MHT-CET : Physics Entrance Exam

### MHT - CET : Physics - Stationary Waves Page 1

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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:

y1 = A sin

 2 p t T

+

 2 p x l

y2 = A sin

 2 p t T

-

 2 p x l

y = y1 + y2

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

1. 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.
2. Stationary waves do not transfer any energy through the medium.
3. 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.
4. The nodes and antinodes are alternately situated. The distance between any two successive nodes (or antinodes) is constant and equal to half the wavelength

 l 2

1. The distance between a node and the adjacent antinode is equal to a quarter wavelength

 l 4

1. The nodes divide the medium into loops or segments of equal lengths.
2. 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.
 l p

=

 l 2

\ l =

 2l P

•

\ 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.

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