JEE Main 2014 Physics Syllabus - Stationary Waves by ednexa.com

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Stationary Waves
Stationary Waves
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
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‘nodes’. The distance between two successive
nodes is λ/2.
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
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in the same phase, while the particles in the
successive loop are out of phase by πc.
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
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because of two reasons. Firstly, the amplitude of
vibration of the particles increases from node to
antinode and decreases from antinode to node.
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.
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Equation of a Stationary Wave
y
1
a sin 2
t x
T
and y
a sin 2
2
t
T
i.e. y = y1 + y2
y a sin 2
t x
T
a sin 2
t
T
x
We know that,
sin A
sin B
2 sin
A
B
2
2 t
2 x
y 2 a sin
cos
T
2 x
put, 2a cos
A
2 t
T
But,1 / T n
y A sin 2 n t
y
A sin
6
A B
cos
2
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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
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
2 x
2 a cos
2 x
x
2
0
, 3 ,5 ,.......
2 2 2
,3
, 5 ,........
2 2
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distance between two successive nodes is,
3
2
2
2
OR 5
3
2
2 x
x
2 x
0,
0,
2
i.e. A = ± 2a
2. At antinodes,
2a cos
2
and so on
2a i.e. cos
2 x
1
, 2 ,3 , ......
, 2 ,3 ,.......
2 2 2
Distance between two successive antinodes is,
2
0
2
or 2
2
2
2
and so on
Hence, distance between successive nodes
and antinodes is
2
or the nodes and antinodes
are equidistant. Also, the distance between
consecutive node and antinode is λ / 4.
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M.C.Q.
Q.1
The phase difference between the particles in successive loops of a stationary wave is
(a.1) 90°
Q.9
(b.1) 45°
(c.1) 180°
(d.1) zero
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.
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
than
the
harmonic.
Vibrations
fundamental
of
frequencies
frequency
which
present are called overtones.
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higher
are
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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
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.
1.
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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.
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Q.65 In the experiment of a simple pendulum, the oscillations of the simple pendulum are actually
(a.65) Free oscillations
(c.65) Damped harmonic oscillations
(b.65) Forced oscillation
(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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