Wave Interference: Beats

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14 Νοε 2013 (πριν από 4 χρόνια και 8 μήνες)

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1

Wave Interference:
Beats

2

Beats

Previously we considered two interfering
waves with the same
w
. Now consider
two different frequencies.

When waves of two slightly different
frequencies arrive at a point, a detector
(ear?) at that point is subjected to two
different sinusoidal signals. The
superposition of those two signals
produces “beats”.

3

Beat frequency

1 2
1 2 2 1
cos( ) cos( )
1 1
2 cos cos
2 2

m m
m
s s t s t
s t t
w w
w w w w
 
   

2 cos'cos

m
s t t
w 
 

1 2 1 2
1 1
'
2 2
,

w w w  w w
   
4

What we perceive

-2
-1
0
1
2
0
50
100
150
200
250
Time (sec)
amplitude (m)
T
beat

T

1 2
cos'1 -1
2'.
We hear a tone at the average frequency
whose amplitude varies.
The amplitude reaches a maximum
whenever is or , so
the perceived beat frequency is
t
w
w w w

 
5

Doppler (frequency) shift

When the source and receiver are in
relative motion, wave fronts get
compressed or stretched in time.

6

Doppler shift for moving source

If the detector and medium are stationary,

v

is the speed of wave (343 m/s for sound in air),

v
s

is the speed of the source,

f

is the frequency of the wave as emitted and

f ’

is the detected (Doppler Shifted ) frequency.

What do the signs tell us?

'
S
v
f f
v v

7

Doppler shift for moving detector

If the source and medium are stationary,

'
D
v v
f f
v

v

is the speed of wave (343 m/s for sound in air),

v
D

is the speed of the detector,

f

is the frequency of the wave as emitted and

f
’ is the detected (Doppler Shifted ) frequency.

8

Putting source and detector motion
into one equation

We can combine these if we measure
everything in a fixed medium.

'
D
s
v v
f f
v v

D Toward

D Away

S Toward

S Away

9