CHAPTER 10: Mechanical Waves (4 Hours)

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CHAPTER 10 Mechanical Waves(4 Hours)
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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.

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Water waves spreading outward from a source.
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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
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• 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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10.2 Types of waves
mechanical wave
progressive or travelling wave
stationary wave
transverse progressive wave
longitudinal progressive wave
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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.
• 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.

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.
• 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.

Figure 10.4
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
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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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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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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.
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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

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10.3 Properties of waves

• 10.3.1 Sinusoidal Wave Parameters
• Figure 10.7 shows a periodic sinusoidal waveform.

Figure 10.7
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• 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).
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• 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.

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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
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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.

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10.3.2 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
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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
  particles vibration at P is
• Figure 10.10 shows three particles in the wave
  profile that propagates to the right.

Figure 10.10
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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
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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
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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
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10.3.3 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

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• Thus the graph of displacement, y against
  distance, x is

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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

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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 t5s and x0.15cm
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Solution a. By comparing thus i.
ii. iii. The period of the motion is
with
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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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Solution b. Therefore the graph of
displacement, y against distance, x in the
range of 0?? x ? ? is
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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
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• 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.
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Solution a. From the graph, By using the
formula of wave speed, thus b. The expression
is given by
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10.3.4 Equation of a particles 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
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10.3.5 Equation of a particles acceleration in
wave

• By differentiating the equation of particles
  velocity in the wave, thus
• The equation of the particles 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 0.500 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
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Solution By applying the general equation of
displacement for wave,
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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
0.500 s is
and
where vy in cm s?1 and x in centimetres and t in
seconds
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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
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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. 0.327
cm, 8.71 cm, 0.433 Hz 0.0377 m s?1 ?0.192 cm
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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.

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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,

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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.
• It occurs when y1 and y2 have the same
  wavelength, frequency and in phase.

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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

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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.

Figure 10.16
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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 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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• 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 10.17
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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

and
where
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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
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• Therefore
• Nodes
• The point with minimum displacement 0
• Therefore

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
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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
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• Graph of displacement-distance (y-x)

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• Production of stationary wave

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10.4.4 Differences between progressive and
stationary waves
Progressive wave Stationary wave

• Wave profile move.

• Wave profile does not move.

• Particles between two adjacent nodes vibrate with
  different amplitudes.

• All particles vibrate with the same amplitude.

• 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 gt0) 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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Solution By comparing thus a. i. Nodes??
particles with minimum displacement, y 0
with
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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
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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
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Solution Therefore the displacement, y against
distance, x graph is
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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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