1. CHAPTER 10: Mechanical Waves (4 Hours)
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CHAPTER 10: Mechanical Waves (4 Hours)

2. Learning Outcome: 10.1 Waves and energy (1/2 hour)
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Learning Outcome:
10.1 Waves and energy (1/2 hour)
At the end of this chapter, students should be able to:
Explain the formation of mechanical waves and their relationship with energy.

3. Water waves spreading outward from a source.

4. 10.1 Waves and energy Mechanical waves
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10.1 Waves and energy
Waves is defined as the propagation of a disturbance that carries the energy and momentum away from the sources of disturbance.
Mechanical waves
is defined as a disturbance that travels through particles of the medium to transfer the energy.
The particles oscillate around their equilibrium position but do not travel.
Examples of the mechanical waves are water waves, sound waves, waves on a string (rope), waves in a spring and seismic waves (Earthquake waves).
All mechanical waves require
some source of disturbance,
a medium that can be disturbed, and
a mechanism to transfer the disturbance from one point to the next point along the medium. (shown in Figures 10.1a and 10.1b)

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Figure 10.1a
Figure 10.1b

6. Learning Outcome: 10.2 Types of waves (1/2 hour)
At the end of this chapter, students should be able to:
Describe
transverse waves
longitudinal waves
State the differences between transverse and longitudinal waves.

7. transverse progressive wave longitudinal progressive wave
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10.2 Types of waves
mechanical wave
progressive or
travelling wave
stationary wave
transverse progressive wave
longitudinal progressive wave

8. 10.2 Types of waves Progressive wave 10.2.1 Transverse waves
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10.2 Types of waves
Progressive wave
is defined as the one in which the wave profile propagates.
The progressive waves have a definite speed called the speed of propagation or wave speed.
The direction of the wave speed is always in the same direction of the wave propagation .
There are two types of progressive wave,
a. Transverse progressive waves
b. Longitudinal progressive waves.
Transverse waves
is defined as a wave in which the direction of vibrations of the particle is perpendicular to the direction of the wave propagation (wave speed) as shown in Figure 10.3.
direction of vibrations
direction of the propagation of wave
particle
Figure 10.3

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Examples of the transverse waves are water waves, waves on a string (rope), e.m.w. and etc…
The transverse wave on the string can be shown in Figure 10.4.
Longitudinal waves
is defined as a wave in which the direction of vibrations of the particle is parallel to the direction of the wave propagation (wave speed) as shown in Figure 10.5.
Figure 10.4
particle
direction of vibrations
direction of the propagation of wave
Figure 10.5

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Examples of longitudinal waves are sound waves, waves in a spring, etc…
The longitudinal wave on the spring and sound waves can be shown in Figures 10.6a and 10.6b.
Figure 10.6a

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Sound as longitudinal waves
Longitudinal disturbance at particle A resulting periodic pattern of compressions (C) and rarefactions (R).
Figure 10.6b

12. Pm
-Pm P0
P(pressure) A -A
(a)
(b)
(c)
(d) P’

13. Figure (a) and (b)
When the tuning fork is struck, its prongs vibrate, disturbing the air layers near it.
When the prongs vibrate outwards, it compresses the air directly in front of it. This compression causes the air pressure to rise slightly. The region of increased pressure is called a compression.
When the prongs move inwards, it produces a rarefaction, where the air pressure is slightly less than normal. The region of decreased pressure is called a rarefaction.
As the turning fork continues to vibrate, the “compression” and “rarefaction” are formed repeatedly and spread away from it.

14. Figure (c)
The figure shows the displacement of the air particles at particular time, t .
At the region of maximum compression and rarefaction, the particle does not vibrate at all where the displacement of that particle is zero.
Figure (d) – graph of pressure against distance
Compression region
The particles are closest together hence the pressure at that region greater than the atmospheric pressure (P0).
Rarefaction region
The particles are furthest apart hence the pressure at that region less than the atmospheric pressure (P0).

15. Differences between transverse and longitudinal waves
Transverse wave
Longitudinal wave
Particles in the medium vibrate in directions perpendicular to the directions of travel of the wave.
Particles in the medium vibrate in directions parallel to the directions of travel of the wave.
Crest and trough are formed in the medium.
Compression and rarefaction occur in the medium.

16. Learning Outcome: 10.3 Properties of waves (2 hours)
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Learning Outcome:
10.3 Properties of waves (2 hours)
At the end of this chapter, students should be able to:
Define amplitude, frequency, period, wavelength, wave number .
Analyze and use equation for progressive wave,
Distinguish between particle vibrational velocity,
and wave propagation velocity,
Sketch graphs of y-t and y-x

17. 10.3 Properties of waves 10.3.1 Sinusoidal Wave Parameters
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10.3 Properties of waves
Sinusoidal Wave Parameters
Figure 10.7 shows a periodic sinusoidal waveform.
Figure 10.7

18. Amplitude, A
is defined as the maximum displacement from the equilibrium position to the crest or trough of the wave motion.
Frequency, f
is defined as the number of cycles (wavelength) produced in one second.
Its unit is hertz (Hz) or s1.

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Period, T
is defined as the time taken for a particle (point) in the wave to complete one cycle.
In this period, T the wave profile moves a distance of one wavelength, . Thus =
Period of the wave
Period of the particle on the wave
and
Its unit is second (s).

20. Wavelength,
is defined as the distance between two consecutive particles (points) which have the same phase in a wave.
From the Figure 10.7,
Particle B is in phase with particle C.
Particle P is in phase with particle Q
Particle S is in phase with particle T
The S.I. unit of wavelength is metre (m).
Wave number, k
is defined as
The S.I. unit of wave number is m1.

21. is defined as the distance travelled by a wave profile per unit time.
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Wave speed, v
is defined as the distance travelled by a wave profile per unit time.
Figure 10.8 shows a progressive wave profile moving to the right.
It moves a distance of  in time T hence
Figure 10.8
and

22. The S.I. unit of wave speed is m s1.
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The S.I. unit of wave speed is m s1.
The value of wave speed is constant but the velocity of the particles vibration in wave is varies with time, t
It is because the particles executes SHM where the equation of velocity for the particle, vy is
Displacement, y
is defined as the distance moved by a particle from its equilibrium position at every point along a wave.

23. 10.3.2 Equation of displacement for sinusoidal progressive wave
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Equation of displacement for sinusoidal progressive wave
Figure 10.9 shows a progressive wave profile moving to the right.
From the Figure 10.9, consider x = 0 as a reference particle, hence the equation of displacement for particle at x = 0 is given by
Figure 10.9

24. For example, the particles at points O and P.
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Since the wave profile propagates to the right, thus the other particles will vibrate.
For example, the particles at points O and P.
The vibration of particle at lags behind the vibration of particle at O by a phase difference of  radian.
Thus the phase of particle at P is
Therefore the equation of displacement for particle’s vibration at P is
Figure shows three particles in the wave profile that propagates to the right.
Figure

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From the Figure 10.10, when  increases hence the distance between two particle, x also increases. Thus 
Phase difference ( )
distance from the origin (x)
and

26. The wave propagates to the right :
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Therefore the general equation of displacement for sinusoidal progressive wave is given by
The wave propagates to the right :
The wave propagates to the left :
where

27. The wave propagates to the right :
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Some of the reference books, use other general equations of displacement for sinusoidal progressive wave:
The wave propagates to the right :
The wave propagates to the left :

28. 10.3.3 Displacement graphs of the wave
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Displacement graphs of the wave
From the general equation of displacement for a sinusoidal wave,
The displacement, y varies with time, t and distance, x.
Graph of displacement, y against distance, x
The graph shows the displacement of all the particles in the wave at any particular time, t.
For example, consider the equation of the wave is
At time, t = 0 , thus

29. Thus the graph of displacement, y against distance, x is
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Thus the graph of displacement, y against distance, x is

30. Graph of displacement, y against time, t
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Graph of displacement, y against time, t
The graph shows the displacement of any one particle in the wave at any particular distance, x from the origin.
For example, consider the equation of the wave is
For the particle at x = 0, the equation of the particle is given by
hence the displacement-time graph is

31. Example 10.1 : A progressive wave is represented by the equation
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Example 10.1 :
A progressive wave is represented by the equation
where y and x are in centimetres and t in seconds.
a. Determine the angular frequency, the wavelength, the period,
the frequency and the wave speed.
b. Sketch the displacement against distance graph for progressive
wave above in a range of 0 x   at time, t = 0 s.
c. Sketch the displacement against time graph for the particle
at x = 0 in a range of 0 t  T.
d. Is the wave traveling in the +x or –x direction?
e. What is the displacement y when t=5s and x=0.15cm

32. iii. The period of the motion is with
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Solution :
a. By comparing
thus i.
ii.
iii. The period of the motion is
with

33. a. iv. The frequency of the wave is given by
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Solution :
a. iv. The frequency of the wave is given by
v. By applying the equation of wave speed thus
b. At time, t = 0 s, the equation of displacement as a function of
distance, x is given by

34. b. Therefore the graph of displacement, y against distance, x in
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Solution :
b. Therefore the graph of displacement, y against distance, x in
the range of 0 x   is

35. c. The particle at distance, x = 0 , the equation of displacement as
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Solution :
c. The particle at distance, x = 0 , the equation of displacement as
a function of time, t is given by
Hence the displacement, y against time, t graph is

36. d) e)

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Example 10.2 :
Figure 10.11shows a displacement, y against distance, x graph after time, t for the progressive wave which propagates to the right with a speed of 50 cm s1.
a. Determine the wave number and frequency of the wave.
b. Write the expression of displacement as a function of x and t for
the wave above.
Figure

38. By using the formula of wave speed, thus
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Solution :
a. From the graph,
By using the formula of wave speed, thus
b. The expression is given by

39. 10.3.4 Equation of a particle’s velocity in wave
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Equation of a particle’s velocity in wave
By differentiating the displacement equation of the wave, thus
The velocity of the particle, vy varies with time but the wave velocity ,v is constant thus
and
where

40. 10.3.5 Equation of a particle’s acceleration in wave
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Equation of a particle’s acceleration in wave
By differentiating the equation of particle’s velocity in the wave, thus
The equation of the particle’s acceleration also can be written as
and
where
The vibration of the particles in the wave executes SHM.

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Example 10.3 :
A sinusoidal wave traveling in the +x direction (to the right) has an amplitude of 15.0 cm, a wavelength of 10.0 cm and a frequency of 20.0 Hz.
a. Write an expression for the wave function, y(x,t).
b. Determine the speed and acceleration at t = s for the
particle on the wave located at x = 5.0 cm.
Solution :
a. Given
The wave number and the angular frequency are given by

42. By applying the general equation of displacement for wave,
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Solution :
By applying the general equation of displacement for wave,

43. b. i. The expression for speed of the particle is given by
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Solution :
b. i. The expression for speed of the particle is given by
and the speed for the particle at x = 5.0 cm and t = s is
and
where vy in cm s1 and x in centimetres and t in seconds

44. b. ii. The expression for acceleration of the particle is given by
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Solution :
b. ii. The expression for acceleration of the particle is given by
and the acceleration for the particle at x = 5.0 cm and t =
0.500 s is
and
where ay in cm s2 and x in centimetres and t in seconds

45. Exercise 10.1 : 1. A wave travelling along a string is described by
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Exercise 10.1 :
1. A wave travelling along a string is described by
where y in cm, x in m and t is in seconds. Determine
a. the amplitude, wavelength and frequency of the wave.
b. the velocity with which the wave moves along the string.
c. the displacement of a particle located at x = 22.5 cm and
t = 18.9 s.
ANS. : cm, 8.71 cm, Hz; m s1; 0.192 cm

46. Learning Outcome: 10.4 Superposition of waves (1 hour)
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Learning Outcome:
10.4 Superposition of waves (1 hour)
At the end of this chapter, students should be able to:
State the principle of superposition of waves and use it to explain the constructive and destructive interferences.
Explain the formation of stationary wave.
Use the stationary wave equation :
Distinguish between progressive waves and stationary wave.

47. 10.4 Interference of waves 10.4.1 Principle of superposition
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10.4 Interference of waves
Principle of superposition
states that whenever two or more waves are travelling in the same region, the resultant displacement at any point is the vector sum of their individual displacement at that point.
For examples,

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Interference
is defined as the interaction (superposition) of two or more wave motions.
Constructive interference
The resultant displacement is greater than the displacement of the individual wave.
It occurs when y1 and y2 have the same wavelength, frequency and in phase.

49. Destructive interference
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Destructive interference
The resultant displacement is less than the displacement of the individual wave or equal to zero.
It occurs when y1 and y2 have the same wavelength, frequency and out of phase

50. 10.4.2 Stationary (standing) waves
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Stationary (standing) waves
is defined as a form of wave in which the profile of the wave does not move through the medium.
It is formed when two waves which are travelling in opposite directions, and which have the same speed, frequency and amplitude are superimposed.
For example, consider a string stretched between two supports that is plucked like a guitar or violin string as shown in Figure
Figure

51. 10.5.1 Characteristics of stationary waves
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When the string is pluck, the progressive wave is produced and travel in both directions along the string.
At the end of the string, the waves will be reflected and travel back in the opposite direction.
After that, the incident wave will be superimposed with the reflected wave and produced the stationary wave with fixed nodes and antinodes as shown in Figure
Node (N) is defined as a point at which the displacement is zero where the destructive interference occurred.
Antinode (A) is defined as a point at which the displacement is maximum where the constructive interference occurred.
Characteristics of stationary waves
Nodes and antinodes are appear at particular time that is determined by the equation of the stationary wave.

52.  = 2  (the distance between adjacent nodes or antinodes)
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From the Figure 10.17,
The distance between adjacent nodes or antinodes is
The distance between a node and an adjacent antinode is
 = 2  (the distance between adjacent nodes or antinodes)
The pattern of the stationary wave is fixed hence the amplitude of each particles along the medium are different. Thus the nodes and antinodes appear at particular distance and determine by the equation of the stationary wave.
Figure

53. 10.4.3 Equation of stationary waves
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Equation of stationary waves
By considering the wave functions for two progressive waves,
And by applying the principle of superposition hence
and
where

54. A cos kx Explanation for the equation of stationary wave
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Explanation for the equation of stationary wave
A cos kx
Determine the amplitude for any point along the stationary wave.
It is called the amplitude formula.
Its value depends on the distance, x
Antinodes
The point with maximum displacement = A
where
and

55. The point with minimum displacement = 0 Antinodes are occur when
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Therefore
Nodes
The point with minimum displacement = 0
Antinodes are occur when
where
and
Nodes are occur when

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sin t
Determine the time for antinodes and nodes will occur in the stationary wave.
Antinodes
The point with maximum displacement = A
Therefore
where
and
Antinodes are occur when the time are

57. The point with minimum displacement = 0
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Nodes
The point with minimum displacement = 0
Therefore
At time , t = 0, all the points in the stationary wave at the equilibrium position (y = 0).
where
and
Nodes are occur when the time are

58. Graph of displacement-distance (y-x)
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Graph of displacement-distance (y-x)

59. Production of stationary wave
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Production of stationary wave

60. 10.4.4 Differences between progressive and stationary waves
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Differences between progressive and stationary waves
Progressive wave
Stationary wave
Wave profile move.
Wave profile does not move.
All particles vibrate with the same amplitude.
Particles between two adjacent nodes vibrate with different amplitudes.
Neighbouring particles vibrate with different phases.
Particles between two adjacent nodes vibrate in phase.
All particles vibrate.
Particles at nodes do not vibrate at all.
Produced by a disturbance in a medium.
Produced by the superposition of two waves moving in opposite direction.
Transmits the energy.
Does not transmit the energy.

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Example 10.4 :
Two harmonic waves are represented by the equations below
where y1, y2 and x are in centimetres and t in seconds.
a. Determine the amplitude of the new wave.
b. Write an expression for the new wave when both waves are
superimposed.
Solution : a.
b. By applying the principle of superposition, thus

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Example 10.5 :
A stationary wave is represented by the following expression:
where y and x in centimetres and t in seconds. Determine
a. the three smallest value of x (x >0) that corresponds to
i. nodes
ii. antinodes
b. the amplitude of a particle at
i. x = 0.4 cm
ii. x = 1.2 cm
iii. x = 2.3 cm

63. a. i. Nodes particles with minimum displacement, y = 0 with
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Solution :
By comparing
thus
a. i. Nodes particles with minimum displacement, y = 0
with

64. a. ii. Antinodes particle with maximum displacement, y = 5 cm
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Solution :
a. ii. Antinodes particle with maximum displacement, y = 5 cm
b. By applying the amplitude formula of stationary wave, i.
ii.
iii.

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Example 10.6 :
An equation of a stationary wave is given by the expression below
where y and x are in centimetres and t in seconds. Sketch a graph of displacement, y against distance, x at t = 0.25T for a range of 0 ≤ x ≤.
Solution :
By comparing
thus
and

66. The particles in the stationary wave correspond to Antinode
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Solution :
The particles in the stationary wave correspond to
Antinode
Node
The displacement of point x = 0 at time, t = 0.25(2) = 0.50 s in the stationary wave is
where
and
where
and

67. Therefore the displacement, y against distance, x graph is
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Solution :
Therefore the displacement, y against distance, x graph is

68. Exercise 10.2 : 1. The expression of a stationary wave is given by
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Exercise 10.2 :
1. The expression of a stationary wave is given by
where y and x in metres and t in seconds.
a. Write the expression for two progressive waves resulting the
stationary wave above.
b. Determine the wavelength, frequency, amplitude and velocity
for both progressive waves.
ANS. : 4 m, 30 Hz, 0.15 m, 120 m s1
2. A harmonic wave on a string has an amplitude of 2.0 m, wavelength of 1.2 m and speed of 6.0 m s1 in the direction of positive x-axis. At t = 0, the wave has a crest (peak) at x = 0.
a. Calculate the period, frequency, angular frequency and wave
number.
ANS. : 0.2 s, 5 Hz, 10 rad s1 ,5.23 m1

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THE END…
Next Chapter…
CHAPTER 11 :
Sound wave