Stationary Waves

What are called Stationary Waves?

When two identical waves of same amplitude, same frequency, same speed moving in same path but opposite in directions are superpose on each other, the resultant wave doesn’t travel in either of the directions and is called a stationary waves or standing waves.

Stationary waves are of two types –

1. Transverse stationary waves – Two progressive transverse waves by superposing produce a stationary transverse wave. Example – Waves in Sonometer and Melde’s experiment.
2. Longitudinal stationary waves – Two progressive longitudinal waves by superposing produce a stationary longitudinal wave. Example – Waves in resonance tube, organ pipe and Kundt’s tube.

Equation of Stationary Waves

Equation for stationary waves is derived from the principle of superposition of waves.

Consider about two sinusoidal waves ( y_1 ) and ( y_2 ) of equal amplitude, equal frequencies and equal speeds are travelling in a long string in opposite directions.

Then, their equations can be represented as –

1. y_1 = A \sin \left ( \omega t - k x \right )
2. y_2 = A \sin \left ( \omega t + k x \right )

According to the principle of superposition, the resultant wave will be –

y = ( y_1 + y_2 ) = [ A \sin ( \omega t - k x ) + A \sin ( \omega t + k x ) ]

Because, \quad [ \sin ( A + B ) + \sin ( A - B ) = 2 \sin A \cos B ]

So, \quad y = 2 A \sin \omega t \cos k x

= ( 2 A \cos k x ) \sin \omega t

Therefore, the general equation representing a stationary wave is \quad y = ( 2 A \cos k x ) \sin \omega t

Comparing it with general equation of a wave, we will get –

• Angular frequency of wave is ( \omega )
• Amplitude of wave is ( A' = 2 A \cos k x )

Position of Nodes in Stationary Waves

Amplitude of a stationary wave is given by –

A' = 2 A \cos k x

At the position of nodes, amplitude or displacement is zero. Therefore, nodes will occur at the points where –

\cos k x = 0 = \cos \left ( \frac {\pi}{2} \right ), \ \cos \left ( \frac {3 \pi}{2} \right ), \ \cos \left ( \frac {5 \pi}{2} \right ) \ ..... etc.

Therefore, for the positions of nodes, the satisfying condition will be –

k x = \left [ n + \left ( \frac { 1 }{ 2 } \right ) \right ] \pi

Where, ( n ) is an integer and can take values of ( n = 0, \ 1, \ 2, \ .... ) etc.

Also, propagation constant of a wave \quad k = \left ( \frac {2 \pi}{\lambda} \right ) .

So, \quad \left ( \frac { 2 \pi }{ \lambda } \right ) x = \left [ n + \left ( \frac { 1 }{ 2 } \right ) \right ] \pi

Or, \quad x = \left ( \frac { \lambda }{ 2 } \right ) \left [ n + \left ( \frac { 1 }{ 2 } \right ) \right ]

= \left ( 2n + 1 \right ) \left ( \frac { \lambda }{ 4 } \right )

Therefore, position of Nodes are \quad x = \left ( \frac { \lambda }{ 4 } \right ), \left ( \frac { 3 \lambda }{ 4 } \right ), \left ( \frac { 5 \lambda }{ 4 } \right ), ........ etc.

Therefore, in stationary waves, position of nodes occur at odd multiples of \left ( \frac {\lambda}{4} \right )

Position of Anti-nodes in Stationary Waves

The position of anti-nodes or the points of maximum displacement will be at points where –

\cos k x = \pm 1

Or, \quad \cos kx = ( \cos 0 ), \ ( \cos \pi ), \ ( \cos 2 \pi ) \ ..... so on.

Therefore, for the positions of anti-nodes, the satisfying condition will be –

k x = ( n \pi )

Where, ( n ) is an integer and can take values of ( n = 0, \ 1, \ 2, \ .... ) etc.

Solving in similar manner, we will get –

\left ( \frac { 2 \pi }{ \lambda } \right ) x = ( n \pi )

Or, \quad x = n \left ( \frac { \lambda }{ 2 } \right )

So, position of anti-nodes are \quad x = 0, \left ( \frac { \lambda }{ 2 } \right ), \left ( \lambda \right ), \left ( \frac { 3 \lambda }{ 2 } \right ), ...... so on.

Therefore, in stationary waves, position of anti-nodes occur at integer multiples of \left ( \frac {\lambda}{2} \right )

Properties of Stationary Waves

Stationary waves have following properties –

1. A stationary wave cannot be formed from two independent waves travelling in different mediums. Actually a stationary wave is produced when a progressive wave and its reflected wave are superposed. Hence, a finite medium with boundaries is essential for formation of it.
2. The stationary wave repeated itself in the same fixed position.
3. Some medium particles remain at rest i.e., they have zero displacement. Their positions are called nodes.
4. Some medium particles suffer maximum displacement. Their positions are called anti-nodes.
5. The distance between two successive nodes and anti-nodes is \left ( \frac {\lambda}{2} \right ) .
6. The positions of nodes and anti-nodes do not change with time.
7. Transfer of energy not takes place along the medium in either direction.
8. Stationary waves may be transverse or longitudinal waves.

Formation of Stationary Waves

Stationary waves are produced by superimposing of two similar waves having following characteristics –

1. Two superposing waves must travel through a single medium.
2. They must have same frequency, same amplitude and same speed.
3. The two superposing waves must travel in opposite directions.

Principle of Superposition of Waves

Formation of stationary waves is based on the principle of superposition of waves. It is stated as –

When a number of waves travel through a medium simultaneously, the resultant displacement of any particle of the medium at a given time is equal to the algebraic sum of the displacements due to individual waves.

Let, y_1, \ y_2, \ y_3, \ .... \ y_n are the displacements of a medium particle due to the effect of different waves. Then –

By the principle of superposition, the resultant displacement of that particle will be –

y = y_1 + y_2 + y_3 + .... + y_n

Superposition of two waves lead to formation of three different effects as follows –

1. INTERFERENCE WAVES – When two waves of same frequency moving with same speed in the same direction in same medium superpose on each other, they will produce interference waves.
2. STATIONARY WAVES – When two waves of same frequency moving with same speed in the opposite directions in same medium superpose on each other, they will produce stationary waves.
3. BEATS – When two waves of different frequencies moving with same speed in the same direction in same medium superpose on each other, they will produce beats.