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Stationary Waves

Stationary Waves

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The resultant waves formed due to superposition of

two exactly identical progressive waves, having same

amplitude, wavelength and speed, travelling in the

same medium, along the same path, but in opposite

directions are called stationary waves.

Characteristics of Stationary waves

1. Stationary waves are formed due to superposition

of two exactly identical waves travelling through the

same medium, along the same path but, in

opposite directions.

2. When stationary waves are set up in a medium, the

particles at some points are permanently at rest

(i.e. amplitude is zero). Such points are called

‘nodes’. The distance between two successive

nodes is λ/2.

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3. When stationary waves are set up in a medium, the

particles at some points vibrate with maximum

amplitude. Such points are called ‘antinodes’.

Distance between two successive antinodes is λ /2.

4. Nodes and antinodes are alternately situated. The

distance between any node and successive

antinode is λ /4.

5. In stationary waves loops are formed between two

successive nodes. All the particles in one loop are

in the same phase, while the particles in the

successive loop are out of phase by πc.

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6. Stationary waves are periodic in time and space.

7. Stationary waves do not transfer energy through

the medium.

8. In stationary waves, all the particles except those at

nodes, vibrate with same period as that of the

interfering waves.

9. The amplitude of vibration is different for different

particles and that increases from node to antinode.

10. A loop is formed between two successive nodes

because of two reasons. Firstly, the amplitude of

vibration of the particles increases from

node to antinode and decreases from antinode to

node.

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Secondly, all the particles reach their maximum

displacement at a time.

11. In case of stationary waves formed due to

interference of longitudinal waves, displacement of

the particle at the node is zero. But, pressure at the

nodes changes between maximum to minimum. At

antinodes, the displacement is maximum, but, the

pressure remains constant. Hence, displacement

nodes are called pressure antinodes and

displacement antinodes are called pressure nodes.

Equation of a Stationary Wave

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i.e. y = y1 + y2

We know that,

Hence, the resultant wave is also an S.H.M. with

same period T. But, in this equation the term ± x/λ

is absent. Thus, the resultant wave is not a

progressive wave. It is travelling neither along + ve

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X - axis nor along - ve X-axis.

Such a steady or localized wave is called

Stationary Wave or Standing Wave. Transfer of

energy is 0.

Amplitude of the resultant wave varies between

± 2a

1. At nodes, i.e. A = 0

distance between two successive nodes is,

2. At antinodes, i.e. A = ± 2a

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Distance between two successive antinodes is,

Hence, distance between successive nodes

and antinodes is or the nodes and antinodes

are equidistant. Also, the distance between

consecutive node and antinode is λ / 4.

M.C.Q.Q.1    The phase difference between the particles in successive loops of a stationary wave is

          (a.1) 90°       (b.1) 45°      (c.1) 180°     (d.1) zero

Q.9    Find the ‘wrong’ statement from the following:

          The equation of a stationary wave is given by , where y and x are in cm and t is in second.

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Then for the stationary wave,

          (a.9) Amplitude=3 cm              (b.9) Wavelength=5cm             (c.9) Frequency=20 Hz   (d.9) Velocity=2 m/s

Harmonics and Overtones

The fundamental frequency along with its

integral multiples are called harmonics. The

fundamental frequency itself is the first

harmonic.

Vibrations of frequencies higher than the

fundamental frequency which are actually

present are called overtones.

Free Oscillations

The body starts oscillating about its mean position,

on its own. Such vibrations are called free

oscillations. The frequency of vibrations of the body

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is called ‘natural frequency of vibrations’ of the

material of the body.

Examples

1. Simple pendulum

2. Tuning fork

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3. Stretched string

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Damped oscillations

Amplitude of free oscillations decreases because in

every oscillation a little amount of energy is utilized to

overcome the air resistance. So, the oscillations are

called ‘damped oscillations’.

Forced oscillations

the body is forced to oscillate with a new frequency.

Hence, such oscillations are called ‘forced

oscillations’.

The oscillations in the body due to an external

periodic force whose frequency is different from

natural frequency of vibration of the body are

called forced vibrations.

e.g.

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2.

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Resonance

Amplitude of forced vibrations is inversely proportional

to the difference between the natural frequency of

vibrations of the body and the frequency of the external

force (called forcing frequency or driving frequency).

Thus, as the driving frequency approaches the natural

frequency, the amplitude of vibration goes on

increasing. Finally when the driving frequency matches

exactly with the natural frequency, we get maximum

amplitude of vibration. This effect is called resonance.

Getting maximum amplitude of forced

vibrations due to synchronization or matching

between natural frequency and driving frequency is

called resonance.

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e.g.

1.

2.

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3.

Q.65  In the experiment of a simple pendulum, the oscillations of the simple pendulum are actually

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          (a.65) Free oscillations                                         (b.65) Forced oscillation

          (c.65) Damped harmonic oscillations          (d.65) Resonant oscillations

Q.66  When a regiment of soldiers have to cross a suspension bridge, they are ordered to

          (a.66) March in steps                                            (b.66) Break the steps

          (c.66) Stand in attention                                       (d.66) Stand at ease

Q.67  In the case of forced vibrations, the resonance becomes very sharp, when the

          (a.67) Restoring force is very small                       (b.67) Applied periodic force is small

          (c.67) Damping force is small                               (d.67) Quality factor is small

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