When two progressive waves of same amplitude and wavelength travelling along a straight line in opposite directions superimpose on each other, stationary waves are formed.

*Stationary waves*

*When two progressive waves of same amplitude
and wavelength travelling along a straight line in opposite directions
superimpose on each other, stationary waves are formed.*

*Analytical method*

Let us consider a
progressive wave of amplitude *a* and
wavelength travelling in the
direction of X axis.

y_{1}
= a sin 2π( t/T - x/ λ ) ????(1)

This
wave is reflected from a free end and it travels in the negative direction of X
axis, then

y_{2}
= a sin 2π( t/T - x/ λ ) ????(2)

According
to principle of superposition, the resultant displacement is

y
= y_{1} + y_{2}

=a[sin
2π( t/T - x/ λ ) + sin 2π( t/T + x/ λ ) ]

=a[2
sin(2πt/T) cos(2πx/ λ)]

∴ y = 2a cos(2πx/ λ)
sin(2πt/T) ?????.(3)

This
is the equation of a stationary wave.

(i)
At points where x = 0, λ/2 , λ, 3λ/2. the values of cos(2πx/ λ)= ?1

∴ A = + 2a. At these
points the resultant amplitude is maximum. They are called antinodes (Fig.
7.13).

(ii)
At points where x = λ/4, 3 λ/4, 5 λ/4 ?????? the values of cos (2πx / λ) = 0

A
= 0. The resultant amplitude is zero at these points. They are called nodes
(Fig.).

The
distance between any two successive antinodes or nodes is equal to λ/2 and the distance between an antinode and a
node is λ/4.

(iii)
When t = 0, 2, T/2,T,3T/2,2T,???then sin(2πx / T) = 0 the displacement is zero.

(iv)
When t = T/4 , 3T/4, 5T/4, etc??.. sin(2πt/T) = ? 1, the displacement is
maximum.

*Characteristics of stationary waves*

1. The waveform remains
stationary.

2. Nodes and antinodes
are formed alternately.

3. The points where
displacement is zero are called nodes and the points where the displacement is
maximum are called antinodes.

4.
Pressure changes are maximum at nodes and minimum at antinodes.

5.
All the particles except those at the nodes, execute simple
harmonic motions of same period.

6. Amplitude of each
particle is not the same, it is maximum at antinodes decreases gradually and is
zero at the nodes.

7. The velocity of the
particles at the nodes is zero. It increases gradually and is maximum at the
antinodes.

8. Distance between any two
consecutive nodes or antinodes is

equal to ^{λ}_{2} , whereas the distance between a node and its
adjacent antinode is equal to ^{λ}_{4} .

9. There is no transfer
of energy. All the particles of the medium pass through their mean position
simultaneously twice during each vibration.

Particles in the same
segment vibrate in the same phase and between the neighbouring segments, the
particles vibrate in opposite phase.

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