__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 –

**Transverse stationary waves**– Two progressive*transverse waves*by superposing produce a stationary transverse wave. Example – Waves in Sonometer and Melde’s experiment.**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 –

- y_1 = A \sin \left ( \omega t - k x \right )
- 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 –

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

- Two superposing waves must travel through a single medium.
- They must have same frequency, same amplitude and same speed.
- 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 –

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