1.

Stationary
Waves


When
two progressive waves of same amplitude, period and speed, travelling through the same medium, along the same
path but in opposite directions interfere, the resultant wave produced is
called a stationary wave.
Since the resultant pattern appears to be standing, they are also termed
as standing waves. There is no transfer of energy along the medium due to
stationary waves.
The points where the resultant amplitude of the wave is zero, i.e.
particles vibrate with minimum amplitude, are called nodes.
The points, where the resultant amplitude of the wave is maximum, i.e.
particles vibrate with maximum amplitude, are called antinodes.
An
analytical treatment of stationary waves:



y = y_{1} + y_{2}
y = R sin

2 p t


T


,

where R = 2A cos

2 p x


l


where R: Amplitude of
the resultant, T: Time period.
Position of nodes: x =

l


4


,

3 l


4


,

5 l


4


Antinodes: x = 0,

l


2


,

l

,

3 l


2


Distance between two successive nodes or antinodes =

l


2


Distance between a node and the successive antinode =

l


4




2.

Properties
of Stationary Waves


 Stationary
waves are produced due to the interference between two exactly identical
progressive waves travelling through a
medium in opposite directions. The velocities of
the progressive waves being equal and opposite, the resultant
velocity is zero. Hence stationary waves travel neither in
the forward nor in the backward direction.
 Stationary
waves do not transfer any energy through the medium.
 The
points, where particles vibrate with minimum (i.e. zero) amplitude,
are called nodes while points, where particles vibrate with
maximum amplitude, are called antinodes.
 The
nodes and antinodes are alternately situated. The distance between
any two successive nodes (or antinodes) is constant and equal to
half the wavelength
 The
distance between a node and the adjacent antinode
is equal to a quarter wavelength
 The
nodes divide the medium into loops or segments of equal lengths.
 At
any instant, all the particles in one loop are in the same phase while
the particles in the adjacent loops are in opposite phases.





3.

Modes
of Vibrations on a Stretched String


The
different ways of vibration of a stretched string are termed as the modes
of vibration of a string.
 If
a stretched wire (or string) is plucked at some point, a disturbance
is produced there. This disturbance travels along the string in the
form of transverse waves.
If T is tension applied to the wire,
m is mass per unit
length (linear density) of wire,
velocity of the transverse wave
V =


 If
the string is stretched between two rigid supports, the disturbance
produced due to plucking travels along the wire from one end and
gets reflected at the other end. The interference of the incident
and reflected waves gives rise to stationary transverse waves such
that nodes are formed at each support.
 In
the simplest form of vibration, a node (N) is formed at each end of
the wire and an antinodes (A) is formed at the centre.
The length of wire is equal to the distance between two successive
nodes.
\ l =

l


2


or l = 2 l.


Frequency of vibrations n =

V


l




V =


, l = 2 l




\ n =

1


2l






This is the smallest frequency of vibration of the wire
and is called the fundamental frequency.

 The
wire can vibrate in different ways, with the condition that nodes
are produced at the fixed ends of the wire.
 If
p loops are formed on a
stretched wire of length l with tension T,
Length of a loop = distance between two consecutive nodes.
\ frequency of vibration n =

p


2l



 The
fundamental frequency is produced for p = 1 (single loop); substituting integral
values for p yields the
corresponding integral multiples of the fundamental frequency or
harmonics.



