz
• adjust tension until beats disappear
Doppler Effect
• true for both sound and electromagnetic
waves (light, microwaves etc.)
• police radar uses this principle: microwaves
with frequency f directed at oncoming car
and reflected microwaves have higher value
f ` due to relative motion
• frequency shift (f `-f) produces “beats” or
modulation from which the speed is found
• consider a source of sound S and a receiver
D which have speeds vS and vD relative to
the surrounding air
Beats
• Displacements due to the two waves at a particular
location are
• p1=p0 sinω1t and p2=p0 sinω2t
• resultant p(t) = p0 [sinω1t + sin ω2t ]
= p0 [2 cos{(ω1t- ω2t)/2} sin {(ω1t+ ω2t)/2}]
• let ∆ω = (ω1- ω2)
ωav = (ω1+ ω2)/2
•
p(t) = [2p0 cos(0.5∆ωt)] sinωavt
• amplitude modulated “carrier” at average ωav=2πfav
• maximum intensity when cos(.5∆ωt) = ±1=>t=2π/∆ω
• ωbeat = ∆ω = (ω1- ω2) = 2π(f1 -f2) = 2πfbeat
beats
• fbeat = f1 -f2
Doppler Effect
• Consider a travelling wave y(x,t)=ymsin(kx- ωt-φ)
• it has frequency f= ω/2π which is the frequency you
hear if you are at rest with respect to the source
• if you are moving at speed v away or towards the
source, then you hear a different frequency
• similarly, if the source is moving
• e.g. a stationary f= 1000 Hz source
• if you approach at 120 km/hr, you hear a higher
frequency ~ 1096 Hz
• if you are receding, you hear a lower f~ 904 Hz
e.g. A passing siren
Doppler Effect
S emits sound in all directions
with frequency f, wavelenth λ,
and speed v= λ f
All points on a wavefront
have the same phase
Wavefronts are a distance
λ apart
frequency f ` detected
at D is the rate at
which wavefronts arrive
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16.107 L01 Jan 25/02
Doppler Effect
Consider first the case vD=0
In time t, wavefronts move distance vt
# of wavelengths in this distance= vt/λ
rate of detection f ` = #/t = (vt/λ)/t
=v/λ =f
same as the source!
Doppler Effect
Consider now the case vD≠0
In time t, wavefronts move distance
vt to right but detector also moves
distance vDt to the left
Wavefront distance relative to D is
vt +vD t
# of wavelengths in this distance
# = (vt +vD t) / λ
rate of detection f ` = #/t
=(vt +vD t) / λ t = (v +vD) / λ
=[(v +vD) /v](v/ λ)
f ` = f [(v + vD) /v] > f
higher than the source!
Doppler Effect
detector moving
•
•
•
•
•
f ` = f [(v ± vD) /v]
v is speed of wave relative to air
vD is speed of detector relative to air
f is frequency of fixed source
higher approaching
If moving away,
f ` = f [(v - vD) /v] < f
lower than the source!
Let source move at vS≠0 towards stationary detector vD=0
Motion of S changes the wavelength of sound emitted
T=1/f is time between
emission of wavefronts
W1 and W2
In time T, W1 moves
distance vT and source
moves distance vST
At end of time T, W2 is
emitted
Distance bewteen W1 and W2
on right is λ ` = vT - vST
D detects f ` = v/ λ ` = f [v/(v-vS)]>f
higher frequency!
If source receding
f `= f [v/(v+vS)]<f
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