Wave Motion Theory Part1

Physics Helpline
L K Satapathy
Wave Motion Theory 1
Wave function
Formation of Waves in a string
Sinusoidal Wave Functions
Condition for Wave Function
Particle velocity and slope

Physics Helpline
L K Satapathy
Wave Motion :
It is a kind of disturbance which travels through a material medium due to
repeated periodic vibration of the particles of the medium about their mean
positions, the disturbance being handed over from one particle to the next without
any net transport of the medium.
Transverse Waves :
Particles of the medium execute SHM about their mean position
in a direction perpendicular to the direction of propagation of the
wave. It travels through a medium in the form of crests and
troughs.
Longitudinal Waves :
Particles of the medium execute SHM about their mean position
in the direction of propagation of the wave. It travels through a
medium in the form of compressions and rarefactions.

Physics Helpline
L K Satapathy
Formation of a Wave in a stretched string :
String fixed at P and held by
hand at O. Set into periodic
motion with time period T.
(i) After ¼ time period
(ii) After ½ time period
(iii) After ¾ time period
(iv) After one time period
(O moved up and disturbance reached A)
(O moved down to mean position, A moved up and
disturbance reached B)
(O moved down, A moved down, B moved up and
disturbance reached C)
(O moved up, A moved down, B moved down, C
moved up and disturbance reached D)
O P
O P
O P
O P
O PT
4T
2T
3 4T
 0t 
x  0 /4 /2 3/4 
O A B C D
4OA AB BC CD
OD


   
 

Physics Helpline
L K Satapathy
Wave Function : A wave pulse is moving along +ve x-direction with speed (v) as
shown in figure. Particles of the medium oscillate in SHM along y–direction
O Px
Displacement of particle at position x and time t = y(x ,t)
It is a function of both x and t . We need to define this function.
 At O , displacement = y(0,t) (depends on t only)
(0, ) ( ) ... (1)y t f t We write
The same displacement occurs at P at time (t + x/v)
[ When position increases by x , time increases by x/v ]
( , ) (0, )y x t x v y t  

Physics Helpline
L K Satapathy
 Eqn. of a wave propagating in the +ve x–direction is given by
( , ) ( ) ... (3)y x t f vt x 
 Eqn. of a wave propagating in the – ve x-direction is given by
( , ) ( ) ... (4)y x t f vt x 
 In general , we write ( , ) ( ) ... (5)y x t f vt x 
Using eqn.(1) we get ( , ) ( ) . . . (2)y x t f t x v  
Since v is const. we can write ( , ) ( )y x t f vt x 
Similarly , displacement y(x .t) at P was at O at time (t – x/v)
[ When position decreases by x , time decreases by x/v ]
( , ) (0, )y x t y t x v  

Physics Helpline
L K Satapathy
Condition for Wave Function :
(2) ( , ) ( )y x t f t x v  
Partially differentiating w.r.to t
( ).(1) ( ) . . . (6)
y
f t x v f t x v
t

    

Again partially differentiating w.r.to t
2
2
( ).(1) ( ) . . . (7)
y
f t x v f t x v
t

    

Partially differentiating w.r.to x
1 1
( ).( ) ( ) . . . (8)
y
f t x v f t x v
x v v

      

Again partially differentiating w.r.to x
2
2 2
1 1 1
( ).( ) ( ) . . . (9)
y
f t x v f t x v
x v v v

      


Physics Helpline
L K Satapathy
2
2
(7) ( )
y
f t x v
t

  

2
2 2
1
(9) ( )
y
f t x v
x v

  

2 2
2 2 2
1
. . . (10)
y y
x v t
 
 
 
[particle velocity]
. . . (11)
y y
v
t x
 
  
 
Also [slope]
Relation between particle velocity and slope
(6) ( )
y
f t x v
t

  

1
(8) ( )
y
f t x v
x v

   

Necessary condition for a
function to represent a wave
Particle velocity = – (Wave velocity)(Slope)

Physics Helpline
L K Satapathy
O
v
Consider a wave propagating along +ve x-direction
Also from the figure it can be seen that slope at O is +ve
 A crest has passed point O.
 Particle velocity = – (+ve)(+ve) = – ve
 It has to be followed by a trough
Understanding from graph
 Wave velocity (v) is +ve
(11)
y y
v
t x
 
  
 
Alternate method :
 Particle at O will move downward
 Particle velocity at O is – ve

Physics Helpline
L K Satapathy
Sinusoidal Wave Functions :
(2) ( , ) ( )y x t f t x v  
 We write sin ( ) . . . (12)y A t x v 
Relations to be used :
2 2
, 2K T K v
T T K
   
   

       
(12) sin sin( ) . . . (13)y A t x y A t Kx
v

 
 
      
 
2 2
(13) sin sin2 . . . (14)
t x
y A t x y A
T T
 

 
   
        
   
 
2 2
(14) sin sin . . . (15)y A t x y A vt x
T
  
 
 
      
 
[ /v = K]

Physics Helpline
L K Satapathy
O x
Illustration :
The figure shows a sinusoidal wave , propagating in
the positive x-direction in a string at t = 0. Then
which of the following equations represents the wave.
( ) sin( ) ( ) sin( )a y A t kx b y A kx t    
( ) cos( ) ( ) cos( )c y A t kx d y A kx t    
Soln. : For the wave in the fig. y = 0 at x = 0 and t = 0
Again , (11)  particle velocity = – (wave velocity)(slope)
Both wave velocity and slope are +ve at x = 0  particle velocity is – ve
( ) cos( ) cos( ) ( 0)
y
a A t kx A t ve at x
t
   

      

[ (a) is incorrect]
( ) cos( ) cos( ) ( 0)
y
b A kx t A t ve at x
t
   

        

[ (b) is correct]
[ (c) & (d) are incorrect]For both (c) & (d) , y = A cos (0) = A [at x = 0 & t = 0]

Physics Helpline
L K Satapathy
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Wave Motion Theory Part1

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