IB – PHYSICS (Core)– Oscillation and Wave
Topic 4.1 Kinematics of simple harmonic motion
Oscillations occur when a system is disturbed from a position of stable equilibrium. This
displacement from equilibrium changes periodically over time. Thus, Oscillations are said to be
periodic, and display periodic motion.. Some examples of oscillations are shown in figure 1
below.
Fig 1 : Some basic examples of Oscillations
Notice that for a system to be oscillating, the shape of the displacement - time graph does not
matter. The only Property that matters is that the motion is periodic.
Basic properties of Oscillating Systems
Amplitude of the oscillation: The amplitude of the oscillation is the parameter that varies with
time and this resides on the y-axis of the oscillation graphs. In figure 1, the amplitude of the
oscillation is the maximum displacement of the object from its equilibrium position.
Time Period (T) of the oscillation: The time period of the oscillation is simply the time taken
for the oscillation to repeat itself. That is, it is the time between successive oscillations of the
system.
Frequency: The frequency of the oscillating system is simply the number of compete
oscillations that happen in 1 second. So,
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The units of frequency are cycles per second which are given the name Hertz
Angular Frequency
The Angular Frequency of a system is the rotational analogue to frequency. It is given the
symbol ω and is measured in radians per second (rads-1). It is defined by the equation
but,
and so is related to frequency by
Phase
The Phase of an oscillation is the amount the oscillation lags behind, or leads in front of a
reference oscillation. For example, take a sine oscillation of maximum amplitude, A, and angular
frequency, ω, and also a cosine oscillation of maximum amplitude, A, and angular frequency, ω
as in figure 3.
Figure 3 : Diagram to show the phase of two oscillations
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Now, here in figure 3 we can take the sine wave to be our reference oscillation. It can be seen
from the diagram that the cosine wave lags behind the sine wave by π/2 (1/4 of a wavelength).
So, we can say that the two waves are out of phase by π/2 or that there is a phase difference of
π/2. Oscillations can have phase differences of any multiple of π. However, if they have a phase
difference of either 0 or 2π they are said to be in phase.
Harmonic Oscillator
A harmonic oscillator is a system which, when displaced from its equilibrium position,
experiences a restoring force, F, proportional to the displacement, x according to Hooke's law:
where k is a positive constant.
If F is the only force acting on the system, the system is called a simple harmonic oscillator,
and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point,
with constant amplitude and a constant frequency (which does not depend on the amplitude).
In addition to its amplitude, the motion of a simple harmonic oscillator is characterized by its
period T, the time for a single oscillation, its frequency, f, the reciprocal of the period f = 1⁄T (i.e.
the number of cycles per unit time), and its phase, φ, which determines the starting point on the
sine wave. The period and frequency are constants determined by the overall system, while the
amplitude and phase are determined by the initial conditions (position and velocity) of that
system. Overall then, the equation describing simple harmonic motion is
x = Acosωt.
Alternatively a sine can be used in place of the sine with the phase shifted by π⁄2.
The general differential force equation for an object of mass m experiencing SHM is:
where k is the spring constant which relates the displacement of the object to the force applied to
the object. The general solution for this equation is given above.
Angular frequency in circular motion is the rate of change of angle. It is measured in radians
per. second. Since 2π radians is equivalent to one complete rotation in time period T:
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The time period of an oscillation is the time taken to repeat the pattern of motion once. In
general:
However, depending on the type of oscillation, the value of ω changes. For a mass on a spring:
The frequency of the oscillations given by:
.
Velocity and Acceleration of a simple harmonic oscillator.
The displacement of a simple harmonic oscillator is:
x = Acosωt
Velocity is the rate of change of displacement, so:
Acceleration is the rate of change of velocity, so:
The velocity and acceleration oscillate with a quarter and half a period delay from the
displacement.
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The velocity and acceleration of a simple harmonic oscillator oscillate with the same frequency
as the position but with shifted phases. The velocity is maximum for zero displacement, while
the acceleration is in the opposite direction as the displacement.
4.2 Energy in simple harmonic motion
The kinetic energy K of the system at time t is
and the potential energy is
The total mechanical energy of the system therefore has the constant value
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Examples of Simple harmonic motion:
1. Simple Pendulum
A pendulum is a weight suspended from a pivot so it can swing freely.
When a pendulum is displaced from its resting equilibrium position, it is subject to a restoring
force due to gravity that will accelerate it back toward the equilibrium position. When released,
the restoring force combined with the pendulum's mass causes it to oscillate about the
equilibrium position, swinging back and forth. The time for one complete cycle, a left swing and
a right swing, is called the period
Figure : The Simple Pendulum
If a pendulum of mass m attached to a string of length L is displaced by an angle
vertical (see figure below),
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it experiences a net restoring force due to gravity:
F = - mgsin
.
For small angles, sin
, providing is expressed in radians (try it on your calculator for
0.1,0.5,1.0 radians). In terms of radians,
=
𝜃 = 𝑠⁄𝑙 radians
where s is the arc length and L is the length of the string. Thus, for small displacements, s , the
restoring force can be written:
𝑚𝑔
𝐹 = −(
𝑙
)𝑠
Since the restoring force is proportional to the displacement, the pendulum is a simple harmonic
oscillator with ``spring constant'' k = mg/L . The period of a simple pendulum is therefore:
𝑚
𝑇 = 2𝜋√
𝑘
or, 𝑇
= 2𝜋√
𝑙
𝑔
Note:

In this small angle approximation, the amplitude of the pendulum has no effect on the
period. This is what makes pendulums such good time keepers. As they inevitably lose
energy due to frictional forces, their amplitude decreases, but the period remains
constant.
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A Particle In A Bowl:
Consider a particle of mass m, placed inside a frictionless of radius r. It is in the
equilibrium when at its bottom most position of the bowl. When the particle is displaced
from the bottom most position to a position P it starts executing SHM. The forces on the
particle at P are shown in figure.
The force trying to bring the particle back in its position is given by,
F =ma = -mg sinθ
From which, a= -g sinθ, using θ = x/r we get,
𝑎 = −𝑔𝑠𝑖𝑛
𝑥
𝑟
The particle starts oscillating but the oscillation is not simple harmonic. Assuming θ to be
small we approximate sinθ = θ and the expression for acceleration changes to,
𝒙
𝒂 = −𝒈 𝒓
𝒈
or 𝒂 = −𝝎𝟐 𝒙, where 𝝎𝟐 = 𝒓 ,
So the particle executes SHM only for very small amplitude. The time period is given by,
𝑟
𝑇 = 2𝜋√
𝑔
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Damped SHM
In physics, damping is any effect that tends to reduce the amplitude of oscillations in an
oscillatory system, particularly the harmonic oscillator. In real oscillators friction, or damping,
slows the motion of the system. Depending on the friction coefficient, the system can:



Oscillate with a frequency smaller than in the non-damped case, and an amplitude
decreasing with time (underdamped oscillator).
The system returns to equilibrium as quickly as possible without oscillating. This is often
desired for the damping of systems such as doors. (critically damped)
Decay exponentially to the equilibrium position, without oscillations (overdamped
oscillator).
4.3 Forced Oscillations and Resonance:
If an oscillator is displaced and then released it will begin to vibrate. If no more external
forces are applied to the system it is a free oscillator and its frequency is called its
natural frequency. If a force is continually or repeatedly applied to keep the oscillation
going, it is a forced oscillator.
The to and fro motion of a body about a mean position is called oscillation. When a body
execute oscillations under the action of external periodic force, Those oscillations are called
forced oscillations.
When one of the two bodies of same natural frequency is set into vibration, the other body also
vibrates with larger amplitude under the influence of the first one. This phenomenon is called
Resonance.
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Take two hollow boxed A and B, open it one side. Place two boxes with their opened sides
facing each other with some distance apart. Mount two tuning forks of same natural frequencies
one each on the two boxes. Vibrate the tuning fork on the box A, the tuning fork on box B also
begins to vibrate due to resonance. When A is vibrated, the air inside the box A vibrates and
these vibrations are transferred into the box B and in turn vibrated the tuning fork on B.
Examples and Applications on Forced Oscillations and
Resonance
When solders are crossing a suspension bridge, they are asked to break their steps. This is
because, when the frequency of the marching coincides with the natural frequency of the bridge,
the bridge vibrates with larger amplitude and collapses due to resonance.
A radio is tuned to obtain a clear sound, such that the frequency of the radio has to coincide with
the frequency of the incoming electro-magnetic waves.
We can find the velocity of sound using resonance phenomenon with resonating air-column
experiment.
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4.4 Wave Characteristics:
In mathematics and science, a wave is a disturbance that travels through space and time, usually
by transference of energy. Waves are described by a wave function that can take on many forms
depending on the type of wave. A mechanical wave is a wave that propagates through a medium
due to restoring forces produced upon its deformation.
Waves travel and transfer energy from one point to another, often with no permanent
displacement of the particles of the medium—that is, with little or no associated mass transport.
They consist instead of oscillations or vibrations around almost fixed locations. Imagine a cork
on rippling water, it would bob up and down staying in about the same place while the wave
itself moves outward. When we say that a wave carries energy but not mass, we are referring to
the fact that even as a wave travels outward from the center (carrying energy of motion), the
medium itself does not flow with it.
Categories of Waves
Waves come in many shapes and forms. While all waves share some basic characteristic
properties and behaviors, some waves can be distinguished from others based on some
observable (and some non-observable) characteristics. It is common to categorize waves based
on these distinguishing characteristics.
Longitudinal versus Transverse Waves versus Surface Waves
One way to categorize waves is on the basis of the direction of movement of the individual
particles of the medium relative to the direction which the waves travel. Categorizing waves on
this basis leads to three notable categories: transverse waves, longitudinal waves, and surface
waves.
A transverse wave is a wave in which particles of the medium move in a direction perpendicular
to the direction which the wave moves. Transverse waves are always characterized by particle
motion being perpendicular to wave motion.
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Crest and Trough
The section of the wave that rises above the undisturbed position is called the crest. That section
which lies below the undisturbed position is called the trough. These sections are labeled in the
following diagram:
Wavelength - Distance between two successive crest or trough and is represented as lambda λ.
Frequency – Number of complete waves generated per second and its unit is Hertz (HZ).
Amplitude – Height of a crest or the depth of a trough measured from the undisturbed position
of what is carrying the wave.
A longitudinal wave is a wave in which particles of the medium move in a direction parallel to
the direction which the wave movesLongitudinal waves are always characterized by particle
motion being parallel to wave motion.
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A sound wave traveling through air is a classic example of a longitudinal wave.
The compressions are regions of high air pressure while the rarefactions are regions of low air
pressure. The diagram below depicts a sound wave created by a tuning fork and propagated
through the air in an open tube. The compressions and rarefactions are labeled.
Wave Fronts and Rays
Waves, whether it be electomagnetic or other, are conveniently described in terms of wave
fronts. A wave front is the line or surface defined by adjacent protions of a wave that are in
phase.--If an arc is drawn along one of the crests of a circular water wave moving out from a
point source, all the particles on the line will be in phase.
The curvature of a short segment of a spherical or circular wave front is small. The segment may
be approximated as a linear wave front or a plane wave front, just as we assume the surface of
the Earth to be locally flat.
In a uniform medium wave fronts propagate outward from the source at a wave speed
characteristic of the medium. For example, the speed of light travels fastest in a vacuum:
The geometrical description of a wave in terms of wave fronts tends to neglect the fact that the
wave is actually sinusoidal. The concept of a ray simplifies the wave description even further. A
ray is a line drawn perpendicular to to a series of wave fronts and pointing in the direction of
propagation. A beam of light can be simplified and represented by a group of parallel rays or just
a sinlge ray.
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4.5 Wave properties:
Reflection of wave:
Waves carry energy and momentum, and whenever a wave encounters an obstacle, they are
reflected by the obstacle. This reflection of waves is responsible for echoes, radar detectors, and
for allowing standing waves which are so important to sound production in musical instruments.
At a fixed (hard) boundary, the displacement remains zero and the reflected wave changes its
polarity (undergoes a 180o phase change).
At a free (soft) boundary, the restoring force is zero and the reflected wave has the same polarity
(no phase change) as the incident wave.
Refraction of Wave:
Refraction of waves involves a change in the direction of waves as they pass from one medium
to another. Refraction, or the bending of the path of the waves, is accompanied by a change in
speed and wavelength of the waves. Speed of a wave is dependent upon the properties of the
medium through which the waves travel. So if the medium (and its properties) are changed, the
speed of the waves is changed. The most significant property of water which would affect the
speed of waves traveling on its surface is the depth of the water. Water waves travel fastest when
the medium is the deepest. Thus, if water waves are passing from deep water into shallow water,
they will slow down. The decrease in speed will also be accompanied by a decrease in
wavelength. So as water waves are transmitted from deep water into shallow
water, the speed decreases, the wavelength decreases, and the direction
changes.
This boundary behavior of water waves can be observed in a ripple tank if the
tank is partitioned into a deep and a shallow section. If a pane of glass is
placed in the bottom of the tank, one part of the tank will be deep and the other
part of the tank will be shallow. Waves traveling from the deep end to the
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shallow end can be seen to refract (i.e., bend), decrease wavelength (the wavefronts get closer
together), and slow down (they take a longer time to travel the same distance). When traveling
from deep water to shallow water, the waves are seen to bend in such a manner that they seem to
be traveling more perpendicular to the surface. If traveling from shallow water to deep water, the
waves bend in the opposite direction.
Diffraction of wave:
Diffraction involves a change in direction of waves as they pass through an opening or around a
barrier in their path. Water waves have the ability to travel around corners, around obstacles and
through openings. This ability is most obvious for water waves with longer wavelengths.
Diffraction can be demonstrated by placing small barriers and obstacles in a ripple tank and
observing the path of the water waves as they encounter the obstacles. The waves are seen to
pass around the barrier into the regions behind it; subsequently the water behind the barrier is
disturbed. The amount of diffraction (the sharpness of the bending) increases with increasing
wavelength and decreases with decreasing wavelength. In fact, when the wavelength of the
waves are smaller than the obstacle, no noticeable diffraction occurs.
Diffraction of water waves is observed in a harbor as waves bend around small boats and are
found to disturb the water behind them. The same waves however are unable to diffract around
larger boats since their wavelength is smaller than the boat. Diffraction of sound waves is
commonly observed; we notice sound diffracting around corners, allowing us to hear others who
are speaking to us from adjacent rooms.
Interference of Waves:
Wave interference is the phenomenon which occurs when two waves meet while traveling
along the same medium. The interference of waves causes the medium to take on a shape which
results from the net effect of the two individual waves upon the particles of the medium. To
begin our exploration of wave interference, consider two pulses of the same amplitude traveling
in different directions along the same medium. Let's suppose that each displaced upward 1 unit at
its crest and has the shape of a sine wave. As the sine pulses move towards each other, there will
eventually be a moment in time when they are completely overlapped. At that moment, the
resulting shape of the medium would be an upward displaced sine pulse with an amplitude of 2
units. The diagrams below depict the before and during interference snapshots of the medium for
two such pulses. The individual sine pulses are drawn in red and blue and the resulting
displacement of the medium is drawn in green.
This type of interference is sometimes called constructive interference. Constructive
interference is a type of interference which occurs at any location along the medium where the
two interfering waves have a displacement in the same direction. In this case, both waves have
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an upward displacement; consequently, the medium has an upward displacement which is greater
than the displacement of the two interfering pulses. Constructive interference is observed at any
location where the two interfering waves are displaced upward. But it is also observed when both
interfering waves are displaced downward. This is shown in the diagram below for two
downward displaced pulses.
In this case, a sine pulse with a maximum displacement of -1 unit (negative means a downward
displacement) interferes with a sine pulse with a maximum displacement of -1 unit. These two
pulses are drawn in red and blue. The resulting shape of the medium is a sine pulse with a
maximum displacement of -2 units.
Destructive interference is a type of interference which occurs at any location along the
medium where the two interfering waves have a displacement in the opposite direction. For
instance, when a sine pulse with a maximum displacement of +1 unit meets a sine pulse with a
maximum displacement of -1 unit, destructive interference occurs. This is depicted in the
diagram below.
In the diagram above, the interfering pulses have the same maximum displacement but in
opposite directions. The result is that the two pulses completely destroy each other when they are
completely overlapped.
Conditions for sustained interference:
(1) The sources must be coherent (i.e., they must maintain a
constant phase relationship with one another).
(2) The sources must be monochromatic (i.e., of a single
wavelength).
(3) The linear superposition principle is applicable.
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Young's Double-Slit Experiment
Slits S1 and S2 serve as a pair of coherent light sources.
A visible pattern of bright and dark parallel bands (called
fringes) are produced on screen C.
Bright bands: produced by constructive interference. Dark bands: produced by
destructive interference.
Path difference of two waves:
= r2 - r1 = d sinθ
Condition for constructive interference:
Pathdifferenced sinθmλ(m = 0, 1, 2, …)
Condition for destructive interference:
Pathdifferenced sinθ(m1/2)λ (m = 0, 1, 2, …)
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