Waves

1. REVIEW LECTURE-4
UNIT-4
(WAVES)
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2.  Progressive or travelling wave are those wave which
transfers energy from one point to another point by a
periodic disturbance.
 For example; Waves in a string , Waves on a water
surface
 Transverse Waves and Longitudinal Waves are types of
progressive waves.
PROGRESSIVE WAVES

3. Transverse Waves:
 “The particles of the medium vibrate at right angles to
direction of propagation of the wave.”
 Transverse wave is not produced or possible in gases.
 Crests and troughs are produced.
Crest:
 It is the peak of the portion of the wave above its
equilibrium level in a transverse wave.
 Distance between 2 consecutive crests is 

4. Trough:
 It is the peak of the portion of the wave below its
equilibrium level in a transverse wave.
 Distance between 2 consecutive troughs is 
Longitudinal Waves:
 “The particles of the medium vibrate along the direction
of propagation of the wave”.
 Compressions and rarefactions are produced.
 Longitudinal waves are possible in all media i.e., solid,
liquid and gas.

5.  “Periodic waves are those, which are repeated in regular
interval of time”.
 Periodic wave may be transverse or longitudinal.
Transverse Periodic waves:
 For transverse periodic waves the displacement of the
medium is perpendicular to the direction of propagation
of the wave.
 A transverse periodic wave can be demonstrated by the
mass-spring experimental setup.
 In fluids, transverse waves die out very quickly and usually
cannot be produced at all.
PERIODIC WAVES

6. Wavelength ():
It is the distance between any two consecutive crests or two
consecutive trugh
Time Period (T):
It is the time in which one wave cycle of a wave is passed
through a cetin point
Frequency (f):
The numbers of waves passing through a certain point in
second.
Amplitude (A):
In progressive wave, the maximum displacement of a
vibrating particle from the mean level to the peak point of
crest or trough
Characteristics of Wave Motion

7.  Wave velocity = frequency x wavelength v = f 
 The relation v = f holds good for any type of wave
motion – transverse or longitudinal.
 When a given wave passes through same medium, speed
of wave remain constant.
If f’ = 2f then v’ = 2f /2 = v
 When a given wave passes from one medium to the
other, its frequency does not change.
From denser medium to rare medium v decreases 
decreases
From rare medium to denser medium v increases 
increases
Wave Velocity (v)

8. In phase points:
In phase points have same state of vibrations.
 The particles in the wave separated by a distance which
integral multiple of λ i.e. n λ are in phase with each other.
Out of phase points:
 Out of phase points have different state of vibrations.
 The particles separated by a distance which is odd
multiple of
𝜆
2
i.e 𝑛 +
1
2
𝜆 = 2𝑛 + 1
𝜆
2
are out phase to each
other.
 Relation between phase difference and path difference:
𝜑 =
2𝜋𝑥
𝜆

9.  In longitudinal periodic waves the displacement of the
medium is parallel to the propagation of the wave.
 A wave in a "slinky" is a good visualization. Sound waves
in air are longitudinal waves.
LONGITUDINAL PERIODIC WAVES

10. 𝑉 =
𝐸
𝜌
𝑉 ∝ 𝐸
Esolid > Eliquid > Egases
Vsolid > Vliquid > Vgases
𝐸
𝜌
>> 1
Newton’s Formula:
𝑣 =
𝐸
𝜌
Where E is the modulus of elasticity of the medium and  is its density.
 For Solids: Modulus of elasticity
E = Young’s modulus of elasticity = Y
 v =
𝑌
𝜌
 For liquids: Modulus of elasticity
E = Bulk modulus of elasticity = B
 v =
𝐵
𝜌
SPEED OF SOUND IN AIR

11.  For gases:
Newton assumed that the propagation of longitudinal wave is
an isothermal process (temperature remains constant). In this
case, modulus of elasticity
Ei = P (Ei is isothermal bulk modulus)
 v =
𝑃
𝜌
= 280ms-1
% error = 16%
 The experimental results did not confirm to Newton’s
assumption.
 Laplace corrected the formula by arguing that sound waves
travel adiabatically. Hence,
Ea = P (Ea is adiabatic bulk modulus)
𝑣 =
𝛾𝑃
𝜌
=
𝛾𝑅𝑇
𝑀
=
𝛾𝐾𝑇
𝑚
= 333ms-1

12. Effects on the speed of sound in a Gas:
 Effect of pressure:
V  Po
Effect of density:
 𝑣 ∝
1
𝜌

𝑣1
𝑣2
=
𝜌2
𝜌1
Effect of temperature:
𝑣 ∝ 𝑇 ⇒
𝑣1
𝑣2
=
𝑇1
𝑇2
𝐾
Speed of sound at ToC vt = vo + 0.61 t
 If temperature is increased by 1oC than speed of sound increase by 0.61ms-
Effect of moisture:
 The presence of moisture in the air reduces the resultant density of air.
 Vwet > Vdry air

13.  “If two or more waves propagate simultaneously in a
medium, then the resultant displacement is given by the
vector sum of displacement due to individual waves.”
 If the displacement given by the various waves to the
particle are 𝑦1, 𝑦2, 𝑦3, . . . . . . . . . 𝑦𝑛 , then the resultant
displacement of the particle is 𝑦 = 𝑦1 + 𝑦2 + 𝑦3+. . . . . . . . . 𝑦𝑛
 Different phenomenon due to principle of superposition
are
(a) Interference (b) Beats (c) Stationary waves
PRINCIPLE OF SUPERPOSITION / SUPERPOSITION OF SOUND WAVES

14.  “Superposition (mixing up) of two identical sound waves
while passing through same medium propagating along
same direction is called their interference.”
Interference of Sound

15. Constructive Interference:
 “crest on crest or trough on trough”
 Path difference = n where n = 0,  1,  2,………
 s = 0,  ,  2…. (series of path difference)
  = 0, 2, 4 , 6 …..(series of phase difference)
 Echoing zone is region of constructive interference
Destructive Interference:
 “ crest on trough
 Path difference = 𝑛 +
1
2
𝜆where n = 0,  1,  2,………
 s = 0,  /2,  3/2…. (series of path difference)
  = , 3, 5…..(series of phase difference)
 Silence zone is region of destructive interference

16.  “Super position of two identical waves traveling opposite to
each other in the same medium simultaneously, gives rise to
stationary or standing waves”
 Points of constructive interference are called antinodes while
points of destructive interference are called nodes as shown
STATIONARY WAVES / STANDING WAVES

17.  Amplitude is maximum at antinodes and minimum (zero) at nodes.
 Nodes are stationary points whereas antinodes are points that
vibrate with maximum amplitude.
 Two consecutive nodes or antinodes are separated by distance equal
to λ/2 and an antinode and its consecutive node by λ/4.

18.  The speed of transverse wave in a stretched string is given by 𝑣 =
𝑇
𝑚
Where T and m are respectively the tension and mass per unit length of
string.
 𝜆1 = 2𝑙 and frequency 𝑓1 =
𝑣
2𝑙
=
1
2𝑙
𝑇
𝑚
. This frequency is called the
fundamental note or first harmonic.
 If the string vibrates in two loops, then 𝜆 = 𝑙 and 𝑓2 =
𝑣
𝑙
= 2𝑓1 . This
frequency is called the first overtone or second harmonic.
 If the string vibrates in three loops, then
3𝜆3
2
= 𝑙 → 𝜆3 =
2𝑙
3
 𝑓3 =
3𝑣
2𝑙
= 3𝑓1. This frequency is called the second overtone or
third harmonic.
 Both the odd and even harmonics are emitted from a stretched string.
is
𝑓𝑛 = 𝑛𝑓1, 𝑤ℎ𝑒𝑟𝑒 𝑛 = 1,2,3. . . . . . .
STATIONARY WAVES IN A STRETCHED STRING/FUNDAMENTAL
FREQUENCY AND HARMONICS

19.  “An organ pipe is a pipe that sets in vibration the air enclosed
in it when the air is blown into it. As a result, sound is produced
in it”.
 Organ pipes are of two types – closed end organ pipe and
open end organ pipe.
 An open end organ pipe has both its ends open.
A closed end organ Pipe has one of its ends closed and the
other open.
 In a closed end pipe a node is always formed at the closed
end and an antinode is formed at the open end.
 Longitudinal stationary waves are formed in an organ pipe.
STATIONARY WAVES IN AIR COLUMNS

20. Open end organ pipe:
 If 𝑙 =
𝜆1
2
thenλ1 = 2𝑙 ,  Frequency, 𝑓1 =
𝑣
𝜆1
=
𝑣
2𝑙
This frequency is called the fundamental note or first harmonic.
 If l = 2 then𝑓2 =
𝑣
𝜆2
=
𝑣
𝑙
= 2𝑓1
This frequency is called the second harmonic or first overtone.
 If 𝑙 =
3𝜆3
2
then 𝜆3 =
2𝑙
3
,  Frequency, 𝑓3 =
𝑣
𝜆3
=
3𝑣
2𝑙
= 3𝑓1
This frequency is called the third harmonic are produced in an open-end organ pipe. That is
𝑓𝑛 = 𝑛𝑓1, 𝑤ℎ𝑒𝑟𝑒𝑛 = 1,2,3. . . . . . .
 The sound emitted by an open-end organ pipe is musical.
 No. of harmonics in open pipe = 2 x No. of harmonics in closed pipe

21. Close end organ pipe:
 If the length of the pipe 𝑙 =
𝜆1
4
𝑡ℎ𝑒𝑛𝜆1 = 4𝑙 ,  Frequency, 𝑓1 =
𝑣
𝜆1
=
𝑣
4𝑙
This frequency is called the fundamental note or first harmonic.
 If 𝑙 =
3𝜆2
4
then 𝜆2 =
4𝑙
3
,  Frequency, 𝑓2 =
𝑣
𝜆2
=
3𝑣
4𝑙
= 3𝑓1
This frequency is called third harmonic or first overtone.
 If 𝑙 =
5𝜆3
4
then 𝜆3 =
4𝑙
5
,  Frequency, 𝑓3 =
𝑣
𝜆3
=
5𝑣
4𝑙
= 5𝑓1
This frequency is called fifth harmonic or second overtone.
fn = nf1 (where n = 1,3,5,7,….)

22.  “Apparent change in pitch (frequency) of sound is due to
relative motion of source and observer”.
 Doppler effect is applicable for both sound and light waves.
 Doppler effect is independent of distance between source
and observer.
CASES
 Apparent frequency of sound heard by a person moving
towards a stationary source with speed ‘u’ is given as;
𝑓′
=
𝑣+𝑢𝑜
𝑣
𝑓 𝑓′
> 𝑓 or 𝜆′
=
𝑣
𝑣+𝑢𝑜
𝜆 , 𝜆′ < 𝜆
 Apparent frequency of sound heard by a listener moving away
from a stationary source with speed ‘u’ is given as;
𝑓′
=
𝑣−𝑢𝑜
𝑣
𝑓 𝑓′
< 𝑓 or 𝜆′
=
𝑣
𝑣−𝑢𝑜
𝜆, 𝜆′ > 𝜆
Doppler Effect (Frequency Shift)

23.  Apparent frequency of sound heard by stationary listener due to source
moving towards him at speed ‘u’ is given as;
𝑓′ =
𝑣
𝑣−𝑢𝑠
𝑓 𝑓′ > 𝑓 or 𝜆′ =
𝑣−𝑢𝑠
𝑣
𝜆 , 𝜆′ < 𝜆
 Apparent frequency of sound heard by stationary listener due to source
moving away from him at speed ‘u’ is given as;
𝑓′ =
𝑣
𝑣+𝑢𝑠
𝑓 𝑓′ < 𝑓 or 𝜆′ =
𝑣+𝑢𝑠
𝑣
𝜆 , 𝜆′ > 𝜆

24. Applications of Doppler’s effect:
 Ships and submarine (sonar devices)
 Bats (for traveling)
 Radar (for detection)
 Determining velocity of a star w.r.t earth
 To monitor blood flow in major arteries.
 When a star is moving away from Earth then wavelength of
light increases and red shift of spectrum is observed.
 When a star is moving towards the Earth then wavelength of
light decreases and blue shift of spectrum is observed.

25. Stationary waves of frequency 300 Hz are formed in a
medium in which the velocity of sound is 1200
metre/sec. The distance between a node and the
neighboring antinode is
(a) 2 m
(b) 3 m
(c) 1 m
(d) 4 m
QUESTION-1

26. The velocity of sound in a gaseous medium is 330 ms–1.
If the pressure is increased 4-times without change in
temperature, the velocity of sound in gas is:
(a) 330 ms–l
(b) 660 ms–1
(c) 165 ms–1
(d) 1320 ms–1
QUESTION-2

27. At what temperature, the velocity of sound will be double
its value at 273 K?
(a) 2  273 K
(b) 4  273 K
(c) 8  273 K
(d) 16  273 K
QUESTION-3

28. Length of a string tied to two rigid supports is 40 cm.
Maximum length (wavelength in cm) of a stationary wave
produced on it, is
(a) 20
(b) 10
(c) 80
(d) 40
QUESTION-4

29. A source of sound moves towards a stationary observer
with a speed one third that of sound. If the frequency of
the sound from the source is 100 Hz, the apparent
frequency of the sound heard by the observer is
(a) 67 Hz
(b) 100 Hz
(c) 150 Hz
(d) 75 Hz
QUESTION-5

30.  “Vibratory motion is that in which a body moves to and
fro about a fixed position along same path.”
 For example; Motion of simple pendulum
 Simple harmonic motion (SHM) is a special type of
vibratory motion in which;
(i) 𝑎 ∝ 𝑥 (ii) a is directed towards mean
position.
 Restoring force is always directed towards mean position
hence assigned negative sign.
SIMPLE HARMONIC MOTION

31.  “Instantaneous displacement is distance covered by body at
any instant from mean position.”
 “Periodic motion is that which repeats itself after equal time
intervals.”
 “Vibration is one complete round trip of a body about its
mean position.”
 “Time period is defined as time taken by vibrating body to
complete its one vibration and denoted by T.”
 “Frequency is number of vibrations per second” and
by
𝑓 =
1
𝑇
 “Amplitude is maximum distance from mean position.”
CHARACTERISTICS OF SIMPLE HARMONIC MOTION

32.  It consists of a heavy point mass suspended from a rigid support by
of almost weightless and inextensible string.
 Motion of simple pendulum is S.H.M if there is no damping.
 Time period and frequency of simple pendulum are given as;
𝑇 = 2𝜋
𝑙
𝑔
and 𝑓 =
1
2𝜋
𝑔
𝑙
 A second pendulum has following characteristics;
SIMPLE PENDULUM

33.  K.Eins =
1
2
k x𝑜
2 1 −
𝑥2
𝑥𝑜
2
𝐾. 𝐸 𝑚𝑎𝑥 =
1
2
𝑘𝑥𝑜
2
It is at mean position.
(K.E) min = 0 It is at extreme position.
𝐾. Eins = 𝐾. 𝐸 max 1 −
𝑥2
𝑥𝑜
2
 𝑃. 𝐸ins =
1
2
kx2
𝑃. 𝐸 max =
1
2
kx𝑜
2 It is at extreme position.
(P.E)min = 0 It is at mean position.
 Total energy of system =
1
2
kx𝑜
2
energy remain conserve in SHM.
ENERGY CONSERVATION IN SHM

34. If length of a pendulum becomes four times, then its
time period will become
(a) Four times
(b) Six times
(c) Eight time
(d) Two times
QUESTION-1

35. The distance covered by a body in one complete
vibration is 20cm. What is the amplitude of body
(a) 10cm
(b) 5cm
(c) 15cm
(d) 7.5cm
QUESTION-2