Physics 1025F
Vibrations & Waves
OSCILLATIONS
Dr. Steve Peterson
Steve.peterson@uct.ac.za
UCT PHY1025F: Vibrations & Waves
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Chapter 11: Vibrations and Waves
Periodic motion occurs when an object
vibrates or oscillates back and forth over the
same path
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Periodic Motion
Periodic motion, processes that repeat, is one of the
important kinds of behaviours in Physics
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Equilibrium and Oscillation
Equilibrium position – position where net force is zero
Restoring force – force acting to restore equilibrium
Oscillation – periodic motion governed by a restoring force
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Equilibrium and Oscillation
A graph or motion that has the form of a sine or cosine
function is called sinusoidal. A sinusoidal oscillation is
called simple harmonic motion (SHM)
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Simple Harmonic Motion
SHM is characterised by…
Amplitude A: maximum distance of
object from equilibrium position
Period T: time it takes for object to complete one complete
cycle of motion; e.g. from x = A to x = −A and back to x = A
Frequency ƒ: number of complete cycles or vibrations per
unit time
Displacement x: is the distance measured
from the equilibrium point
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𝑇=
𝑓
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Simple Harmonic Motion
SHM occurs whenever the net force along direction of 1D
motion obeys Hooke’s Law
- (i.e. force proportional to displacement and always directed
towards equilibrium position)
Not all periodic motion over the same path can be classified
as SHM
Initially, we will look at the horizontal mass-spring system as
a representative example of SHM
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Hooke’s Law Review
Fs   kx
spring
force
k is the
spring constant
x is the displacement of the mass m from its equilibrium
position (x = 0 at the equilibrium position)
The negative sign indicates that the force is always directed
opposite to displacement (i.e. restoring force towards
equilibrium)
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Example: Hooke’s Law
A prosthetic leg contains a spring to absorb shock as the
person is walking. If an 80 kg man compresses the spring
by 5 mm when standing with his full weight on the
prosthetic, what is the spring constant (k)?
How far would the spring compress for a 100 kg man?
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Horizontal Mass on a Spring
From Newton II, for a mass-spring system:
Fn et   kx  m a
a 
k
x
m
For a horizontal mass-spring system & all other cases of
SHM, acceleration depends on position
Since acceleration is not constant in SHM standard
“equations of motion” cannot be applied
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Example: SHM
V&S Example 13.2: A 0.350-kg object attached to a spring of
force constant 1.30 x 102 N/m is free to move on a
frictionless horizontal surface. If the object is released from
rest at x = 0.10 m, find the force on it and its acceleration at
x = 0.10 m, x = 0.05 m, x = 0 m, x = -0.05 m, and x = -0.10 m.
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The Simple Pendulum
SHM occurs whenever the net force along direction of
1D motion obeys Hooke’s Law
For a pendulum, the restoring force is
F net   mg sin 
Does this motion qualify as simple harmonic motion?
A. Yes
B. No
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The Simple Pendulum
A pendulum only exhibits SHM if it is restricted to
small-angle oscillations (< 10°). For such small angles
(in radians), we get the small-angle approximation,
where
sin   
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The Simple Pendulum
Using the small-angle
approximation, the restoring
force becomes
F net   mg sin    mg 
The pendulum displacement
(the arclength s) is proportional
to the angle s  L 
giving
F net
 mg 
  mg
 
s
L
 L 
s
UCT PHY1025F: Vibrations & Waves
Linear restoring force
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Energy in a Mass-Spring System
The potential energy of a spring (Section 6-4):
The kinetic energy of the mass (Section 6-3):
Therefore the total energy of the spring-mass system is:
This total energy is conserved (assuming no friction, etc…)
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Energy in Simple Harmonic Motion
• Energy is all PE when 𝒙 = 𝑨
• Total energy is
• Energy is all KE when 𝒙 = 𝟎
• Total energy is conserved, so
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Example: Energy of Spring
A 4.0 kg mass attached to a horizontal spring with stiffness
400 N/m is executing simple harmonic motion. When the
object is 0.1 m from equilibrium position it moves with 2.0
m/s.
• Calculate the amplitude of the oscillation
• Calculate the maximum velocity of the oscillation
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Energy in Simple Harmonic Motion
Conservation of energy allows the calculation of the
velocity of an object attached to a spring at any position in
its motion:
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SHM and Uniform Circular Motion
• The velocity of the rotating object is equal to the
maximum velocity of the object in SHM.
• The circle circumference is 2𝜋𝐴 and the rotation time is
𝑇, thus
2𝜋𝐴
𝑣𝑚𝑎𝑥 =
= 2𝜋𝐴𝑓
𝑇
• From energy, we have: 𝑣𝑚𝑎𝑥 = 𝑘 𝑚 𝐴
• Combining them gives:
𝑇 = 2𝜋
UCT PHY1025F: Vibrations & Waves
𝑚
𝑘
OR
𝑓=
1
2𝜋
𝑘
𝑚
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Simple Harmonic Motion
• The position, velocity and acceleration are all sinusoidal
• The frequency does not depend on the amplitude
• The object’s motion can be written as
 2 t 
y ( t )  A cos 

 T 
v y ( t )   v m ax
 2 t 
sin 

 T 
 2 t 
a y ( t )   a m ax cos 

T


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Example: SHM
Giancoli Example 11-7: The displacement of an object is
described by the following equation, where x is in meters
and t is in seconds: 𝑥 = 0.30 m cos 8.0𝑡 .
Determine the oscillating object’s (a) amplitude, (b)
frequency, (c) period, (d) maximum speed, and (e)
maximum acceleration.
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The Simple Pendulum (Review)
Using the small-angle
approximation, the restoring
force becomes
F net   mg sin    mg 
The pendulum displacement
(the arclength s) is proportional
to the angle 𝒔 = 𝑳𝜽
giving
F net
 mg 
  mg
 
s
L
 L 
s
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Frequency of Simple Pendulum
Simple harmonic motion is based on the
restoring force obeying Hooke’s Law, so let’s
compare the pendulum force to Hooke’s law.
Fnet
 mg 
 
s
 L 
Fnet   kx
𝑚𝑔
,
𝐿
If we take 𝑘 =
then our frequency
equation becomes:
f 
1
k
2
m
f 
1
g
2
L
And the period equation becomes:
UCT PHY1025F: Vibrations & Waves
T  2
L
g
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Frequency and Period
Two observations:
– The frequency and period of oscillation
depend on physical properties of the oscillator.
• Spring: Mass & Spring Constant
• Pendulum: Length
– They do not depend on the amplitude of the oscillation.
• Pendulum frequency does not depend on mass
The pendulum depends only on 𝐿 and 𝑔 making it a useful timing device
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Damping & Resonance
• Damped harmonic motion happens when energy is
removed (by friction, or design) from the oscillating
system.
• Resonance occurs when energy is added to an oscillator
at the natural frequency of the oscillator.
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Natural Frequency
All systems have a natural frequency, the frequency at
which a system will oscillate if left by itself.
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Resonance
Resonance occurs when energy is added to an oscillator at
the natural frequency of the oscillator.
If an external force of this frequency is applied, the
resulting SHM has huge amplitude!
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The Wave Model
The basic properties of waves (the wave model) cover
aspects of wave behaviour common to all waves.
A wave is the motion of a disturbance.
Waves carry energy & momentum without the physical
transfer of material.
A traveling wave is an organized disturbance with a welldefined wave speed.
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Two Types of Waves: Mechanical
Mechanical Waves
… require some source of disturbance and a medium that
can be disturbed with some physical connection or
mechanism through which adjacent portions can
influence each other (e.g. waves on a string, sound, water
waves)
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Two Types of Waves: Electromagnetic
Electromagnetic Waves
... don’t require a medium and can travel in a vacuum
(e.g. visible light, x-rays etc)
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Making a wave
A wave pulse can be created with a single ‘snap’ on a rope
• Energy is transmitted from one point on the rope to the
next
A periodic (continuous) wave can be created by wiggling
the rope up and down continuously
• Energy is continuously being transmitted along the rope
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Types of Mechanical Travelling Waves
Transverse waves:
In a transverse wave, each element that is disturbed moves in a direction
perpendicular to the wave motion.
Longitudinal waves:
In a longitudinal wave, the elements of the medium undergo displacements
parallel to the motion of the wave. A longitudinal wave is also called a
compression wave.
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Some definitions…
•
•
•
•
•
•
crests and troughs are the high and low points of a wave
amplitude, 𝐴, is the height of a crest (depth of a trough)
wavelength, 𝜆, is the distance between crests (troughs)
frequency, 𝑓, is the number of cycles per unit time
period, 𝑇, is the length of a cycle
wave velocity, 𝑣, is the velocity the wave crest travels
𝝀
𝒗 = = 𝝀𝒇
𝑻
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Waves on a String and in Air
Waves on a string (transverse waves)
are propagated by the difference in
directions of the tensions.
Sounds waves (longitudinal waves)
are pressure waves.
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Wave Speed: String
Both waves on a string and sound waves require a medium
and the properties of the medium determine the speed of
the wave.
Ts
For wave on a string, the speed is given by: v 

where 𝑇𝑠 is the tension in the string and
𝜇 is the linear mass density: 𝜇 = 𝑚 𝐿
Observations:
- Wave speed increases with increasing tension
- Wave speed decreases with increasing linear density
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The Principle of Superposition
Two travelling waves can meet and pass through each other
without being destroyed or even altered.
Principle of Superposition
- when two waves pass
through the same point,
the displacement is the
sum of the individual
displacements
Pulses are unchanged after
the interference.
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Constructive Interference
Constructive:
Two waves, 1 and 2, have the
same frequency and amplitude
and are “in phase.”
The combined wave, 3, has the
same frequency but a greater
amplitude.
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Destructive Interference
Destructive:
Two waves, 1 and 2, have the
same amplitude and frequency
but one is inverted relative to
the other (i.e. they are 180°
“out of phase”)
When they combine, the
waveforms cancel.
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Wave Pulse Reflection
Just like light reflects off water or an echo bounces off a
cliff, a wave pulse on a string will reflect at a boundary.
Whenever a traveling pulse reaches a boundary, some or all
of the pulse is reflected.
There are two types of boundaries:
- Fixed end
- Loose end
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Reflection of Pulses – Fixed End
When a pulse is reflected from a
fixed end, the pulse is inverted, but
the shape and amplitude remains the
same.
Think about Newton’s 3rd law at the
boundary point.
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Reflection of Pulses – Free End
When reflected from a free end,
the pulse is not inverted, again the
shape and amplitude remains the
same.
Think about Newton’s 3rd law at the
boundary point.
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Pulse Refection at a Discontinuity
A discontinuity can act like a fixed or a free end depending
on how the medium changes.
Low to high linear mass
density acts like fixed end
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High to low linear mass
density acts like free end
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Standing Waves
When a travelling wave reflects back on itself, it creates
travelling waves in both directions.
The wave and its reflection interfere according to the
Principle of Superposition.
The wave appears to stand still, producing a standing wave.
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Standing Waves on a String
A simple example of a standing wave is a wave on a string,
like you will see in Vibrating String practical.
The mechanical oscillator creates a traveling wave that is
reflected off the fixed end and interferes with itself.
The result is a series of nodes and antinodes, with the exact
number depending on the oscillating frequency.
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Standing Waves on a String
Nodes are points where the amplitude is 0. (destructive
interference)
Anti-nodes are points where the amplitude is maximum.
(constructive interference)
Distance between two successive nodes is ½ λ.
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Standing Waves on a String
The figure shows the “n = 2” standing
wave mode.
The red arrows indicate the direction of
motion of the parts of the string.
All points on the string oscillate together
vertically with the same frequency, but
different points have different
amplitudes of motion.
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Standing Wave on a String
There are restrictions to a standing
wave on a string.
1. Two ends of the string are fixed,
so 𝑥 = 0 and 𝑥 = 𝐿 must be
nodes.
2. Standing waves spacing is 𝜆 2
between nodes, so the nodes
must be equally spaced.
As a result, standing waves will only
form at particular modes, which have
numbers, i.e. 𝑚 = 1, 𝑚 = 2, etc.
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Standing Wave on a String
Each mode has a specific
wavelength.
For 𝑚 = 1, the wavelength is:
𝜆1 = 2𝐿.
In general, the wavelength for
a standing wave on a string is:
𝝀𝒎 =
𝟐𝑳
𝒎
for 𝑚 = 1, 2, 3, 4, …
Note: The mode number (𝑚)
is equal to the number of antinodes.
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Standing Wave on a String
The standing wave on a string can exist only if it has one of
these wavelengths: 𝜆𝑚 = 2𝐿 𝑚.
We can also calculate the
frequency of the standing
wave:
𝑣
𝑣
𝑣
𝑓𝑚 =
=
=𝑚
𝜆𝑚 2𝐿 𝑚
2𝐿
for 𝑚 = 1, 2, 3, 4, …
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Standing Wave on a String
𝑣
.
2𝐿
The first mode is called the fundamental frequency: 𝑓1 =
All other modes have a frequency that are multiples of this
fundamental frequency: 𝑓𝑚 = 𝑚𝑓1 .
The fundamental frequency (𝑓1 ) is known as the first
harmonic, 𝑓2 is the second harmonic, 𝑓3 is the third
harmonic, etc …
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