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 12 cos cos

2 2

m m

m

s s t s t

s t t

2 cos ' cos ms t t

1 2 1 2

1 1'

2 2 ,

4

What we perceive

-2

-1

0

1

2

0 50 100 150 200 250

Time (sec)

am

pli

tude

(m)

Tbeat

T1 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

5

Doppler (frequency) shift

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

Doppler link

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),

vs 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

vf f

v v

7

Doppler shift for moving detector

If the source and medium are stationary,

' Dv vf f

v

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

vD 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 vf f

v v

D TowardD Away

S Toward

S Away

9

Radar Guns: Detecting Doppler shift

Police radar uses electromagnetic waves (v=3x108 m/s), therefore the relative change in frequency due to reflection from an object travelling (100 m/s) is small.

The best way to detect the difference between the emitted and reflected waves is to add them together in your detector and observe the beats.