Interference of Sound Interference of Sound WavesWaves
InterferenceInterference
2 waves, of the same frequency; out of phase.
Eg. y1=A0sin (kx - t) y2=A0sin (kx - t +)
Then yR=ARsin(kx-t+/2),
and the resultant amplitude is AR=2A0cos(½.
Identical waves which travel different distances will arrive out of phase and will interfere, so that the resultant amplitude varies with location.
Quiz
y
You are located at position y, where you can hear a loud sound - the first maximum in intensity from two speakers.The speakers are then connected ‘out of phase’ (difference of π). What will you hear?
A) no change – same loud soundB) no soundC) something between ‘no sound’ and ‘loud sound’
Example:
Two sources, in phase; waves arrive at P by paths of different lengths:
At P:
1 0 1
2 0 2
sin( )sin( )
y A kx ty A kx t
detectorS1
S2
Px1
x2
Phase difference :
Then (using trig), at detector:
1 2 1 2( ) ( ) ( )kx t kx t k x x k x
2 radians
k x x
1 2
0(2 cos )sin( ( ) )
2 2 2R
x xy A k t
Define x to be the path difference
=
kx terms don’t cancel!
ExampleExample
A pair of speakers is separated by 3.0m and driven by thesame oscillator. The listener walks perpendicular from a point on the centerline 8m away to a distance of 0.35m before reaching the first minimum in sound intenstiy.What is the frequency of the oscillator?
8.0m 0.35m3m
Solution
Intensity IIntensity I
I = Power per unit area Units: W / m2
(the area is measured perpendicular to the wave velocity)Intensity is proportional to (resultant amplitude)2 , since I=P/A and power is proportional to A2
Two sources, each with amplitude Ao, intensity Io ,
phase difference
otant amplitude 2 cos( )
2Rresul A A
2 2
o max4 cos ( ) cos ( )
2 2RI I I
Notes:
1) Maximum IR is 4 x IO
2) Maxima when = 0, 2π, 4π, 6π , …
3) Minima (zero intensity) when = π, 3π, 5π , …
x = ± λ/2, ± 3λ/2, ± 5λ/2,…
x = 0, ± λ, ± 2λ,…
Note: The sources are in phase !!!
2 speakers, same intensity, in phase; f = 170 Hz (so = 2.0 m when speed of sound is 340 m/s)
At the position x=9 m, find the intensity in terms of the
intensity of a single speaker
6 m
x
The detector
Example
I think I hear
something!
Solution
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