Waves


   
 
Analytical Treatment of the Formation of Stationary Waves
Case (i) Reflection occurring at a free end
In this case, there will be no reversal of phase due to reflection. Let the incident wave be represented by the equation
 
 
The reflected wave will have the same amplitude, velocity and wavelength. It can be represented by
 
 
The resultant displacement of the particle subjected to these two disturbances simultaneously is, y = y1 + y2
 
 
 
 
 

This represents a sine wave with amplitude   . Thus, the amplitude of the resultant wave is a function of x.

 
 
 
Differentiating equation (iii) with respect to time, we get
 
 
Differentiating again with respect to time, we get
 
 
 
Differentiating equation (iii) with respect to x, we get
 
 
Changes with respect to position
 
a) Positions of the antinodes
 
Consider the positions, where
 
 
 
 
 
 
Since the resultant amplitude R is maximum, these points correspond to the antinodes. It may be noted that the antinodes correspond to   i.e., zero strain. Thus, antinodes are obtained at those points  corresponding to
 
 
 
 
The antinodes are formed at those points corresponding to etc. The result that x = 0 corresponds to an antinode, indicates that the open end of an organ pipe or the free end of a stretched string corresponds to an antinode.
 
b) Positions of the nodes
 
Consider the positions, where
 
 
Then y = 0
 
R = 0
 
 
 
 
Since y = 0, these positions correspond to nodes. The strain represented by will be maximum at these points.
 
 
 
 
From the above analysis, it follows that two consecutive nodes or antinodes are separated by l/2. A node and the adjacent antinodes are separated by l/4. Between two nodes, an antinode is formed and vice versa.
 
Case (ii) Reflection occurring at a fixed end
In this case, there will be a reversal of phase due to reflection. Let the incident wave be represented by the equation.
 
 
The reflected wave will have the same magnitude of the amplitude, velocity and wavelength. Due to reversal of phase (i.e., a crest returning as a trough or vice versa) the sign of A is reversed in the expression for y2.
 
 
The resultant displacement of the particle subjected to these two displacements simultaneously, is
 
y = y1 + y2
 
 
Using the identity
 
 
 
 
 
This represents a simple harmonic wave whose amplitude has the magnitude
 
 
The velocity of the particle at any instant of time is given by
 
 
Acceleration is given by
 
 
The strain or the compression is given by
 
 
Changes with respect to position
 
a) Positions of the antinodes
 
Consider those positions where
 
 
 
 
 
 
Since the resultant amplitude is maximum, these points correspond to antinodes.
 
 
 
 
antinodes are obtained at points corresponding to
 
 
 
b) Positions of the nodes
 
Consider the positions where
 
 
Then y = 0
 
R = 0
 
 
 
 
Since the resultant amplitude is zero, these positions correspond to nodes.
 
 
 
 
 
Therefore, the fixed end corresponds to a node.
 
  
     
   
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