PHYSICS
CHAPTER 10
CHAPTER 10:
Mechanical Waves
(4 Hours)
1
PHYSICS
CHAPTER 10
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.
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CHAPTER 10
Water waves spreading outward from a source.
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CHAPTER 10
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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PHYSICS
CHAPTER 10
Figure 10.1a
Figure 10.1b
5
PHYSICS
CHAPTER 10
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.
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CHAPTER 10
10.2 Types of waves
mechanical wave
progressive
or
travelling wave
transverse
progressive
wave
stationary wave
longitudinal
progressive
wave
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CHAPTER 10
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.
10.2.1 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
particle
Figure 10.3
direction of the propagation
of wave
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PHYSICS


CHAPTER 10
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.

v
A
Figure 10.4
10.2.2 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.
particle
direction of vibrations
Figure 10.5
direction of the propagation
of wave
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PHYSICS


CHAPTER 10
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.
C
R
C
R
C
R
A
C
R
C

v
Figure 10.6a
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CHAPTER 10
Sound as longitudinal waves
C R C R C R C RC R C R C R C R C R C R

v
Figure 10.6b

Longitudinal disturbance at particle A resulting periodic
pattern of compressions (C) and rarefactions (R).
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PHYSICS
CHAPTER 10

v
(a)
R
C
R
C
C
R
R
C
(b)
y


A
(c)
0
x
-A
P(pressure)
Pm
P’
(d)
P0
-Pm
x
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CHAPTER 10
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.
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PHYSICS
CHAPTER 10
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).
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CHAPTER 10
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.
PHYSICS
CHAPTER 10
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,
yx, t   A sint  kx

dy
Distinguish between particle vibrational velocity, v y 
dt
and wave propagation velocity, v  f .

Sketch graphs of y-t and y-x
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CHAPTER 10
10.3 Properties of waves
10.3.1 Sinusoidal Wave Parameters

Figure 10.7 shows a periodic sinusoidal waveform.

B
C
Q
S
P

T

Figure 10.7
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CHAPTER 10
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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PHYSICS
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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
1
T
f
Its unit is second (s).
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CHAPTER 10
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

k
2

The S.I. unit of wave number is m1.
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CHAPTER 10
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.

v


Figure 10.8
It moves a distance of  in time T hence
distance
v
time
v

T
and
v  f
1
T
f
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PHYSICS


CHAPTER 10
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
v y  A cost   
Displacement, y
 is defined as the distance moved by a particle from its
equilibrium position at every point along a wave.
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CHAPTER 10
10.3.2 Equation of displacement for sinusoidal
progressive wave

Figure 10.9 shows a progressive wave profile moving to the
right.
y (displacem ent)

v
A
y
O
A

x
P
x (distance from origin)
Figure 10.9
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
yt   A sin t
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PHYSICS


CHAPTER 10
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 t  

Therefore the equation of displacement for particle’s
vibration at P is y t  A sin t  






Figure 10.10 shows three particles in the wave profile that
propagates to the right.
Δ  2
Δ  
O
Figure 10.10
x
P
x
Q
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PHYSICS

CHAPTER 10
From the Figure 10.10, when  increases hence the distance
between two particle, x also increases. Thus
Phase difference
( )

distance from the
origin (x)
x

x

2 
 2 
   x
  
  kx
and
 2 

k
  
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PHYSICS

CHAPTER 10
Therefore the general equation of displacement for sinusoidal
progressive wave is given by
The wave propagates to the right :
yx, t   A sin t  kx
The wave propagates to the left :
yx, t   A sin t  kx
where
yx,t  : displaceme nt of the particle as a
function of x and t
A : Amplitude of the wave
ω : angular frequency
k : wave number
x : distance from the origin
t : time
 : phase angle
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PHYSICS

CHAPTER 10
Some of the reference books, use other general equations of
displacement for sinusoidal progressive wave:
The wave propagates to the right :
yx, t   A sin kx  t 
The wave propagates to the left :
yx, t   A sin kx  t 
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PHYSICS
CHAPTER 10
10.3.3 Displacement graphs of the wave

From the general equation of displacement for a sinusoidal
wave,
y  A sin t  kx
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
y  A sin t  kx
At time, t = 0 , thus y  A sin  0  kx
y  A sin  kx
y   A sin kx
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PHYSICS

CHAPTER 10
Thus the graph of displacement, y against distance, x is
y

v
A
0
A

2

3
2
2
x
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CHAPTER 10
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
y  A sin t  kx

For the particle at x = 0, the equation of the particle is
given by
y  A sin t 
y  A sin t  k 0
hence the displacement-time graph is
y
A
0
A
T
2
T
3T
2
t
2T
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PHYSICS
CHAPTER 10
Example 10.1 :
A progressive wave is represented by the equation
yx, t   2 sin t  x 
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
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Solution :
a. By comparing
yx, t   2 sin t  x  with yx, t   A sin t  kx
thus
i.
ii.
iii. The period of the motion is
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PHYSICS
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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
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PHYSICS
CHAPTER 10
Solution :
b. Therefore the graph of displacement, y against distance, x in
the range of 0 x 
 is
y (cm)
x (cm)
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PHYSICS
CHAPTER 10
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
y (cm)
t (s)
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PHYSICS
CHAPTER 10
d)
e)
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PHYSICS
CHAPTER 10
Example 10.2 :
y (cm)
3
0
x (cm)
1.0
2.0
3
Figure 10.11
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.
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PHYSICS
CHAPTER 10
Solution : v  0.5 m s 1
2
a. From the graph,   1.0 10
m
By using the formula of wave speed, thus
b. The expression is given by
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PHYSICS
CHAPTER 10
10.3.4 Equation of a particle’s velocity in wave

By differentiating the displacement equation of the wave, thus
dy
and y  A sin t  kx
vy 
dt
d
v y   A sin t  kx
dt
v y  A cost  kx

where v y : velocity of the particle in the wave
The velocity of the particle, vy varies with time but the wave
velocity ,v is constant thus
vy  v
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CHAPTER 10
10.3.5 Equation of a particle’s acceleration in wave

By differentiating the equation of particle’s velocity in the wave,
thus
dv y
and v y  A cos t  kx
a 
y
dt
d
a y   A cost  kx
dt


a y   A 2 sin t  kx
where a y

: accelerati on of the particle in the wave
The equation of the particle’s acceleration also can be written as
a y   y
2
The vibration of the particles
in the wave executes SHM.
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PHYSICS
CHAPTER 10
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 = 0.500 s for the
particle on the wave located at x = 5.0 cm.
Solution : A  15.0 cm; λ  10.0 cm; f  20.0 Hz
a. Given y 0,0  15.0 cm
The wave number and the angular frequency are given by
 
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PHYSICS
CHAPTER 10
Solution : A  15.0 cm; λ  10.0 cm; f  20.0 Hz
By applying the general equation of displacement for wave,
42
PHYSICS
CHAPTER 10
Solution : A  15.0 cm; λ  10.0 cm; f  20.0 Hz
b. i. The expression for speed of the particle is given by
and
where vy in cm s1 and x in centimetres and t in seconds
and the speed for the particle at x = 5.0 cm and t = 0.500 s is
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PHYSICS
CHAPTER 10
Solution : A  15.0 cm; λ  10.0 cm; f  20.0 Hz
b. ii. The expression for acceleration of the particle is given by
and
where ay in cm s2 and x in centimetres and t in seconds
and the acceleration for the particle at x = 5.0 cm and t =
0.500 s is
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PHYSICS
CHAPTER 10
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. : 0.327 cm, 8.71 cm, 0.433 Hz; 0.0377 m s1; 0.192 cm
yx, t   0.327 sin 2.72t  72.1x 
45
PHYSICS
CHAPTER 10
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 :
y  A coskx sin t

Distinguish between progressive waves and stationary
wave.
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PHYSICS
CHAPTER 10
10.4 Interference of waves
10.4.1 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,
A

y2
A

y1
2A
  
y  y1  y2  A  A  2 A
A

y2
A

y1
t 0
t  t1
t  t2
A

y1
A

y2
  
y  y1  y2  A  A  0
A

y2
A

y1
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PHYSICS
CHAPTER 10
10.4.2 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.
  

y

y2

y1
y  y1  y2
x
0

It occurs when y1 and y2 have the same wavelength, frequency
and in phase.
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PHYSICS
CHAPTER 10
Destructive interference
 The resultant displacement is less than the displacement of
the individual wave or equal to zero.
y
0


y2   
y  y1  y2  0
x

y1
It occurs when y1 and y2 have the same wavelength, frequency
and out of phase
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CHAPTER 10
10.4.2 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
10.16.
N
A
N
A
N
A
N
Figure 10.16
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PHYSICS
CHAPTER 10
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 10.16.
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.



10.5.1 Characteristics of stationary waves

Nodes and antinodes are appear at particular time that is
determined by the equation of the stationary wave.
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PHYSICS
N
N
A

4

2


Figure 10.17
From the Figure 10.17,
 The distance between adjacent nodes or antinodes is



CHAPTER 10
N
A
A
N

2
The distance between a node and an adjacent antinode is
4
 = 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.
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PHYSICS
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10.4.3 Equation of stationary waves

By considering the wave functions for two progressive waves,

And by applying the principle of superposition hence
y1 x, t   a sin( t  kx)
y2 x, t   a sin( t  kx)
 

y  y1 x, t   y2 x, t 
y  a sin t  kx  a sin t  kx
y  asin t cos kx  cos t sin kx
 asin t cos kx  cos t sin kx
y  2a sin t cos kx
y  A cos kx sin t and A  2a
where
A : amplitude of the stationary wave
a : amplitude of the progressiv e wave
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PHYSICS
CHAPTER 10
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
A cos kx  A
cos kx  1
kx  cos 1 1
kx  0,  ,2 ,3 ,...
kx  m where m  0,1,2,3,...
2
m
and k 
x

k
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PHYSICS
CHAPTER 10
Therefore

Nodes

m
x   
2
Antinodes are occur when

3
x  0, ,  , ,...
2
2
The point with minimum displacement = 0
A cos kx  0 1
kx  cos 0
 3 5
kx  , , ,...
2 2 2
n
kx   where n  1,3,5,...
2
2
n and
k
x
Therefore

2k
n
x   
4
Nodes are occur when
x
 3 5
,
4 4
,
4
,...
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PHYSICS
CHAPTER 10

sin t

Determine the time for antinodes and nodes will occur in
the stationary wave.
Antinodes


The point with maximum displacement = A
A sin t  A
Therefore
n
t   T
4
sin t  1 1
t  sin 1
 3 5
t  , , ,...
2 2 2
n
where n  1,3,5,...
t 
2
2
n
and  
t
T
2
Antinodes are occur when the
time are T 3T 5T
t
,
4 4
,
4
,...
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PHYSICS

CHAPTER 10
Nodes

The point with minimum displacement = 0
A sin t  0
sin t  0 1
t  sin 0
t  0,  ,2 ,3 ,...
t  m where m  0,1,2,3,...
2
and  
t
T

m
Therefore
m
t   T
2

Nodes are occur when the time
are
T
3T
t  0, , T , ,...
2
2
At time , t = 0, all the points in the stationary wave at the
equilibrium position (y = 0).
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PHYSICS
CHAPTER 10
Graph of displacement-distance (y-x)
y
T
t
4
A
0
A
A
T
t  0, , T
2

4

2
3
4

5 3
4 2
7
4
2
N
A
N
A
N
N
A
A
x
3T
t
4
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PHYSICS
CHAPTER 10
Production of stationary wave
t 0
T
t
4
T
t
2
3T
t
4
t T
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PHYSICS
CHAPTER 10
10.4.4 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.


Neighbouring particles vibrate
with different phases.

Particles between two adjacent
nodes vibrate with different
amplitudes.
Particles between two adjacent
nodes vibrate in phase.

All particles vibrate.


Produced by a disturbance in
a medium.

Particles at nodes do not vibrate
at all.
Produced by the superposition of
two waves moving in opposite
direction.

Transmits the energy.

Does not transmit the energy.
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PHYSICS
CHAPTER 10
Example 10.4 :
Two harmonic waves are represented by the equations below
y1 x, t   3 sin t  x 
y2 x, t   3 sin t  x 
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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PHYSICS
CHAPTER 10
Example 10.5 :
A stationary wave is represented by the following expression:
y  5 cos x sin t
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
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PHYSICS
CHAPTER 10
Solution :
By comparing
y  5 cos x sin t
with
y  A cos kx sin t
thus
a. i. Nodes particles with minimum displacement, y = 0
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PHYSICS
CHAPTER 10
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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PHYSICS
CHAPTER 10
Example 10.6 :
An equation of a stationary wave is given by the expression below
y  8 cos 2x sin t
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
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PHYSICS
CHAPTER 10
Solution :
The particles in the stationary wave correspond to
 Antinode
where

and
Node
where
and
The displacement of point x = 0 at time, t = 0.25(2) = 0.50 s in the
stationary wave is
66
PHYSICS
CHAPTER 10
Solution :
Therefore the displacement, y against distance, x graph is
y (cm)
x(cm)
A
N
A
N
A
67
PHYSICS
CHAPTER 10
Exercise 10.2 :
1. The expression of a stationary wave is given by
y  0.3 cos 0.5x sin 60t
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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PHYSICS
CHAPTER 10
THE END…
Next Chapter…
CHAPTER 11 :
Sound wave
69