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Chapter 15: - WAVES (Hours: 10, Marks: 9)
PART A – Answer ALL the Questions
[I (question numbers: 1 – 10) main of Annual Exam Question Paper]
ONE MARK QUESTIONS

1
...

2
...

3
...
Does, wave carry energy?
Yes
5
...
What are matter waves?
The wave associated with moving material
particles are called matter waves
...
Name the kind of wave that is employed in
the working of an electron microscope?
Matter waves associated with electrons
...
Define amplitude of a wave
...

9
...

It is the time taken by a wave to move
through a distance of wavelength during
wave propagation
...
Define frequency of a wave
...

11
...

The distance between two consecutive
particles of medium which are in the same
state of vibration (phase) is called as
wavelength
...
Define wave velocity
...

13
...

14
...

It is the number of waves that can be
accommodated per unit length
...
How propagation is constant related to
wavelength of a wave?

...
Name the factors which determine the
speed of a propagation of an
electromagnetic wave?
Permittivity and permeability of the medium
17
...
Name the quantities associated with a
wave that changes when a wave travels
from one medium to another
...
What is sound?
Sound is a form of energy that produces a
sensation of hearing
...
How is sound produced?
Vibrating bodies surrounded by a material
medium produces sound
...
Why do we see the flash of lightening
before we hear the thunder?
Because speed of light is much greater that
the speed of sound
...
What is a stationary wave?
When two progressive waves of equal
amplitude, frequency and speed traveling in
a medium along the same line but in opposite
direction
superimpose,
the
resulting
waveform appears to be stationary pattern,
such a wave is called stationary wave
...
How much energy is transported by a
stationary wave?
Zero
24
...
What is an antinode?
Antinodes are certain location (position) in a
stationary wave, where the particle of the
medium
vibrates
with
maximum
displacement
...
What is a segment (or) loop in a stationary
wave?
The wave form (or) region between two
consecutive nodes in a stationary wave is
called a loop (or) segment
...
What is the length of a loop in a stationary
wave in terms of wavelength?
1

vibration (applied force) becomes equal to
natural frequency of the vibrating body
...

This phenomena is called “Resonance”
40
...

41
...

42
...

43
...
Give the formula for speed of transverse
wave on a stretched string
...
How much is the distance between a node
and its neighbouring antinode?

...
How much is the distance between a node
and its neighbouring node
...

30
...
What happens to a wave, if it meets a
boundary which is not completely rigid?
A part of the incident wave gets reflected
and part of incident wave gets transmitted
into the other medium (assuming that
boundary)
32
...
What is the phase angle between the
incident wave and the wave reflected at an
open boundary?
No phase change (or) zero
...
Give the relation between phase difference
and path difference
...
Where T is the tension in the string
μ is the linear mass density
45
...
61 ms-1 per degree centigrade rise in
temperature
46
...

47
...
e
...
How does the velocity of sound in air vary
with pressure?
Velocity of sound in air is independent of
pressure provided temperature remains
constant
...
Give the dimensional formula for
propagation constant
...
The fundamental frequency of a closed
pipe is 80Hz
...

35
...

36
...

37
...

38
...

39
...
Calculate the wavelength of a wave whose
angular wave number is 10π radian m-1?

55
...
Give the relation between time period and
frequency of a wave
...

52
...
08m
...

57
...
What is the distance between two
consecutive antinodes in a stationary wave
of wavelength 2m?
1m
59
...
i
...
With
what
velocity
does
an
electromagnetic wave travel in vacuum?
3 x 108 ms-1

antinode is

...
How is the frequency of an air column in
an open pipe related with the temperature
of air?
Frequency of air column in an open pipe is
directly proportional to square root of its
absolute temperature (
√ )
54
...
What will be its
velocity at 4 atmospheric pressure?
330 ms-1

PART B - Answer any FIVE (out of EIGHT) of the following questions
...


5
...

Consider a wave traveling with a velocity „v‟ let
𝞶 be its frequency & l be its wavelength
...

By the definition of wave velocity we have

1
...

The waves that requires material medium for
their propagation (transmission) are called as
mechanical waves
...
Waves on a surface of
water, sound waves, seismic waves etc
2
...

The waves that do not require material medium
for their propagation are called as nonmechanical waves
...
Radio waves, light
waves, x-rays etc
...
What are longitudinal waves? Give two
examples
The waves in which the particles of the medium
oscillate parallel (along) to the direction of wave
propagation are called longitudinal waves
...

Sound waves, waves set up in air column
4
...

The waves in which the particles of the medium
oscillate perpendicular to the direction of wave
propagation are called Transverse waves
...

Light waves, waves on the surface of water,
waves on a string
...

𝞶
...
At which positions (or) locations of the
stationary wave, the pressure changes are
maximum and minimum
...

7
...

Displacement is maximum antinode and
displacement is minimum at node
...
Give any two applications of Doppler’s
effect?
Sonography, echocardiogram and speed of
vehicles

PART C - Answer any FIVE (out of EIGHT) of the following questions
...
Give the differences between mechanical and
a non-mechanical (electromagnetic) waves
No Mechanical wave
Non-mechanical
wave
1
Requires
a Do not require a

2
3

material medium
for
their
propagation
Particles of the
medium oscillate
...

Electric
and
magnetic
field

i
...

ii
...

iii
...

However, the particles of the matter
themselves do not move away and they
perform only simple harmonic motion
about their mean position
...

Waves in a homogeneous medium travel
with constant velocity at a given
temperature
...

v
...

vi
...

vii
...

viii
...
Wave propagation
can be transverse on a liquid surface and
on the strings
...

4
...

Pressure: Let p be the pressure and v be the
volume of a given mass „m‟ of the gas
...

Are
always
transverse
in
nature
...

5
Doppler effect is Doppler effect is
asymmetric
symmetric
Eg
...
Light waves
...
What are beats? Give the theory of beats
...
The periodic waxing
(rise) and waning (fall) in the intensity of sound
due to superposition of two sound waves of
nearly the same frequencies (but not equal)
traveling in the same direction are called beats
...

(
)
...

&


...

If | ω1 & ω2 | < < ω1, which means ωa >> ωb
Here the amplitude and hence intensity of
resultant wave is maximum (longest) when the
term
= t1 (or) – 1
i
...
, the intensity of the resultant wave waxes
and wanes with a frequency which is

If the temperature of the gas remains constant,
Then, according to Boyle‟s law, we have

...

Since m is constant we get
But, by newton‟s Laplace formula, velocity of
sound in gas is given by
As γ is constant and

Since
𝞶
The beat frequency is given by,

√
...
Therefore velocity of sound remains
constant
...
Then, from perfect gas equation we have
( )
If „r‟ is the density of gas, then

3
...

Characteristics of a progressive mechanical
wave:
4

temperature and pressure, experimental
observation shows that humid air is less dense
than dry air
...

But, by Newton Laplace formula, we have
v is velocity of sound in gas is given by

(1), becomes
( )
...
(From equation (2))

√
...

Humidity: Humidity is a measure of water
content in air
...
Hence, the velocity of sound in gas
depends directly on humidity

PART D (Theory) - Answer any FOUR (out of SIX) of the following questions
...
Give the differences between progressive and
stationary waves
...
It
velocity
called remains localized
...

stationary wave
...

wave
4
Different particles All particles lying
over a distance in a loop have
same phase at a
at a given instant of given instant of
time
...

5
No particles in the The particles at
medium
are node
are
completely at rest
...

6
There is a net There is no net
transfer of energy transfer of energy
in the direction of across any section
propagation
of of the medium
...
y(x,t) = 2A
i
...

i
...
y(x,t) = f(x)
...
Give the differences between longitudinal
and transverse waves
...

(perpendicular) to
the direction of the
propagation of the
waves
...

rarefaction
...

density)
...
mechanical
(or)
non
mechanical
wave
...

5
They can travel in They can travel in
solids, liquids and solids and on the
gases
Surface of liquids,
if the waves are
mechanical
...

polarized
...
013 × 105 Pascal (Nm-2)
r = 1
...

density
Eg
...
Eg
...
Light wave
...
Write Newton’s formula for speed of sound
in a gas
...

Velocity of sound waves is determined by the
elastic and inertial properties of the medium and
in a given medium sound travel as a longitudinal
wave
...
Thus, the
value of velocity of sound in gas obtain by
newton‟s formula does not agree with the
experimental value and this has a discrepancy of
about 16%
...
However, newton‟s formula needs a
correction and this correction was given by
Laplace
...
This is because, the
vibration (compression & rarefaction) of layer
of air are so rapid, that there is hardly any time
for the exchange of heat between the layers and
also air is a bad conductor of heat
...
e

( )

Where E is the modulus of elasticity and r is the
density of the medium
For solids, E = Y (Young‟s modulus) and for a
gaseous medium E = B (bulk modulus)
...


( ) [From equation (1)]

√

√

Differentiating equation (5) we get

The Bulk modulus of the medium is defined by
( )

And hence from equation (3) we get B = γp
...


...


This is in close agreement with the experimental
value
...
Mention the characteristics of a stationary
wave
...

Stationary wave remains localized
between two fixed points
...
e
...
In a stationary wave, there exist certain
points called Nodes and antinodes
...


Differencing; Dp
...
Dv = 0 or
And hence from equation (3) we have B = P
Therefore equation (2) becomes
√

( )

√

When sound waves travel through a gas,
alternate compression and rarefaction are
produced
...

Thus, propagation of sound waves in a gaseous
medium is accompanied by continuous changes
in pressure and volume and newton assumed
that this change in pressure and volume takes
place under ISOTHERMAL condition that is at
constant temperature
...


Similarly at a time
, the source emits
th
(n+1) crest and this crest reaches the observer

(λ) [i
...
, ]
iv
...

The distance two consecutive nodes (or)
antinodes is equal to half the wavelength

Hence in a time interval of (
)
observers detector counts n crest and this time

(

at a time

Always a node exists between two
successive antinodes and vice-versa
...
e
...

vi
...

There is no net transfer of energy across
any segment of the stationary wave
...
What is Doppler Effect? Derive an expression
for the apparent frequency when a source
moves towards a stationary listener
...

(Convention – Take the direction from the
observer to the source as the positive direction
of velocity)
Consider a source S producing the sound waves
...


)

v
...


(

)
...


If VS is small compared to the speed of the wave
V, taking binomial expansion to terms in first
order in
(

& neglecting higher power we get,
)

Note: For a source approaching the observer, we
get
(

) (That is by replacing VS by -VS)

6
...

The apparent change in the frequency of the
sound heard by the listener due to relative
motion between the source producing the sound
and the listener is called as Doppler Effect
...

Let the speed of the waves of frequency 𝞶 and
period To, both measured by an observer at rest
with respect to the medium be V and we assume
that the observer has a detector that counts every
time a crest reaches it
...

At S2, the source emits the second crest this
crest reaches the observer at

(

When the observer is moving with a velocity
Vo‟ towards a stationary source, the source and
medium are approaching at a speed of Vo and
the speed with which the wave approaches is
Vo+V
...
Now, since the observer is moving,
the velocity of the wave relative to the observer
is (V+Vo) and therefore the first crest reaches

at
Similarly, at a time t = nTo, the source emits
(n+1)th crest and this crest reaches the observer
(

at a time

the observer at a time

)

At a time t = To, both the observer and source
has moved to other new position O2 and S2
respectively
...

At S2, the source emits second crest, this crest
reaches the observer at a time

Hence in a time interval of (
– ) observer‟s
detector counts n crest and this time interval is
(

given by *

)

+

(

)

Therefore, observer records the time period of
the waves as T given by
(

*

)

+ (

)

(


...


At a time t = nTo, the source emits (n+1)th crest
and this reaches the observer at time


...


(

)

(

(

)
...


)

Therefore, observer records the time period of
the wave as T given by

)
(

)

(

(
(

Note: For an observer moves away from the
source at rest we get


...
What is Doppler Effect? Derive an expression
for the apparent frequency when the source
and listener are moving in the same direction
...

(Convention – Take the direction from the
observer to the source as the positive direction
of velocity)
Consider a source S producing the sound waves
...

Let the source and the observer be moving with
a speed of VS & Vo respectively as shown below

)
(

)

(

)


...


*

(

)

(

)

*

+
...


Thus the frequency 𝞶 observed by an observer is
given by

*

+
...
Discuss different modes of vibration on a
stretched string
...
when such a string is plucked at
any part of its length, the transverse wave of
velocity V, frequency 𝞶, & wavelength λ travel
towards each end of the wire and gets reflected
at the fixed ends, this reflected wave superpose
with the incident wave, forming a stationary
wave, such that always at the fixed ends, nodes
are formed and the string oscillates in such a
way that it is divided into an integral number of
equal loops which is characterized by a set of
8

Open pipe – A pipe which is open at both ends
such that always antinodes are formed at their
open ends is called as open pipe
...
This air column is set into
oscillation by holding a vibrating tuning fork at
one of its ends
...
The first three nodes of oscillations
are as shown below
...


These stationary wave formed is given by
Where
gives its amplitude
Therefore the positions of nodes are given by
sin Kx = 0 and hence Kx = nπ
Where n = 0, 1, 2, 3 …
Since

; we get

Where n = 0, 1, 2, 3 …
In the same way, the positions of the antinodes
are given by Sin Kx = 1
& hence

(

)

Where n = 0, 1, 2, 3…
Since

; we get

These stationary wave formed is given by
(

)
Where
gives its amplitude
Therefore the positions of nodes are given by
sin Kx = 0 and hence Kx = nπ
Where n = 0, 1, 2, 3 …

Where n = 0, 1, 2, 3…
Taking x = 0 and x = L as the positions of node,
the condition for node at x = 0 is already
satisfied & at x = L, the condition for node
requires that the length L is related to

Since

Where n = 0, 1, 2, 3 …
In the same way, the positions of the antinodes
are given by Sin Kx = 1

wavelength λ by
Thus the possible wavelengths of stationary
wave, formed in different modes of oscillations
are given by

; we get

& hence

where n = 1, 2, 3 …

(

)

Where n = 0, 1, 2, 3…

The corresponding frequencies given by

Since

, where n = 1, 2, 3 …

; we get

(

)

Where n = 0, 1, 2, 3…
Since at both the ends, antinodes are formed in
different modes of oscillation, the possible
wavelengths of stationary waves formed
indifferent modes of oscillation is given by

Thus, for fundamental mode of oscillation (or) I
harmonic, n = 1 and corresponding frequency is
given by
For second mode of oscillation n = 2 and
corresponding frequency is given by

where n = 1, 2, 3 …

This mode is called as I overtone (or) II
harmonic
...

Therefore, : : : …
...

9
...


given by
For second mode of oscillation n = 2 and
corresponding frequency is given by
9

This mode is called as I overtone (or) II
harmonic
...

Therefore, : : : …
...

Thus, open pipe produces both odd & even
harmonics
...
e the ratio of frequencies of the
overtones to that of fundamental frequency are
both even and odd natural numbers
...
Discuss different modes of vibration (first
three harmonics) produced in a closed pipe
...

Consider a closed pipe of length L, which
encloses certain specific amount of air called as
air column
...
These longitudinal waves of frequency 𝞶
& wavelength λ travel with a velocity of v
through the pipe and gets reflected at the other
end, because the other end acts like a boundary
and this reflected wave traveling in opposite
direction superpose with incident wave forming
a stationary wave such that a node is formed at
the closed end and an antinode at the open end,
which is characterized by a set of natural
frequencies called as normal modes of
oscillation
...


Where
gives its amplitude
Therefore the positions of nodes are given by
sin Kx = 0 and hence Kx = nπ
Where n = 0, 1, 2, 3 …
Since

; we get

Where n = 0, 1, 2, 3 …
In the same way, the positions of the antinodes
are given by Sin Kx = 1
(

& hence

)

Where n = 0, 1, 2, 3…
Since

(

; we get

)

Where n = 0, 1, 2, 3…
Taking the closed end of the pipe to be x = 0,
the condition for node is satisfied and the other
end of the pipe to be x = L where an antinode is
formed, requires that the length L to be related
to wavelength λ given by

(

) for n =

0, 1, 2, 3………
...

Similarly, for third mode of oscillation n =2 &
corresponding frequency is given by
This is called as II overtone (or) V harmonic
and so on………
...
= 1 : 3 : 5:………
Thus, a closed pipe produces odd harmonics i
...
,
the ratio of frequencies of overtone to that of
fundamental frequency are odd natural numbers

These stationary wave formed is given by
$$$$$$$$$$$$$$$$

10



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