CBSE Physics/ Lakshmikanta Satapathy/ Phenomenon of Beats/ Doppler Effect/ Determination of frequency of tuning fork/ Loading and filing tuning fork

1. Physics Helpline
L K Satapathy
Wave Motion Theory-5
Phenomenon of Beats
Doppler Effect
S1
O
S2 Sn+1
t=nTt=0 t=T t=nTt=Tt=0
On+1O2O1
vS vO
S
L

2. Physics Helpline
L K Satapathy
Wave Motion Theory-5
Beats : The periodic variation in the intensity of sound due to the superposition of
two sound waves of slightly different frequencies is known as beats.
 At a given position , time variation of displacements are given by
As the two waves are of different frequencies, the phase difference will change
continuously with time. Hence we have omitted the phase constant in the
equations.
1 1 1
2 2 2
sin sin2 . . . (1)
sin sin2 . . . (2)
y A t A f t
and y A t A f t
 
 
 
 
Let the two interfering waves be 1 1 2 2sin ( ) & sin ( )y A t x v y A t x v    
If the frequencies of the interfering waves be f1 and f2 , then what we hear is a
sound of the average frequency = ½ (f1 + f2 ) whose intensity varies slowly with a
frequency (f1 - f2 ) which is known as Beat Frequency.

3. Physics Helpline
L K Satapathy
Wave Motion Theory-5
1 2 1 2sin2 sin2y y y A f t A f t    
 The resultant displacement
1 2 1 2
2 sin2 .cos2
2 2
f f f f
A t t 
    
    
   
1 2
sin2 . . . (3)
2
f f
y R t
 
   
 
 Resultant amplitude R is given by
1 2
2 cos2
2
f f
R A t
 
  
 
 1 22 cos . . . (4)R A f f t  
 Resultant amplitude varies with time.

4. Physics Helpline
L K Satapathy
Wave Motion Theory-5
Intensity of resultant 2
I R
1 2 1 2 1 2 1 2
2 1 1 3 5
, , . . .
2 2 2 2
n
t t
f f f f f f f f

   
   
   1 2 1 2cos 1f f t f f t n       
1 2 1 2 1 2
1 2
0 , , , . .
n
t t
f f f f f f
   
  
 Beat period & Beat frequency
1 2 1 2
1 1
0bt
f f f f
  
 
1 2bf f f 
Also I is min. when    1 2 1 2cos 0 (2 1)
2
f f t f f t n

      
 Beat period & Beat frequency
1 2 1 2 1 2
3 1 1
2 2
bt
f f f f f f
  
   1 2bf f f 
 I is max. when

5. Physics Helpline
L K Satapathy
Wave Motion Theory-5
Determination of unknown frequency of a tuning fork :
Let the frequency of known tuning fork A = f
And the frequency of unknown tuning fork B = f 
When sounded together , let the number of beats produced = m  f – f = m
(1) By loading fork B with wax , its frequency f  decreases
Either f – f = m or f – f = m
 f  = f + m or f  = f – m
(a) If f  = f + m , then m = f – f will also decrease
 By loading B , if m decreases , then f  = f + m

6. Physics Helpline
L K Satapathy
Wave Motion Theory-5
(2) By filing fork B , its frequency f  increases
(a) If f  = f + m , then m = f  – f will also increase
 By filing B , if m increases , then f  = f + m
 By filing B , if m decreases , then f  = f – m
It may be noted that we can obtain the value of f 
also by loading or filing the known tuning fork A
 By loading B , if m increases , then f  = f – m
(b) If f  = f – m , then m = f – f  will increase when f  decreases
(b) If f  = f – m , then m = f – f  will decrease

7. Physics Helpline
L K Satapathy
Wave Motion Theory-5
Doppler effect : The apparent change in the pitch of a note due to the relative
motion of the source of sound and the observer is known as Doppler Effect.
S1
O
S2 Sn+1
t=nTt=0 t=T t=nTt=Tt=0
On+1O2O1
vS vO
S
L
velocity v. S is on the left of observer O. Direction from S to O is taken as +ve.
Both S and O move towards right with velocities vS and vO respectively.
Initially , they are at S1 and O1 such that S1O1 = L
Let the source emit the first wave from S1 at time t = 0
Source S is emitting sound of frequency f , wavelength  , time period T and

8. Physics Helpline
L K Satapathy
Wave Motion Theory-5
Velocity of sound relative to observer
 Time at which the first wave reaches the observer is
The 2nd wave is emitted at t = T , when the source is at S2 and observer at O2 such
that
 The time at which the 2nd wave reaches the observer is
Similarly the source emits the (n+1)th wave at t = nT when the source is at Sn+1 and
observer is at On+1 .
 Distance between them is
Time at which the (n+1)th wave reaches the observer is
ov v 
1
o
Lt
v v


2 2 ( )o sS O L v v T  
2
( )o s
o
L v v T
t T
v v
 
 

1 1 ( )n n o sS O L nT v v    
1
( )o s
n
o
L nT v v
t nT
v v
 
 

Velocity of observer relative to source o sv v 

9. Physics Helpline
L K Satapathy
Wave Motion Theory-5
 Time interval in which the observer receives n waves is
 Apparent time period is
1 1
( )o s
n
o o
L nT v v Lt t nT
v v v v
 
   
 
1 o s s
o o
v v v v
nT nT
v v v v
    
         
1 1nt t
T
n
 
 
s
o
v v
T
v v
 
   
1 1 1 1, s
o
v v
T T
f f f v v f
 
         
o
s
v v
f f
v v
 
    
 Apparent frequency is

10. Physics Helpline
L K Satapathy
Wave Motion Theory-5
Case 1(a) : When S moves towards O, which is at rest.
Case 2(a) : When O moves away from S , which is at rest.
OS
Case 1(b) : When S moves away from O, which is at rest.
OS
Case 2(b) : When O moves towards S, which is at rest.
OS
OS
0 , 0s o
s
vv v f f f f
v v
 
         
0 , 0s o
s
vv v f f f f
v v
 
         
0 , 0 o
s o
v v
v v f f f f
v
 
        
 
0 , 0 o
s o
v v
v v f f f f
v
 
        
 
f  f when dist. SO decreases and f  f when dist. SO increases due to motion of S and O

11. Physics Helpline
L K Satapathy
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