Giáo khoa vật lý 12 (Tiếng Anh)

MINITRY OF EDUCATION AND TRANNING
PHYSICS
12
2000
Translated by VNNTU – Dec. 2001
Page 2
TABLE OF CONTENTS
Part I. OSCILLATIONS AND WAVES 8
Chapter I – Mechanical Oscillations 8
§1. Periodic and simple harmonic motions. Oscillation of a mass-spring system. 8
1. Oscillations 8
2. Periodic motion 8
3. Mass-spring system. Simple harmonic motion 8
§2. Exploring a simple harmonic motion 10
Uniform circular motion and simple harmonic motion 10
2. Angular phase and angular frequency of a simple harmonic motion 11
3. Free motion 11
4. Velocity and acceleration in a simple harmonic motion 11
5. Oscillation of a simple pendulum 12
§3. Energy in a simple harmonic motion 13
1. Energy changes during oscillation 13
2. Conservation of mechanical energy during oscillation 14
§4. - §5. The combination of oscillations 15
1. Examples of the combination of oscillations 15
2. Phase-differences between oscillations 15
3. Vector-diagram method 16
4. The combination of two oscillations of same directions and frequencies 16
5. Amplitude and initial phase of the combinatorial oscillation 17
§6. - §7. Underdamped and forced oscillations 18
1. Underdamped oscillation 18
2. Forced oscillation 18

3. Resonance 19
4. Applying and surmounting resonant phenomenon 19
5. Self-oscillation 20
Summary of Chapter I 20
Chapter II – Mechanical wave. Acoustics 22
§8. Wave in mechanics 22
1. Natural mechanical waves 22
2. Oscillation phase transmission. Wavelength. 22
3. Period, frequency and velocity of waves 23
4. Amplitude and energy of waves 23
§9. - §10. Sound wave 24
Sound wave and the sensation of sound 24
2. Sound transmission. Speed of sound 25
3. Sound altitude 25
Timbre 25
5. Sound energy 26
6. Sound loudness 26
7. Sound source – Resonant box 27
§11. Wave interference 28
1. Interferential phenomenon 28
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2. Theory of interference 28
3. Standing wave 29
Summary of Chapter II 31
Chapter III – Electric oscillation, Alternating current 32
§12. Harmonic oscillation voltage. Alternating current 32
1. Harmonic oscillation voltage 32
2. Alternating current 32
3. Root mean square (rms) value of intensity and voltage 33

§13. - §14. Alternating current in a circuit containing only resistance, inductance or capacitance 34
1. Relation between current and voltage 34
2. Ohm’s law for an AC circuit containing only resistance 34
1. Effect of capacitors to the alternating current 34
2. Relation between current and voltage 35
3. Ohm’s law for an AC circuit containing only capacitance 35
1. Effects of an inductor to the alternating current 36
2. Relation between current and voltage 36
3. Ohm’s Law for an AC circuit with inductors 36
§15. Alternating current in an RLC circuit 37
Electric current and voltage in an RLC circuit 37
Relation between current and voltage in an RLC circuit 38
3. Ohm’s Law for an RLC circuit 38
4. Resonance in an RLC circuit 39
§16. Power of the alternating current 39
1. Power of the alternating current 39
2. Significance of the power coefficient 40
§17. Problems on AC circuits 41
Problem 1 41
Problem 2 41
§18. Single-phase AC generator 42
1. Operational principle of single-phase AC generators 42
2. Structure of an AC generator 42
§19. Three-phase alternating current 43
1. Operational principle of three-phase AC power generators 43
2. WYE connection 44
3. Delta connection 45
§20. Asynchronous three-phase motors 45
Operational principle of asynchronous three-phase motors 45
Rotating magnetic field of three-phase current 46

3. Structure of an asynchronous three-phase motor 46
§21. Transformers. Electricity transmission 47
1. Operational principle and structure of transformers 47
2. Transformation of current and voltage via transformer 47
3. Transmission of power 48
§22. Generation of direct current 49
1. Benefits of direct current 49
Half-cycle rectifying method 49
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3. Two half-cycle rectifying method 49
4. Operational principle of DC power generators 50
Summary of Chapter III 50
Chapter IV – Electromagnetic oscillation. Electromagnetic wave 52
§23. Oscillation circuits. Electromagnetic oscillation 52
1. Fluctuation of charges in an oscillation circuit 52
2. Electromagnetic oscillation in an oscillation circuit 53
§24. Alternating current, high-frequency electromagnetic oscillation, and mechanical oscillation 54
1. Electric oscillation in an alternating current 54
2. High-frequency electromagnetic oscillation 54
3. Electromagnetic oscillation and mechanical oscillation 54
§25. Electromagnetic field 57
1. Fluctuated electric field and fluctuated magnetic field 57
2. Electromagnetic field 57
3. Transmission of electromagnetic interaction 57
§26. Electromagnetic waves 58
1. Electromagnetic waves 58
2. Properties of electromagnetic waves 58
3. Electromagnetic waves and wireless communication 58
§27. Transmitting and receiving electromagnetic waves 59

Periodic-oscillation transmitters using transistors 59
2. Open oscillation circuit. Antenna 60
Principles of transmitting and receiving electromagnetic waves 60
§28. - §29. A glance at radio transmitters and receivers 61
1. Principle of oscillation amplification 61
2. Principle of amplitude modulation 62
3. Operational principle of radio transmitters 62
4. Operational principle of radio receivers 63
Summary of Chapter IV 64
Part II. OPTICS 66
Chapter V – Light reflection and refraction 66
§30. Light transmission. Light reflection. Plane mirror 66
1. Light propagation 66
2. Light reflection 67
3. Plane mirror 67
§31. Concave spherical mirrors 68
Definitions 68
2. Reflection of a light in a concave spherical mirror 69
3. Formation of images by concave spherical mirror 69
4. Main focal point. Focal length 70
5. Method to draw an object’s image obtaining from a concave spherical mirror 70
§32. Convex spherical mirrors. Convex spherical mirror equations. Applications of convex spherical
mirrors 72
1. Convex spherical mirror 72
2. Convex spherical mirror equations 72
3. Applications of convex spherical mirrors 74
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§33. Light refraction 75
Light refraction phenomenon 75

2. The law of light refraction 75
3. Index of refraction (refractive index) 76
§34. Total internal reflection 77
Total internal reflection 77
2. Conditions to achieve total internal reflection 78
3. Critical angle 78
4. Applications of total internal reflection 79
§35. Prism 80
1. Definition 80
2. Path of a monochromatic ray through a prism. Angle of deviation 80
3. Prism equations 80
4. Minimum deviation angle 80
§36. Thin lenses 82
1. Definition 82
2. Main focal point. Optical center. Focal length 82
3. Supplemental focal points. Focal plane 83
4. Lens power 84
§37. Image of an object through lenses. Lenses equations 85
1. Observing an object’s image through a lens 85
2. Method to draw an object’s image through a lens 85
3. Lens equation 86
4. Lateral magnification 87
The human eye and optical instruments 91
§38. Camera and the human eye 91
1. Camera 91
2. The human eye 91
§39. Eye’s defects and correcting methods 94
1. Near-sightedness (myopia) 94
Farsightedness (hyperopia) 95
1.State the characteristics of near-sighted eye and the correcting method. 95

§40. Magnifying glass 95
1. Definition 95
2. Near point and infinite point 96
3. Angular magnification 96
§41. Microscope and telescope 98
1. Microscope 98
2. Telescope 99
Chapter VII – The wave-nature of light 103
§42. Light dispersion phenomenon 103
1. Experiment on light dispersion phenomenon 103
2. Experiment on monochromatic light 103
3. Synthesizing white light 104
4. Dependence of the index of refraction of a transparent medium on the color of the light 104
§43. Light interference phenomenon 105
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1. Young's experiment on light interference phenomenon 105
2. Explanation of the phenomenon 105
3. Conclusion 106
1. Describe the experiment on the interference of light? 106
§44. Measuring the wavelength of the light. The wavelength and the color of the light 106
1. Interference fringe distance 106
2. The wavelength and the color of the light 108
§45. Spectrometer. Continuous spectrum 108
1. Relation between the index of refraction of a medium and the wavelength of the light 108
2. Spectrometer 109
3. Continuous spectrums: 109
§46. Line spectrum 110
1. Emission line spectrum 110
2. Absorption line spectrum 111

3. The spectroscopic analysis approach and its advantages 112
§47. Infrared and ultraviolet rays 112
1. Experiments to discover infrared and ultraviolet rays 112
2. The infrared ray 113
3. The ultraviolet ray 113
§48. X-rays 114
1. X-ray tube 114
2. The nature of X-rays 114
3. Properties and uses of X-rays 114
4. Electromagnetic waves scale 115
Chapter VIII – Light quantum 118
§49. The photoelectric effect 118
1. Hertz’s experiment 118
The experiment with a photocell 118
§50. The quantum hypothesis and photoelectric laws 120
1. Photoelectric laws 120
2. The quantum hypothesis 120
3. Explaining photoelectric laws by using the quantum hypothesis 121
4. Wave-particle duality of the light 122
§51. Light dependant resistor and photoelectric battery 123
1. The photoconduction phenomenon 123
2. Light dependant resistor (LDR) 123
3. The photoelectric battery 124
§52. Optical phenomena relating to the quantum property of the light 125
1. The luminescence 125
2. Photochemical reactions 125
§53. Application of the quantum hypothesis to hydrogen atom 126
1. The Bohr model of the atom 126
2. Using the Bohr model to explain hydrogenous line spectrum 127
Summary of Chapter VIII 128

Part III. NUCLEAR PHYSICS 130
Chapter IX – Basic knowledge on the atomic nucleus 130
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§54. Structure of the nucleus. The unit for atomic 130
1. Structure of the nucleus 130
2. Nuclear forces 130
3. Isotopes 130
4. The unified atomic mass unit 131
§55. Radioactivity 132
Radioactivity 132
2. The radioactive decay law 133
§56. Nuclear reactions 134
1. Nuclear reactions 134
2. Conservation laws in nuclear reactions 134
3. Application of conservative laws to radioactivity. Transmutation rules 135
§57. Artificial nuclear reactions. Applications of isotopes 136
1. Artificial nuclear reactions 136
2. Particle accelerators 136
§58. Einstein’s relation between mass and energy 138
1. Einstein’s axioms 138
§59. THE LOSS OF MASS. NUCLEAR ENERGY 140
1. The loss of mass and binding energy 140
Exothermic and endothermic nuclear reactions 140
3. Two exothermic nuclear reactions 141
§60. Nuclear fission. Nuclear reaction plants 142
1. Chain nuclear reactions 142
Nuclear reaction plants 143
§61. THERMONUCLEAR REACTION 144
Supplemental reading:

Primary Particles
145
1. Properties of the primary particles: 145
2. Antiparticles. Antimatter. 146
3. Fundamental interactions. Classification of primary particles. 146
4. Quarks 147
SUMMARY of chapter IX 147
Part IV. PRACTICAL EXPERIMENTAL EXERCISES 150
Experimental exercise 1 – Clarification of the law on the simple pendulum’s oscillation.
Determination of the gravity acceleration 150
Experimental exercise 2 – Determination of Sound wavelengths and frequencies 152
Experimental exercise 3 – The alternating current circuit with R, L, C 154
Experimental exercise 4 – The refraction index of glass 156
Experimental exercise 5 – Observation of light dispersion and interference phenomena 158
COMBINED EXPERIMENTAL EXERCISES 160
Experimental exercise A – Determination of Capacitance and inductance (2 sections) 160
Experimental exercise B – Characteristics and applications of transistors (2 sections) 163
Experimental exercise C – Determination of focal length of lenses (2 sections) 166
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Part I.

OSCILLATIONS AND WAVES
Chapter I –

MECHANICAL OSCILLA TIONS
§1.

P
ERIODIC AND SIMPLE HARMONIC MOTIONS

. O
SCILLATION OF A MASS
-
SPRING SYSTEM
.
1.

Oscillations
Flower stirs in the branch as the wind breezes. Pendulum of the clock swings to the left and right. On the
rippled lake, a small piece of wood bobs and rolls. The string of the guitar vibrates when it is played.
In the examples above, things move in a small space, not too far away from a certain equilibrium
position. The movement likes that is called the
oscillation
.
An oscillation, or vibration, is a limited
motion on a space, repeating back and forth many times around an equilibrium position.
That position often is where thing is at rest (does not move): when there is no wind, a clock does not
work, a smooth lake, non vibrating guitar’s strings.
2.

Periodic motion
Observing the oscillation of a pendulum of the clock, after a certain period of time of 0.5s, it passes
through a lowest position from the left to the right. The oscillation like that is called the
periodic
oscillation
.
Periodic oscillation is the oscillation whose state is repeated as it was after a constant period
of time.
The smallest period of time of T after that states of oscillation are repeated as they were is called
the

period
of periodic oscillation.
The quantity f =
T
1

showing the number of oscillations (i.e. how many times a state of oscillation is
repeated as it were) per unit of time is called the
frequency
. Frequency is usually specified in hertz (Hz).
In the example above, the period of the pendulum is T = 0.5s so its frequency is f =
5.0
1
= 2Hz, it means
that the pendulum carries out 2 oscillations in a second.
The vibration of the guitar’s strings do not permanently maintained. It is damped then ended. But if it is
observed in a very small period of time, it is approximately a
harmonic oscillation
.
3.

Mass-spring system. Simple harmonic motion
Considering a mass-spring system consisted of a small ball of m kg attached rigidly to a spring of
negligible mass, put in the horizontal plane as shown (figure 1.1a). There is a small hole through the ball
so it can be translated along a fixed rod in the same plane.
We choose a datum axis that coincides with the rod, is directed from the left to the right, and the origin O
is the equilibrium position of the ball (position where the ball is at rest). A ball is deflected to the right by
a force
F
then released (figure 1.1b; the spring is not shown). It is observed that the ball moves toward

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the point O, passes through O. This translation is repeated many times, i.e. the ball oscillates around the
equilibrium position O.
This phenomenon is analyzed as following: when the ball is pulled to an ordinate x, forces exert into it
consist of the pulling force
F’
, the elastic force
F
of the spring, the gravity force and the reacting force of
the rod to the ball (these two forces are not shown in the figure). The gravity and reacting force are in the
vertical plane, equal to each other and opposite in the direction so they have no affect on the horizontal
translation of the ball. At the time the ball is released, there is only an elastic force exert on it.
Within the limitation of elasticity of the spring, the force
F
is always proportional with the displacement
x of the ball from the equilibrium position (is also the deflection of the spring), and directs toward the
point O. Since
F
is along the coordinate axis, it can be written as:
F = - kx (1-1)
Here k is the
spring constant
(stiffness) of the spring, and the minus sign indicates that the force F is
acting in opposite direction compared with the deflection x of the ball.
According to Newton’s second law, it can be written as F = ma, or ma = - kx. Thus a =
x
m
k
−

.
It is known that a velocity and acceleration are defined by v =
t
x
∆
∆
and a =
t
v
∆
∆
. If the motion is
investigated in a very small period of time ∆t, then
t
x
∆
∆
becomes a derivative of x with respect to time
variable, v =
x
!
; similarly,
t
v
∆
∆
becomes a derivative of x respecting to time t, a =
v
!
, i.e. a second order

derivative of x with respect to time variable: a =
x
!!
.
Therefore we have
x
!!
=
x
m
k
−
(1-2)
Let ω =
m
k
then
x
!!
+ ω
2
x = 0 (1-2a)
It can be proved having a solution of x = Asin(ωt+ϕ) (1-3)
where A and ϕ are constants and ω =
m
k
.
Really , taking derivative of the displacement x (1-3) with respect to the time variable we have the velocity of the
ball : v =
x

!
=
ω
Acos(
ω
t +
ϕ
) (1-4)
Taking derivative of the velocity v (1-4) with respect to the time, we get the acceleration of the ball:
a =
x
!!
= -
ω
2
Asin(
ω
t +
ϕ
) (1-5)
Replacing the value of x into (1-5) we get:
x
!!
= -
ω
2
x (1-6)
(1-6) has the same format as (1-2a), it shows that (1-3) is the solution of (1-2a), in another way, the equation of the
oscillated ball is x = Asin(
ω

t +
ϕ
).
Since sine function is a periodic function, it is said that the oscillation of the ball (i.e. the oscillation of
the mass-spring system) is a
simple harmonic motion
(SHM). Note that a cosine expression can be
transformed to a sine expression such a way that: Acos(ωt+ϕ) = Asin(ωt+ϕ+π/2)
Therefore, it can be defined that a SHM is an oscillation that can be described by a sinusoidal (or
cosinusoidal) function, where A, ω, ϕ are constants.
In the equation (1-3), x is the
displacement
of the oscillation, showings precisely the deflection of the
ball from the equilibrium position. A is the
amplitude
of the oscillation. It is the maximum value of
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displacement, occurred when sin(ωt+ϕ) has the maximum value of 1. The meanings of ω, ϕ and ωt+ϕ
will be clarified at §2.
It is known that sine function is a periodic function with the period of 2π. Thus, it can be written as
x = Asin(ωt +ϕ) = Asin(ωt +ϕ +2π), or x = Asin[ω(t +
2π
ω
) + ϕ]
It means that the displacement of the ball at time (t +
ω
π
2
) have the same value at time t. The period of

time T =
ω
π
2
is called the
cycle
of the SHM. The reciprocal of T, f =
T
1
=
π
ω
2
is called the
frequency
of
the SHM.
Particularly, for the mass-spring system, we have
T =
ω
π
2
= 2π
k
m
(1-4)
Now the system is taken out from the rod and hung up vertically (figure 1.1c). If the ball is pulled down
then released, it will oscillate in the vertical direction. That is also a mass-spring system. Everything have
been said about a horizontally oscillated spring system can be applied to a vertically oscillated spring
system as well. In this case, the equilibrium position is no longer the point O that corresponds with the

time the spring was not deflected, but is a point O’ that corresponds to the time the spring was deflected
due to the gravity of the ball.
Questions
1. Make statement about the definitions of oscillation, periodic oscillation and harmonic oscillation?
2. Differentiate between periodic and general oscillation, between periodic and harmonic oscillation?
3. Make statement about the definitions of time constant, frequency, displacement, amplitude of
harmonic oscillation?
4. Give more example about oscillation and harmonic oscillation?
§2.

E
XPLORING A SIMPLE HARMONIC MOTION
1.

Uniform circular motion and simple harmonic motion
Let consider a point M moves in a circle of central point O and radius
A (figure 1.2). The angular velocity of point M is ω (measured in
rad/s). A point C in the circle is chosen to be an origin. At the initial
time t = 0, the position of the moving point is M
0
, is specified by an
angle of ϕ. At an arbitrary time, the position of the moving point is M
t
specified by an angle of (ωt + ϕ).
We project the path of point M onto an axis x’x pass through point O
and perpendicular to OC. At time t, the projection of point M onto x’x
axis is point P which has the ordinate of x = OP. Since OP is the
projection of OM
t
onto the x’x axis so we have:

x = OM
t
sin(ωt +ϕ)
x = Asin(ωt +ϕ) (1.8)
(1.8) has the same format as (1.3) so we can conclude that the motion of point P on the x’x axis is a
SHM. In the other wa
y, a simple harmonic oscillation can be considered as the projection of an uniform
circular motion onto any straight line in the same plane.
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2.

Angular phase and angular frequency of a simple harmonic motion
From figure 1.2, the angular (ωt +ϕ) specifies the position of point P at the time t, it is called the
phase
(or angular phase) of the oscillation at the time t. The angle of ϕ specifies the position of P at the initial
time t = 0, and is called the initial phase (or initial angular phase) of the oscillation. The
angular velocity
ω allows us to determine f =
π
ω
2
, which is the number of circle of M in a unit of time, and is also the
number of oscillation of P in a unit of time. We know that f is the frequency of the oscillation, therefore
ω is called the
angular frequency
(circular frequency) of the oscillation. Here ϕ, ω and (ωt +ϕ) are
specified angles and can be measured directly.
In equation (1-3) for the mass-spring system, the quantities ϕ, ω and (ωt+ϕ) have the same names but
they are not the real angles which can be experimentally measured. They are intermediary quantities

which allows determining the frequency and states of the oscillation.
3.

Free motion
Let’s analyze more detail the motion of the mass-spring system described in §1 (figure 1.1).
The maximum displacement the ball can reach is the amplitude A. The time when the ball is released and
start to move is chosen to be the initial time t = 0. At that time x = A. In order to have the equation
x = Asin(ωt +ϕ) satisfied, we must have sin(ωt +ϕ) = 1, and since ωt =0 so ϕ = π/2.
Therefore, the oscillation equation of the ball is x = Asin(ωt +π/2) (1-9)
So we have determined the amplitude, initial phase and the cycle of the oscillation. The amplitude and
initial phase depends on the initial conditions, i.e. the way to excite the oscillation and the way to choose
the space and temporal coordinate. The period depends only on the mass of the ball and the spring
constant, not on other factors. If the initial conditions are changed then the amplitude A and initial phase
ϕ will be changed as well but ω, T are constant.
An oscillation the period of which depends only on the system’s characteristics (here is a mass-spring
system), and not on other stimulating factors, is called a
free oscillation. A system that can implement a
free oscillation by itself is called a
self-oscillation system
. After being stimulated, a self-oscillation
system will proceed with its
own frequency
. The oscillation of a mass-spring system is a free oscillation.
4.

Velocity and acceleration in a simple harmonic motion
As we know from §1,
v =
x
!

= ωAcos(ωt +π/2) = ωAsin(ωt +π) (1-10)
a =
v
!
=
x
!!
= - ω
2
Asin(ωt +π/2) = ω
2
Asin(ωt -π/2) (1-11)
In figure 1.3, there are curves presenting functions (1-9), (1-10) and (1-11). It is observed that after each
cycle T =
ω
π
2
, the values of displacement, velocity and acceleration are in same values as before, and the
behaviors of curves are also unchanged. The phase of the oscillation (ωt +ϕ) not only determines the
position of the oscillating thing but also allow to specify the value and behavior of the velocity and
acceleration.
The phase of the oscillation determine the state of the oscillation
. Similarly,
the initial
phase
ϕ
specifies the initial state of the oscillation
.
When the ball is in the harmonic oscillation, its velocity and acceleration fluctuate following a sinusoidal
or cosinusoidal function, i.e. they

fluctuate harmonically
with the same frequency of the ball.
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5.

Oscillation of a simple pendulum
A simple pendulum consists of a ball attached at one end of a string. The ball has a mass of m and its
size is very small in comparison with the length of the string. The string is inelastic (constant length) and
has a negligible mass. The ball can be seen as a point of mass m attached to a no-mass string. When it is
hung at point Q, its equilibrium position is QO (figure 1.4). The ball is pushed follow an arc from O to P
corresponding to a deflection angle of α. We only investigate the case in which α is small enough to have
the arc
"
OP coincide with the chord OP and sinα can be approximated as α (in radian). If α ≤10
o
then the
error is not greater than 6/1000.
Thus we have: sinα ≈ α =
l
s
(1-12)
Now, the ball is released and it swings itself. The force acting on the ball include of the gravity
F
t
= m
g
,
the tend force
T

of the string. The force
F
t
is resolved into 2 components:
F
’ in the direction of the string
and
F
is perpendicular to the string. The component
F
’ balances the tend force
T
, hence the ball does not
move in the string direction. The direction of component
F
is tangential to the arc
∩
OP
, but since α is
very small it can be considered lying along the chord OP and direct to point O.
From Newton’s second law, it can be written as :
m
a
=
F
(1-13)
Point O is chosen to be the origin while the chord OP is taken as coordinate axis. Since
a
and
F

are in OP
axis and the direction of
F
is opposite with the ordinate s = OP so
ma = F = -F
t
sinα = - mgα = mg
l
s
, or
a = -
l
g
s; and
s
!!
= -
l
g
s (1-14)
Equation (1.14) has the same format as (1.2). Here
l
g
plays the role of
m
k
, and s plays the role of x.
Therefore (1.14) has the same meaning with (1.2). We can applied the analysis process in §1 and §2 as
well then it can be concluded that the motion of a single pendulum is a harmonic oscillation with angular
frequency ω =

l
g
.
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The time constant of the pendulum is:
T =
ω
π
2
= 2π
l
g
(1-15)
For small oscillation, i.e. with α ≤10
o
,
the cycle of a simple pendulum is not dependent on the oscillation
amplitude
. All the discussion have been made for a mass-spring system in §2 can be applied to the single
pendulum as well.
The period of the single pendulum depends on the gravity constant g. At a specified location to the earth
(g is constant), the oscillation of the single pendulum can be regarded as free oscillation.
Calculations in details have proved that when the ball is moving, the tend force
T
has a magnitude T > F’. The
result is that the ball is exerted by a force of (T - F’) directed to Q. This force cause a centripetal acceleration so
that the ball travels in a circle path while the acceleration in the OP direction maintains a =- gs/l.
In the calculation above, the change of force
T

is not taken in to account, but the result is still valid.
Questions
1. Make statement about definition of phase and initial phase of periodic oscillation?
2. What is angular frequency? What is the relationship between angular frequency ω and the frequency f?
3. What kind of oscillation that can be called free oscillation?
4. Why is the formula (1.15) valid only for small oscillations?
5. The displacement of an object (measured in cm) fluctuates is described by x = 4cos4πt. Calculate the
frequency of this oscillation. Determine the displacement and velocity after it starts to oscillate in 5
seconds?
6. A single pendulum has a period of 1.5s when it oscillates at a place where the gravity constant is
9.8m/s
2
. Determine the length if the string?
7. Determine the time constant of the pendulum in exercise 6 when it is brought to the Moon, knowing
that the gravity constant in the Moon is 5.9 times smaller than the Earth.
Hints: 5) 2Hz; 4cm; 0cm/s; 6) 0.56m; 7) 3.6s.
§3.

E
NERGY IN A SIMPLE HARMONIC MOTION
1.

Energy changes during oscillation
When the ball of the mass-spring system is pulled from point O to point P (figure1.5, the spring is not
shown), the force has done a work to elongate the spring, this work is passed to the ball as potential
energy. At that time, the elastic force of the spring has a maximum value so the potential energy also has
a maximum magnitude.
When the force is not exerted, the spring compressed, the
elastic force directs the ball toward point O. Its velocity is
increasing, the kinetic energy is increasing while the

potential energy is decreasing.
When the ball is at the equilibrium position O, the elastic force and the potential energy are zero, its
velocity and kinetic energy reach maximum value. The ball continues to move due to the inertial motion,
the spring is contracted, the elastic force appears in opposite direction and grows up, and the velocity is
decreasing.
When the ball reaches to point P’, the spring is contracted to the shortest length, the elastic force reaches
the maximum value and the ball is stopped. Its kinetic energy is zero, the potential energy has maximum
magnitude and stop increasing.
After that, the spring is stretching, the elastic force is decreasing, the ball is pushed to point O. Its kinetic
energy is increasing while the potential energy is decreasing.
Translated by VNNTU – Dec. 2001
Page 14
During the oscillation process of the mass-spring system, there is always a transformation between
kinetic and potential energy: when the kinetic energy is increasing then the other is decreasing and vice
versa.
2.

Conservation of mechanical energy during oscillation
We will analyze quantitatively a process of energetic transformation of a mass-spring system.
The kinetic energy of the ball is E
d
=
1
2
mv
2
.
Replacing v by its expression from (1-10): v = ωAcos(ωt +π/2), we have :
E
d

=
1
2
mω
2
A
2
cos
2
(ωt +
2
π
) (1-16)
It was proved that the potential energy of a ball is equal to the work done of the elastic force in order to
bring it from the position x to the equilibrium: E
t
=
1
2
kx
2
Replacing x by its expression from (1-9): x = Asin(ωt +π/2) and replace k by mω
2
, we get
E
t
=
1
2
mω

2
A
2
sin
2
(ωt +
2
π
) (1-17)
(1-16) and (1-17) are the expressions of kinetic and potential energy of the ball at an arbitrary time t, and
the total energy of the ball at this time is
E = E
t
+ E
d
E =
1
2
mω
2
A
2
[sin
2
(ωt +
2
π
) + cos
2
(ωt +

2
π
)]
E =
1
2
mω
2
A
2
= constant (1-18)
The total energy of oscillation is conserved. During the oscillation process, the total energy is unchanged
and proportional with the square of amplitude.
There is a transformation between potential and kinetic energy. Based on (1-18), we rewrite (1-16) and
(1-17) in new forms:
E
d
= Ecos
2
(ωt +
2
π
) (1-19a)
E
t
= Esin
2
(ωt +
2
π

) (1-19b)
Note that the kinetic energy of a single pendulum is dependent on the initial excitation. If it is excited by
a powerful force to make a bigger displacement then the amplitude is bigger hence the total energy is also
bigger. Certainly, we just can increase the amplitude to a limited value within elastic limitation of the
spring.
Questions
1. Describe quantitatively the process of transformation of total energy of a single pendulum?
2. How to increase the total energy of single pendulum and at what value it can be increase?
3. How many time is the total energy of a pendulum changed if its frequency is increased 3 times while
the amplitude is reduced 2 times?
Translated by VNNTU – Dec. 2001
Page 15
§4.

- §5. T
HE COMBINATION OF OSCILLATIONS
1.

Examples of the combination of oscillations
In the real life as well as in science and technology, there are cases in which the oscillation of an object is
a combination of many different oscillations. When the hammock is hung on a ship, it will swing with its
own frequency. However, the ship is also oscillated due to wave. Finally, the oscillation of the hammock
is a combination of two components: its own oscillation and the oscillation of the ship.
Generally, partial oscillations can have different directions, amplitudes, frequencies and phases.
Therefore, it is very complicated and difficult to determine the combined oscillation. We will only deal
with simple situations that are usually encountered in science and technology.
2.

Phase-differences between oscillations
Two oscillations which have the same frequency, generally can have different phases. For example, two

identical mass-spring systems are hung next to each other, they have the same angular frequency of ω.
The balls are pulled to displacements x
1
= A
1
and x
2
= A
2
respectively. At the time t = 0, ball 1 is released
to start moving. At the time when ball 1 passes through its equilibrium position, ball 2 is released and
start traveling.
It takes a quarter of period for ball 1 to travel from position x
1
= A
1
to the equilibrium position. So that
the oscillation of ball 2 is retarded a mount of
4
T
compared with ball 1.
We find the oscillating equations of two ball in the form of
x
1
= A
1
sin(ωt + ϕ
1
)
x

2
= A
2
sin(ωt + ϕ
2
)
As we have known (from §2), the oscillation equation of ball 1 is:
x
1
= A
1
sin(ωt + π/2) (1-20)
For ball 2, at the time t =
4
T
then x
2
= A
2
, thus:
22 2
sin( )
4
T
AA
ω
ϕ
=+
;
2

sin( ) 1
4
+=
T
ω
ϕ
;
2
42
+=
T
ωπ
ϕ
;
2
2
.0
24 2 4
=− =− =
TT
T
πω π π
ϕ
The oscillation equation of ball 2 is
x
2
= A
2
sinωt (1-21)
In general, the phase difference between two oscillations that have the same frequency is:

(ωt + ϕ
1
) - (ωt + ϕ
2
) = ϕ
1
- ϕ
2
The phase difference is a constant quantity and equal to the difference between the initial phases. It is
called the
phase difference
∆ϕ and these two oscillation are called phase-different oscillations. When
∆ϕ = ϕ
1
- ϕ
2
> 0, it is said that oscillation 1 is a lead in phase to oscillation 2 or oscillation 2 is a lag in
phase to oscillation1. When ∆ϕ = ϕ
1
- ϕ
2
< 0, it is said in a contrast way.
In this example, it is said that ball 1 leads ball2 by an angle of
2
π
or ball 2 lags ball 1 by an angle of
2
π
(note that these angles only appear in calculations but are not real angles that can measured by angular
ruler).

The phase difference is a characteristic quantity for the discrepancy between two oscillations that have
the same frequency. If the phase difference is zero, or generally is 2nπ then they are in phase. If it is π or
(2n+1)π then they are out of phase (n is an arbitrary integer, n = 0; ±1; ±2; ±3; )
Translated by VNNTU – Dec. 2001
Page 16
3.

Vector-diagram method
In order to combine two harmonic oscillations with the same directions, frequencies but different
amplitudes and phases, it is usually used a very convenient method called Fresnel’s vector diagram. This
method is based on a property having been discussed in §2: a simple harmonic oscillation can be treated
as the projection of a uniform circular motion on to a straight line in the same plane.
According to this method, each oscillation can be represented by a
vector. Suppose that an oscillation x = A sin(ωt+ϕ) need to be
represented. A horizontal axis (∆) and a vertical axis x’x that
intersects (∆) at point O are built (figure 1.6). A vector
A
whose
origin is at point O, magnitude is proportional with amplitude A and
makes with axis (∆) an angle of initial phase ϕ. At the time t = 0,
vector
A
(its head is M
0
) is rotated in positive direction
(conventionally is counter-clockwise) with angular velocity of ω.
When the head M of vector
A
is projected on to x’x axis then the
motion of the projection P on x’x is a harmonic oscillation. At any

time t, the head of
A
is M, its projection on x’x is P and we have:
x =
OP
= Asin(ωt + ϕ)
That is the simple harmonic oscillation necessary to express. It is said that a simple harmonic oscillation
x = Asin(ωt + ϕ) is represented by a vector
A
.
4.

The combination of two oscillations of same directions and frequencies
Suppose that one object (e.g. a mass-spring system hung in a moving train) simultaneously takes part in
two oscillations of the same directions and the same frequency ω, but they have different amplitudes A
1
,
A
2
as well as initial phases ϕ
1
, ϕ
2
x
1
= A
1
sin(ωt + ϕ
1
) (1-22)

x
2
= A
2
sin(ωt + ϕ
2
) (1-23)
The resultant motion is the combination of two components (1-22) and (1-23). The Fresnel’s vector
diagram method will be applied to find the equation of resultant motion.
Two axes (∆) and x’x are drawn as in figure 1.7. Draw a vector,
namely
A
1
, whose magnitude is proportional with amplitude A
1
makes an angle of ϕ
1
with (∆) (figure 1.7). Similarly, draw
vector
A
2
whose magnitude is proportional with amplitude A
2
makes an angle of ϕ
2
with (∆). Draw vector
A
which is the
resultant vector of
A

1
and
A
2
, this vector makes an angle of ϕ
with (∆).
In figure 1.7, the angle between
A
1
and
A
2
is (ϕ
2
- ϕ
1
) (the phase
difference of two components x
1
and x
2
). Since ϕ
1
and ϕ
2
are
constant then (ϕ
2
- ϕ
1

)

is also constant.
Now rotate
A
1
and
A
2
around point O in positive direction with
the same frequency of ω. Then a trapezoid OM
1
MM
2
is not
deformed since both sides OM
1
, OM
2
and the angle
"
21
MOM
are unchanged. Therefore
A
has a constant
magnitude and rotates around O in positive direction with angular velocity ω of
A
1
and

A
2
.
Since the resultant of projections of components onto an axis is the projection of the resultant vector
projected on that axis. So motion of P (projection of M) on x’x is the combination of P
1
(projection of
M
1
) and P
2
(projection of M
2
) on x’x axis, it is also a harmonic oscillation.
A
is the resultant vector of
A
1
and
A
2
, and it also represents the combined oscillation and its initial phase is ϕ (figure 1.7).
Similarly, if it is necessary to combine various oscillations x
1
, x
2
, x
3
… it is recommended to draw the
resultant vector

A
of
A
1
,
A
2
,
A
3

Translated by VNNTU – Dec. 2001
Page 17
Figure 1.7 is called a vector diagram.
5.

Amplitude and initial phase of the combinatorial oscillation
The equation of resultant motion is x = x
1
+ x
2
= Asin(ωt +ϕ) (1-24)
where A is proportional with the magnitude of amplitude vector
A
.
It is necessary to evaluate A and ϕ in (1-24). For triangle OMM
2
in figure 1.7, we have:
"
22 2

22 22 2
OM OM MM 2OM.MMcosOMM=+ − , or
A
2
= A
1
2
+ A
1
2
– 2A
1
A
2
cos
"
2
OM M
Since
"
2
OM M
and
"
2
MOM
are supplementary angles thus:
cos
"
2

OM M
= - cos
"
2
MOM
= - cos(ϕ
2
- ϕ
1
) (1-25)
From figure1.7 we can deduce that
tgϕ =
MP '
OP '
=
OP
OP '
=
112 2
112 2
A sin A sin
A cos A cos
ϕ
+
ϕ
ϕ
+
ϕ
(1-26)
Finally, the total oscillation is a harmonic oscillation described by (1-24), has the same frequency with

constituent oscillation, has amplitude specified by (1-25) and initial phase determined by (1-26)
From (1-25), the amplitude of total oscillation is dependent on the phase difference (ϕ
2
- ϕ
1
) of
constituent oscillations.
If the constituencies are in phase (ϕ
2
- ϕ
1
= 2nπ) then cos(ϕ
2
- ϕ
1
) = 1, the resultant amplitude has
maximum value of A = A
1
+ A
2
.
If the constituencies are out of phase (ϕ
2
- ϕ
1
= (2n+1)π) then cos(ϕ
2
- ϕ
1
) = -1, the resultant amplitude

has minimum value of A = |A
1
- A
2
|.
If the phase difference is arbitrary then the resultant amplitude satisfies |A
1
- A
2
| < A < A
1
+A
2
.
Questions
1.What is the phase difference?
2.How are in phase oscillations, out of phase oscillation, leading phase oscillation, lagging phase
oscillation?
3.From figure 1.3 in §2, compared with the oscillation of a mass-spring system, are the velocity and
acceleration of ball lagging or leading and how much are they?
4.Briefly state the Fresnel’s vector-diagram method?
5.Two harmonic oscillations have the same direction and the same frequency f = 50Hz, and have the
amplitudes A
1
= 2a, A
2
= a and the initial phases ϕ
1
=
3

π
, ϕ
2
= π.
a) Write the equations of these two oscillations.
b) Draw in the same diagram vectors
A
1
, A
2
, A
, of these two components and of the resultant oscillation.
c) Calculate the initial phase of the resultant.
Hints: 5) ϕ = π/2.
Translated by VNNTU – Dec. 2001
Page 18
§6.

- §7. U
NDERDAMPED AND FORCED OSCILLATIONS
1.

Underdamped oscillation
In research of harmonic oscillation of mass-spring system, simple pendulum and other things, it is
observed that their frequencies and amplitudes are time-independent quantities. It means that they will
repeats back and forth forever, never ended. But in fact, amplitudes of free oscillations will be damped
then ended because generally they move in a certain medium and are effected by frictions. Depending on
how much the friction is, the oscillation will be damped fast or slowly. Such oscillations are called
underdamped oscillations
. An underdamped oscillation does not have harmonic properties, therefore

when talking about the amplitude, frequency, or cycle of an underdamped oscillation, it implies
approximation.
When a mass-spring system oscillates in the air, the air friction make it be damped.
But since it is small so it takes quite a long time to ended. Therefore, if this
oscillation is examined in a short time, the damping is negligible and it can be seen
as a harmonic oscillation.
Let a pendulum fluctuates in a container of water (figure 1.8). The friction of water
is strong enough so it will be damped fairly quickly and it will stop at the
equilibrium position (figure 1.9a).
Replacing with a container of lubricating oil, if its friction is large enough there is
no fluctuation. The ball passes through an equilibrium position (one time only) then
returns and stops there. (figure 1.9b)
If the friction of lubricating oil is much stronger, the
ball even can not passes through the equilibrium
position and stops immediately (figure 1.9c)
In the real life and technology, in some cases the
underdamping is harmful and it is required to
overcome this phenomenon (i.e. for clock pendulums).
In contrast, in some cases it is useful and needed so
people make use of that. For example, we all know
that the surface of the road is not fairly flat, and the
faster the vehicle travels, the more vibrative it is,
hence vehicles and motorcycles must have buffer
springs. When there is a gap, the spring is compressed
or stretched. After that, the frame continues vibrating
like a spring, makes travelers tired and uncomfortable.
In order to make it damped faster, vehicles are
equipped with a special equipment. It consist of a
piston that can travel in a vertical cylinder contain
lubricating oil, piston is assembled to the frame and

the cylinder is mounted to a shaft of wheel. When the
frame vibrates on buffer springs, the piston is also
fluctuated inside the cylinder. The lubricating oil
make the vibration damped faster and so the vibration
of the frames does.
2.

Forced oscillation
In order to make an oscillation not be damped, the simplest way is exerting on it an externally periodic
force. This force supplies energy to the whole system to compensate the frictional losses.
It is known that a mass-spring system and simple pendulum are free oscillations. If there is no friction,
they will oscillate ever and ever with their own frequencies. However, it is ideal. In fact, external
environment exert on the ball a strong or light frictional force F
ms
, make the vibration damped (see the
diagram in figure 1.9).
An externally periodic force is applied to the ball called the
forced force
:
Translated by VNNTU – Dec. 2001
Page 19
F
n
= Hsin(ωt +ϕ)
where H is the amplitude and ω is the frequency of the forced force. Generally, the frequency
of the external excitation f =
2
ω
π
is different from that of the free oscillation f

0
of the ball.
Theoretical calculations resulted in: during a period of initial time ∆t, total vibration of the system is a
complicated, a combination of many free vibrations as well as external excitation. After that, free
vibrations are ended, the ball oscillates due to the external excitation. Its frequency is the frequency of the
external force and the amplitude is dependent on a relationship between the externally excited frequency
f and free frequency f
0
. That is why a vibration after along time is a forced oscillation. If the excitation is
maintained for a long time then the forced vibration is also maintained during that time.
The complicated oscillation time ∆t is always very small compared with the forced oscillating time
afterward. It can be said that after ∆t, the system ‘forgot’ its free vibration. Therefore, in fact, it is usually
studied the forced oscillation after ∆t and it is unnecessary to care about a complicated vibrations during
∆t.
3.

Resonance
This phenomenon can be examined by an experiment (figure 1.10).
A is a pendulum consisted of a mass of m fixed on to the metal rod. N is a
light and thin slab by assemble composite. The frequency f
0
of the pendulum
when it does not assembled slab N is directly determined by a chronometer.
B is another pendulum consisted of a mass of M >> m that can easily slide on
to a thin calibrated metal rod. The frequency f is determined corresponding
to each position of pendulum B on the rod by a chronometer.
These two pendulums, A (that is not assembled to slab N) and B, are hung
next to each other, two rod are joined by a light spring L. Pendulum B is
allowed to swing in the plane that is perpendicular to the plane of paper. The
frequency f of pendulum B is transferred to pendulum A as a forced

excitation by the spring. This force makes pendulum A vibrating, and after a
time, pendulum A has forced oscillation with the frequency of f. Changing in
position of pendulum B results in the change in frequency f as well, and it is
observed that the vibration of pendulum A reaches the maximum value when
f ≈ f
0
, but when f is smaller or greater than f
0
then the amplitude of pendulum
A is decreasing dramatically.
The phenomenon of the amplitude of forced oscillation is increased dramatically to a maximum value
when the frequency of forced excitation is equal to the free frequency of system is called the
resonance
.
Now, slab N is assembled to pendulum A to increase the atmospheric friction. Repeating the whole
process above, it is shown that pendulum A has a resonant oscillation at f = f
0
, but its amplitude is
smaller than that in the case of no assembly with slab N. In this case, because energy provided by forced
excitation is mainly used to compensate frictional losses, thus it does not make the amplitude increase
significantly. The resonance exhibits clearest when the friction is insignificant.
4.

Applying and surmounting resonant phenomenon
Resonance is the most encountered phenomenon in life and technology, it can be harmful and useful for
people.
A child can use a small force to swing an adult’s hammock when the hammock reaches the highest level.
Continuing swing like that after certain time, i.e. exerting on the hammock a periodic force whose
frequency is the same as hammock’s frequency, a child can swing it to higher level, i.e. hammock’s
amplitude is bigger. It is impossible for the child to push a hammock to that level.

In many cases, resonance is harmful and need to be overcome. Every elastic thing are oscillations and
they have their free frequencies. They can be a bridge, frame, shaft…Due to some reasons, they vibrate
Translated by VNNTU – Dec. 2001
Page 20
resonantly with the other (e.g. a big electric generator), and they will vibrate dramatically and can be
broken, collapsed that is the concern of engineers.
At the middle of XIX century, there was a troop paraded on to a bridge, it was vibrated dramatically and
broken, make a lot of human losses. That is because the parading frequency of the troop coincided
accidental with the free oscillating frequency of the bridge and it made resonance. After this accident,
army regulation of some countries do not allow parade on the bridge.
5.

Self-oscillation
There is another way to maintain oscillation, keep it not be damped and there is no external excitation. A
simple example is the pendulum clock.
The pendulum of the clock swings freely with its specified cycle T. Due to the friction with the air and at
the shaft, its fluctuation will be damped if it is not compensated the losses.
When the clock is wound up, it is accumulated a certain amount of potential energy. The spring is related
with the pendulum by a system of cog-wheel and proper mechanism. When the pendulum reaches to the
maximum displacement, after half of cycle, the spring is stretched a little bit and a part of this potential
energy is transferred to the pendulum through agent mechanism. This amount of energy is sufficient to
compensate the frictional losses. Therefore, the pendulum can swing for a long time with the same
frequency and amplitude. In the watch and table clock, spiral pendulum plays the role of the pendulum
clock.
The vibration that can be maintained without external excitation is called self-oscillation. A system, such
as a pendulum clock, consists of oscillating mass, energy source and energy transfer mechanism is called
self oscillation system.
Note that in the forced oscillation, oscillating frequency is the externally excited force and the amplitude
depends on the external amplitude. But in the self oscillation, the frequency and amplitude is unchanged
from their original value as well as free oscillation.

Questions
1. In what conditions the oscillation is underdamped oscillation? How does the amplitude of
underdamped oscillation change?
2. How to make a forced oscillation? Why it has the name like that?
3. What is the resonant phenomenon? When does it happen?
4. A motorcycle is traveling on the road, consequently there is a small gap after a distance of 9m. The
free frequency of the frame of the motorcycle is on the buffer springs is 1.5s. At what speed, the
motorcycle is vibrated most dramatically?
SUMMARY OF CHAPTER I
1.Oscillation is a space limited motion, repeat back and forth around an equilibrium position.
In all kind of oscillations, periodic oscillation is a kind in which the state of motion is repeated as it was
after a certain period of time.
Cycle T is the smallest time after that the state of motion is repeated as it was. Frequency
1
f
T
=
is the
number of oscillation in a unit of time. Its unit is Hertz (Hz).
In all kind of periodic oscillations, harmonic oscillation is described by a sinusoidal or cosinusoidal law:
x = Asin(ωt+ϕ) or x = Acos(ωt+ϕ). The displacement x is the deflection from an equilibrium position.
The amplitude is the maximum displacement. The angular frequency ω is a quantity to specify the
frequency
2
f
ω
π
=
and the cycle
2

T
π
ω
=
of the harmonic oscillation. The phase (ωt+ϕ) specifies the
state of oscillation at any time of time t. The initial phase ϕ specifies the initial state, i.e. at the time t = 0.
Translated by VNNTU – Dec. 2001
Page 21
A harmonic oscillation can be regarded as the projection of a uniform circle motion in the projectile
plane. The angular velocity of the circle motion is the angular frequency of harmonic oscillation. In the
vector diagram method of Fresnel, each oscillation is represented by a vector rotates in the datum plane
in the positive direction, and the total oscillation is the projection of the motion of the head of resultant
vector on a straight line in the same plane.
2. The cycle of a mass-spring system is T = 2π
k
m
depending only on the characteristics of the
pendulum. It is called free oscillation and its cycle is called free cycle. The simple pendulum has a cycle
of T = 2π
l
g
depending on the gravity acceleration. When the simple pendulum is placed in a specific
location (g is constant), it can be seen its fluctuation as free oscillation.
When the pendulum oscillates harmonically, the velocity and acceleration of the ball behave follow a
sine or cosine law, i.e. they behave with the same frequency of the ball.
During the oscillating process, there is a transformation from kinetic energy to potential energy and vice
versa, but the mechanical energy is constant and proportional to the square of amplitude of the ball.
3. The phase difference of two oscillation is the difference of initial phase and is call the phase difference
∆ϕ= ϕ
1

- ϕ
2
. Two oscillations are in phase if ∆ϕ = 2nπ, are out of phase if ∆ϕ = (2n+1)π.
The combination of two harmonic oscillations with the same directions and frequencies but different
amplitudes is a harmonic oscillation with the same frequency. However, the amplitude and initial phase
of total oscillation is dependent on the phase difference of two components. If two components are in
phase then the total amplitude is maximum of A=A
1
+ A
2
. if they are out of phase then it is minimum of
A=| A
1
– A
2
|.
4.In fact, every vibration is underdamped vibration, because the environmental frictions dissipate the
oscillating energy. In order for an oscillating system which has free frequency f
0
is not damped, it is
applied a externally harmonic force of frequency f, called forced excitation. Forced oscillation has the
same frequency f with external excitation. The resonance happens when f = f
0
, the amplitude of forced
oscillation is increased dramatically to the maximum value. Bigger maximum amplitude is, smaller
environmental friction is.
In life and science, technology, the resonance can be either harmful or useful.
Translated by VNNTU – Dec. 2001
Page 22
Chapter II –

MECHANICAL WAVE. A COUSTICS
§8.

W
AVE IN MECHANICS
1.

Natural mechanical waves
When we throw a rock into a still water surface, we can observe a number of circular water waves
spreading out to every direction from the place where the stone is thrown. If we drop a cork or a leaf
down to the water surface, it will also rise up and down in response to the stimulated water waves. But it
only fluctuate in one vertical direction, instead of moving horizontally with the circular water wave.
We can explain the observation as follows. Among the water molecules, there is a coalescent force that
make them united together. When a water molecule, say A, rises up, the coalescent force makes the
nearby molecules to go up also, but a few time later. It is also these forces that helps to draw the water
molecule A back to its previously resting place. These forces acting very much the same role as the
elastic force does in an elastic pendulum. In conclusion, each molecule oscillating in a vertical direction
will tend to make the nearby molecules to oscillate in the vertical mode likewise and this mechanic
makes the oscillation to spread faring away.
Mechanical waves
are mechanical oscillations that
spread
out with time in a material medium.
Note that in mechanical waves only the oscillation states, i.e. the phases of the oscillation, is spreading
away, while the medium’s small mediums are only fluctuating around its original resting balance place.
The water wave is one type of waves that can be observed by normal eye. In reality, using appropriate
equipment, scientists can observe waves in all other types of material – say it in solid, liquid or in gas
form. For example, if dropping some grains of sand into the surface of a wide big iron board, then using a
hammer to smash hard in one far end of the iron surface, we can still see the grains of sand bumping up.

This is because of the waves spreading through the iron board. Unfortunately, we cannot see this type of
waves with bared-eyes.
In the example of the water waves, the direction of the oscillations of the mediums’ elements is
perpendicular to the direction in which the waves travel. Such a wave is called a transverse waves. There
exists another type of wave, known as a longitudinal wave, in which the oscillation of particles of the
medium is along the same direction as the motion of the wave. Longitudinal waves will be discussed in
details in this chapter.
2.

Oscillation phase transmission. Wavelength.
A stone thrown into a water surface can create only a few
small waves, the oscillation will soon die out. To make
better research in mechanical waves, a small equipment is
created to help making the waves last longer. Using a thin
pieces of elastic metal, at one end sticking in a small ball or
needle. Place the metal piece so that the marble slightly
touches the surface of a large water tray (figure 2.1). Then
we just need to flip on the right end of the metal piece to
make the ball harmonically vibrate with period T. Then all
water molecules contacting with the ball will also vibrate
with period T in a relatively long time. On the water
surface, a number of circular waves will start to spread in
all directions.
We can simplified by imagining viewing the water tray through a vertical projection through P. The cut
trace will have the form displayed in figure 2.2. It is the truly form of the water waves in different instant
of time.
Suppose at t = 0 the waves have the form displayed in figure 2.2a. We can see that points A, E and I
vibrate in phase: they all go through their equilibrium positions and move downwards. Points C and G
are out of phase to points A, E and I: they also go through their equilibrium positions but moving
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Page 23
upwards. At the time of t = T/4 (figure 2.2b) the oscillating phase of point A transmitted to point B, and
at different times of t = T/2; (figure 2.2c), t = 3T/4 (figure 2.2d) and t = T (figure 2.2e) the phase
transmitted to points C, D and E respectively. It should be noted that the oscillating phase is transmitted
in a horizontal direction, while water elements only fluctuating vertically.
In figure 2.2 we can see that points A, E and I are always in phase with each other. The distance between
two successive in-phase points along the direction of wave transmission is called the
wavelength
,
denoted by λ (the Greek letter lambda). In general,
those points the distance between which is a multiple
of the wavelength will oscillate in phase
.
The distance between points I and G is a half of the wavelength and they are in opposite phase of each
other. In general,
those points the distance between which is an odd multiple of a half of the wavelength
will oscillating out of phase
.
3.

Period, frequency and velocity of waves
At every points through which the mechanical waves go, the medium elements oscillate with the same
period, which equals to the period T of the wave source. This period is called
the period
of wave. The
reciprocal of the period, f = 1/T, is called
the frequency
of wave.
In the above example, we can see that after each period T the oscillation phase travels through a distance
equal to the wavelength. Thus, we can also say that:

the wavelength is the distance the wave travels in a
period T
.
The speed of wave transmission is called the
wave velocity
. Since in a period T the wave travels through
a distance equal to the wavelength λ, we have the following relation:
λ = v T (2-1)
or λ =
v
f
(2-2)
4.

Amplitude and energy of waves
Once the wave reached a point, it makes the medium elements at that point oscillate with a particular
amplitude. This amplitude is called the
wave amplitude
at the specific point in question.
We have known that the energy of a harmonic oscillation is proportional to the square of its amplitude.
The wave makes the elements oscillating, thus provides them an energy. We say: the process of wave
transmission is also the process of transferring energy.
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For waves that originating from one point and spreading out in a surface, the wave energy is also
stretched in to a circle that keep expanding. Since the length of the circle is proportional to its diameter,
when the wave spreads far away its energy also decreases proportional to the traveling distance. For
waves that originating from one point and spreading out in a space, the wave energy reduces proportional
to the square of the traveling distance.
In an ideal case when the wave is transmitted in one straight line, the wave energy will not be reduced

along the direction of wave propagation and the wave amplitude is the same at every point the wave goes
through.
Questions
1. What is a wave?
2. For a wave, what is transmitted, what is not?
3. Define the transverse wave and the longitudinal wave?
4. State two definitions of wavelength. If the wave velocity is constant, then what is the relationship
between the wavelength and the wave frequency?
5. In figure 2.2, which points oscillate in phase and out of phase to point H?
§9.

- §10. S
OUND WAVE
1.

Sound wave and the sensation of sound
Take a thin and long iron bar, then firmly clamp the below end of it (figure 2.3a).
Flip on the other end of the bar, we can see the iron bar to swinging back and
forth. Lowering the clipped end of the bar, so that the swinging part is shortened.
Again we flip on the free end of the bar. We can observe that the bar also
oscillating but with a faster frequency than before. When the swinging part of the
iron bar is shortened to some extend (that means the iron bar oscillating frequency
has been raised to some specific degree) we start hearing the small little noise
“uh”, i.e. the vibration of the iron bar starts creating sounds. Thus, we conclude,
the oscillating of the iron bar do sometime create sounds, and sometimes do not.
This phenomenon can be explained as follows: When the iron bar swing into one side, it makes the air
layer right before it was compressed, and it also makes the air layer right after it being relaxed. This
compressing and relaxing of the air happens periodically, and has created in the air a mechanical wave
which have a frequency equals to that of the oscillation of the iron bar. This wave spread towards our
ears, and forced our eardrums starting to be compressed and relaxed with the same frequency. It is the

fluctuating of our eardrums at a certain range of frequency that helped us to recognize sounds.
The human ears can detect only oscillations having frequencies in the range from 16Hz to 20,000Hz. Any
oscillation with the frequency in the range of 16Hz – 20,000Hz is called the
sound wave
, or shortly the
sound
. The science that study sounds is called
acoustics
.
Sounds can spread in any medium, say it solid, liquid or gas/air. For example: if we have some
experiences, we can here the sounds of a horse herd trotting or the sounds of a train running far away,
which we cannot hear from the sound spreading through the air. The reason is that, sound waves
spreading in the air was barred by a lot of barricade and was faded away very rapidly.
Mechanical waves with frequency greater than 20,000Hz is called
ultrasonic waves
. Some natural
species can emit and detect ultrasonic. Those waves which has its frequency smaller than 16Hz is called
infrasonic waves
. Human by using appropriate equipment can also create and detect these sounds, and
these has a lot of applications in science and technology.
In terms of physic nature, sound waves, ultrasonic and infrasonic sounds are not different to each other
and to other mechanical waves. The classification is made based on the sensation ability of the human ear
in detecting different waves. This is determined by the physiological characteristics of the human ear.
Thus, in acoustics, scientists do distinguish the physical and physiological characteristics of sounds.
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2.

Sound transmission. Speed of sound
Sound waves can spread in any medium, but its sound-transmission speed (the

speed of sound
) depends
on the elasticity and density of the medium.
In general, the speed of sound in a solid medium is greater than that in the liquid, and spreading speeds in
a liquid medium is greater than that of a gas/air medium. The speeds also change in response to the
change in the temperature of the medium.
Below are speeds of sound in some substances:
Solid and liquid (t = 20
o
C) Gas (in atmospheric pressure)
Carbonized steel 6,100m/s Air (t = 0
o
C) 332 m/s
Iron 5,850 m/s Steam (t = 135
o
C) 494 m/s
Rubber 1,479 m/s
Water 1,500 m/s
Those mediums like cotton or sponge is not good in transmitting sound waves as their elastic
characteristic is bad. They are used as sound-proof material.
Sound waves cannot be transmitted in vacuum. We can prove that by putting an electronic bell into a
glass container of a vacuum pump. When we start to extract the air out of the container, we can here that
the bell’s sounds also start to fading, and when there is no air left in the container, the sounds of the bell
has disappeared.
3.

Sound altitude
Among the sounds that we can detect, there are sounds that its frequency is specified, e.g. the sound of a
singer singing a song, or the sounds of a musical instrument. This is called
musical sounds

. There are
also sounds that do not have specific frequency, like the sound of a diesel engine, or the sound of a herd
of horse running. These sounds are called
interference
. In its nature, these sounds are the combination of
a number of oscillations that having very different frequency and amplitude. We’ll only study the musical
sounds.
With only one tune of a song, but if it was song by either soprano or tenor, can give us very different
experiences. Sounds with different frequencies present us different sound senses. Those sounds that have
a high frequency is called
high-pitch
sounds or
treble
. Those sounds that have lower frequency is called
low-pitch
sounds or
bass
. The pitch of a sound is a physiological characteristic of a sound, it is based
upon a physical characteristic: the sound frequency.
4.

Timbre
Even when two singers sings the same tune in the same
pitch, we can distinguish the sound of each singer. Or when
the guitar, the flute, the clarinet is playing the same musical
tune, we can still make distinctions between those different
musical instruments. Each person, each musical equipment
create sounds that have different characteristics that we, by
our hearing senses, can distinguish. These characteristics of
sounds is called

timbre
.
Timbre is a physiological characteristics of sounds. This
characteristic is based on physical characteristics of sound:
the frequency and the amplitude. Experiments have proved
that when a man, or a musical instrument, produces a sound
wave with frequency f
1
, he or it also produces other sound
waves of frequencies f
2
= 2f
1
, f
3
= 3f
1
, f
4
= 4f
1
, etc.
The sound waves with frequency f
1
is called the
fundamental sound
, or the
first harmonic
. Other sounds