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Superposition of Progressive Waves: Stationary Wave

When two identical progressive waves travelling in a medium with same velocity but in opposite directions along the same straight line are superposed, then give rise to a system of alternative rarefaction and compression that cannot move in any direction of medium. This resultant wave is called stationary or standing wave.

 

They are called stationary because there is no flow of energy along the waves. There are certain points, half a wavelength apart, which are permanently at rest are known as nodes and there are some other points midway between the nodes where the displacement is maximum, known as antinodes.

 

Analytical Treatment of Stationary Waves and Their Properties

 

Consider two plane progressive waves-one travelling along positive X-axis and another along negative X-axis with amplitude a, wave velocity v and wavelength A. So, the superposing

progressive waves are

 

y1=a sin 2π/λ (v t –x). and y2 =a sin 2π /λ(v t  +x)

 

By the principle of superposition, the resultant displacement of a particle at x at time t

will be

 

Y = Y1 + Y2

 

= a sin 2π/λ(v t-x) a sin 2π/λ (v t+x)

 

= 2a cos  2π/λ vt cos  2π/λ x

 

Y = A cos 2πvt/λ

 

A =2a cos 2π/λ x

 

This equation represents a simple harmonic motion of same wavelength of the   superposed wave but not of same amplitude. Moreover, the amplitude, A = 2a cos 2π /λ x is  not a constant. For different values of x, A will have different values. Equation of motion given by equation (1) is not a progressive motion since its phase does not contain any term like (vt-x) or (vt + x). So, equation (1) represents a stationary wave.

 

Position of Nodes

The nodes will be obtained when

 

2π/λ x =0 or Cos or 2π/λ x =(2n±1) π/2

 

X =(2n±1)

 

where n = 0, 1, 2, 3, 4, …

 

So nodes will be obtained at x = ± λ/4, ± 3λ/4 , ± 5λ/4, … the particles at these points will be at rest. The distance between any two successive nodes is -.

 

Position of Antinodes

 

Antinodes are the points of maximum vibration, so the antinodes will be obtained when

 

Cos 2π/λ   x = 1   or   2π/λ  x = + n r

 

x=±2n λ/4

 

So nodes will be obtained at x = 0, ±λ/2 .,±λ,± 3λ/2., … at these points the amplitude of vibration of the particles will be ± 2a. The distance between any to successive antinodes is λ/ 2

 

The distance between one node and next antinode is λ / 4

 

 

Characteristics of Stationary Waves

 

1. Stationary waves are produced when two identical waves travelling along the same straight    line but in opposite direction are superposed.

 

2. Crests and troughs do not progress through the medium but simply appear or disappear at the same place alternatively.

 

3. All the particles, except those at the nodes, follow simple harmonic motion. The amplitude of the oscillation is zero at nodes and maximum at antinodes. The distance between two successive antinode and node is equal to half of wavelength.

 

4. The particle between two successive nodes are in the same phase of vibration while the particles on opposite sides of a node are in opposite phase of vibration.

 

5. Stationary waves can be produced both by longitudinal waves and transverse waves.

 

6. All the particles pass through their mean positions or reach their outermost positions simultaneously twice in a periodic time. 7. There is no advancement of the wave and no flow of energy in any direction.

 

7. There is no advancement of the wave and no flow of energy in any direction.