This document discusses the reflection and transmission of waves at the junction of two strings with different linear densities. It provides equations relating the amplitudes of the incident, reflected, and transmitted waves based on the continuity of displacement and slope at the junction. It also discusses sound as a pressure wave and derives an expression for the speed of sound in a fluid from the definition of pressure as a cosine wave. Finally, it defines the loudness of sound in decibels and calculates differences in loudness for different sound intensities.

1. Physics Helpline
L K Satapathy
Wave Motion Adv. Theory
Amplitudes of reflected & transmitted waves
Sound as a pressure wave
Speed of sound in a fluid
Intensity and loudness of sound

2. Physics Helpline
L K Satapathy
Wave Motion Adv. Theory
Amplitudes of reflected and transmitted waves
The fig. shows two strings AO and BO of equal area of cross-section joined at the
origin O. The composite string is stretched between rigid supports such that a
tension T exists in it.
A transverse wave pulse of amplitude Ai is sent along the string from A . The linear
densities of the parts AO and BO are different such that the speeds of the wave in
AO and BO are v1 and v2 respectively.
OA B
yi
yr
yt
As the pulse reaches O , it is partly reflected along OA and partly transmitted
along OB. Let Ai , Ar and At be the amplitudes of incident, reflected and
transmitted waves respectively.

3. Physics Helpline
L K Satapathy
Wave Motion Adv. Theory
Continuity of displacement at x = 0 gives
sin sin sini r tA t A t A t    
i r ty y y 
. . . (1)i r tA A A  
Continuity of slope at x = 0 gives i try yy
x x x
 
 
  
1 1 2
1 1 2
cos ( ) cos ( ) cos ( )i r tA t x v A t x v A t x v
v v v
  
  
     
            
     
For the incident wave (+ve x-direction)
For the reflected wave (-ve x-direction)
For the transmitted wave (+ve x-direction)
1sin ( )i iy A t x v 
1sin ( )r ry A t x v 
2sin ( )t ty A t x v 

4. Physics Helpline
L K Satapathy
Wave Motion Adv. Theory
1 1 2
2 2
(1) (2) 2 1i t t
v v v
A A A
v v
   
       
   
2
1 2
2
. . . (3)t i
v
A A
v v
 
   
 
2
1 2
2
(1) 1r t i i
v
Also A A A A
v v
 
     
 
2 1
1 2
. . . (4)r i
v v
A A
v v
 
   
 
1 1 2
0 , cos cos cosi r tAt x A t A t A t
v v v
  
  
     
          
     
1
1 1 2 2
. . . (2)
i r t
i r t
A A A v
A A A
v v v v
       

5. Physics Helpline
L K Satapathy
Wave Motion Adv. Theory
Sound as a pressure wave
Consider a sound wave , propagating through a section of the medium
with speed v as shown in the figure.
A BC D
y y+y
x
v A
In the undisturbed state ,
Length of the segment AB = x
Area of cross-section = A
 Volume of the segment V = A.x
When A is displaced to C such that AC = y
B is displaced to D such that BD = y + y
 Increase in length of the segment = y
 Increase in volume of the segment V = A.y
. . . (1)
V A y y
Strain
V A x x
  
  
 

6. Physics Helpline
L K Satapathy
Wave Motion Adv. Theory
Let the equation of the wave is given by sin ( )oy y t x v 
(1) cos ( )oyV y
t x v
V x v


 
     

Again bulk modulus
p V y
B p B B
V V V x
 
      
 
cos ( ) cos ( )o
o
B y
p t x v Or p p t x v
v

     
Where pressure amplitude o
o o
B y
p Bky k
v v
  
   
 
We observe that displacement is represented by a sine wave & pressure by a cosine
wave. Hence a displacement node corresponds to a pressure antinode i.e. where
pressure variation is maximum the displacement is minimum and vice versa.

7. Physics Helpline
L K Satapathy
Wave Motion Adv. Theory
Speed of sound in a fluid M N
p+pp
A
v
x+xx
Consider a sound wave propagating with speed v
through a segment of fluid of area of cross-section A
and bounded by M and N such that MN = x
Let the excess pressures at M and N be p and p + p
 Force towards right at point M = pA
and Force towards left at point N = ( p + p)A
 Net Force on the segment towards right = – A.p
From the previous section , we have sin . . . (1)o
x
y y t
v

 
  
 
cos . . . (2)oB y x
and p t
v v


 
  
 

8. Physics Helpline
L K Satapathy
Wave Motion Adv. Theory
2
2
sin . . sin .o oB y B yx x
p t x t x
v v v v v
 
 
      
               
      
2
2
. .sinoAB y x
F A p x t
v v


 
         
 
Mass of the segment is m = .A.x
 Acceleration
2
2
sin . . . (3)oB yF x
a t
m v v



  
      
2
2
2
(1) sin . . . (4)o
y x
Also a y t
t v
 
  
       
2
2
2 2
(3)&(4) 1o
o
B y B B
y v
v v


  
     

9. Physics Helpline
L K Satapathy
Wave Motion Adv. Theory
Intensity and loudness of sound
For a normal ear , the threshold intensity for audibility is 12 2
10oI W m

And the maximum intensity (threshold of pain) is 2
1I W m
 Loudness of sound of intensity Io is  10 1010log 10log 1 0o
o
o
I
L dB
I
 
   
 
For threshold of pain 12
10 10 1012
1
10log 10log 10log (10 ) 120
10o
I
L dB dB
I 
   
      
  
Loudness of sound of intensity I is defined as 1010log
o
I
L dB
I
 
  
 

10. Physics Helpline
L K Satapathy
Wave Motion Adv. Theory
(iii) Difference in loudness when I2 = 2 I1
1 2
1 10 2 1010log & 10log
o o
I I
L dB L dB
I I
   
    
   
2 1 2
2 1 10 10 10
1
10 log log 10log o
o o o
II I I
L L
I I I I
      
           
       
2
10 10
1
10log 10log 2 10 0.301
I
I
 
    
 
2 1 3.01 3L L dB   
(ii) For I = 100 Io , we have
10 1010log 10log (100) 20 2
o
I
L dB dB bel
I
 
    
 
(i) For I = 10 Io , we have 10 1010log 10log (10) 10 1
o
I
L dB dB bel
I
 
    
 

11. Physics Helpline
L K Satapathy
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