PHYSICS CHAPTER 10

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CHAPTER 10: Mechanical Waves

(4 Hours)

PHYSICS CHAPTER 10

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At the end of this chapter, students should be able to: Explain the formation of mechanical waves and their

relationship with energy.

Learning Outcome:

10.1 Waves and energy (1/2 hour)

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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

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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10.2 Types of waves

progressive or travelling wave

transverse progressive

wave

longitudinal progressive

wave

mechanical wave

stationary wave

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10.2 Types of wavesProgressive 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 wavesb. 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

direction of the propagation of wave

particleFigure 10.3

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v

A

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.

direction of the propagation of wave

direction of vibrations

Figure 10.4

Figure 10.5

particle

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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.

vC R C R C R C R C

A

Figure 10.6a

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C R C R C R C RC R C R C R C R C R C Rv

Longitudinal disturbance at particle A resulting periodic pattern of compressions (C) and rarefactions (R).

Figure 10.6b

Sound as longitudinal waves

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v

Pm

-Pm

P0

R

P(pressure)

A

-A

C C C CR R R

(a)

(b)

(c)

(d)

P’

y

x0

x

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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)

Figure (d) – graph of pressure against distance

• 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.

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).

PHYSICS 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.

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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

Learning Outcome:

10.3 Properties of waves (2 hours)

kxtAtxy sin,

fv dt

dyvy

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10.3 Properties of waves10.3.1 Sinusoidal Wave Parameters Figure 10.7 shows a periodic sinusoidal waveform.

B

C

QP TS

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 s1.

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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

fT

1

Its unit is second (s).

PHYSICS 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

The S.I. unit of wave number is m1.

20

2

k

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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

v

Figure 10.8

time

distancev and

Tv

f

1T

fv

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The S.I. unit of wave speed is m s1. 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.

tAvy cos

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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

10.3.2 Equation of displacement for sinusoidal progressive wave

ent)(displacem y

origin) from (distance xx

yv

A

A

O P

Figure 10.9

tAty sin

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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 10.10 shows three particles in the wave profile that propagates to the right.

t

tAty sin

Δ

O Px

2Δ

Q

xFigure 10.10

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From the Figure 10.10, when increases hence the distance

between two particle, x also increases. Thus

x

Phase difference

( )

x

2

distance from the

origin (x)

k

2

x

2

and

kx

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Therefore the general equation of displacement for sinusoidal progressive wave is given by

kxtAtxy sin,

The wave propagates to the right :

The wave propagates to the left :

kxtAtxy sin,

origin thefrom distance : x

where

frequencyangular : ω wave theof Amplitude : A

number wave: k

time: t

a as particle theofnt displaceme:, txytx and offunction

angle phase :

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Some of the reference books, use other general equations of displacement for sinusoidal progressive wave:

tkxAtxy sin,

The wave propagates to the right :

The wave propagates to the left :

tkxAtxy sin,

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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

10.3.3 Displacement graphs of the wave

kxtAy sin

kxtAy sin

kxAy 0sin kxAy sin kxAy sin

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

0

A

A

y

x

2

2

3 2

v

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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

kxtAy sin

0sin ktAy tAy sin

0

A

A

y

t

2

T T2

3T T2

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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

Example 10.1 :

xttxy sin2,

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Solution :

a. By comparing

thus

i.

ii.

iii. The period of the motion is

kxtAtxy sin,with xttxy sin2,

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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(cm) y

(cm) x

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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 (cm) y

(s) t

PHYSICS CHAPTER 10

d)

e)

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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 s1.

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.

Example 10.2 :

0

3

3

(cm) y

(cm) x0.1 0.2

Figure 10.11

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Solution :

a. From the graph,

By using the formula of wave speed, thus

b. The expression is given by

1s m 5.0 vm 100.1 2

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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

10.3.4 Equation of a particle’s velocity in wave

dt

dyvy kxtAy sinand

kxtAvy cos

kxtAdt

dvy sin

wavein the particle theof velocity : yv

vvy

where

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By differentiating the equation of particle’s velocity in the wave, thus

The equation of the particle’s acceleration also can be written as

10.3.5 Equation of a particle’s acceleration in wave

dt

dva y

y kxtAvy cosand

kxtAay sin2

kxtAdt

day cos

wavein the particle theofon accelerati : ya

yay2

where

The vibration of the particles in the wave executes SHM.

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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

Example 10.3 :

Hz020 cm; 0.01 cm; 0.15 .fλA cm 0.150,0 y

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Solution :

By applying the general equation of displacement for wave,

Hz020 cm; 0.01 cm; 0.15 .fλA

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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

Hz020 cm; 0.01 cm; 0.15 .fλA

and

where vy in cm s1 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

Hz020 cm; 0.01 cm; 0.15 .fλA

and

where ay in cm s2 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. Determinea. 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 s1; 0.192 cm

xttxy 1.7272.2sin327.0,

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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.

Learning Outcome:

10.4 Superposition of waves (1 hour)

tkxAy sincos

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2tt AA

AA

1y

2y

1y

2y

AAAyyy 221 021 AAyyy

1tt A2

0t A A A

A1y

2y

1y

2y

10.4 Interference of waves10.4.1 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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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.

10.4.2 Interference

21 yyy

y

x0

1y2

y

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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

1y

y

x0

021 yyy2y

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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

N A N A N A N

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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

N A N A N A N

2

4

2

4

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By considering the wave functions for two progressive waves,

And by applying the principle of superposition hence

10.4.3 Equation of stationary waves

)sin(,1 kxtatxy )sin(,2 kxtatxy

txytxyy ,, 21

kxtakxtay sinsin kxtkxtay sincoscossin kxtkxta sincoscossin

kxtay cossin2 aA 2tkxAy sincos and

waveeprogressiv theof amplitude : awhere wavestationary theof amplitude : A

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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

AkxA cos1cos kx

1cos 1kx,...3,2,,0 kx

,...3,2,1,0m wheremkx

kx

m

2

kand

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Therefore

Nodes The point with minimum displacement = 0

Therefore

,...2

3,,

2,0

x

2

mx

Antinodes are occur when

0cos kxA 0cos 1kx

,...2

5,

2

3,

2

kx

,...5,3,1n where2

nkx

kx

2

n

2

kand

,...4

5,

4

3,

4

x

4

nx

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

AtA sin 1sin t 1sin 1t

,...2

5,

2

3,

2

t

,...5,3,1n where2

n t

2

nt

T

2and

,...4

5,

4

3,

4

TTTt Tt

4

n 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).

0sin tA 0sin t

0sin 1t,...3,2,,0 t

,...3,2,1,0m where mt

m

tT

2and

,...2

3,,

2,0

TT

Tt Tt

2

m Nodes are occur when the time are

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4

Tt

y

x

A N A N A N A N A

4

3Tt

0

A

A 2

2

3 24

4

34

54

7

0t2

,T

T,

Graph of displacement-distance (y-x)

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Production of stationary wave

0t4

Tt

2

Tt

4

3Tt Tt

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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.

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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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

Example 10.4 :

xttxy sin3,1

xttxy sin3,2

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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

Example 10.5 :

txy sincos5

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Solution :

By comparing

thus

a. i. Nodes particles with minimum displacement, y = 0

txy sincos5 tkxAy sincoswith

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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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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

Example 10.6 :

txy sin2cos8

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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

(cm) y

)cm(x

A N A N A

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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 s1

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 s1 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 s1 ,5.23 m1

txy 60sin5.0cos3.0

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THE END…Next Chapter…

CHAPTER 11 :Sound wave