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4. Oscillations & Waves
Topic OutlineSection Recommend
ed Time Giancoli Sections
4.4 Wave Characteristics 2h 11.7, 11.8, 11.94.5 Wave Properties 2h 11.11, 11.134.1 Simple Harmonic Motion 2h 11.1, 11.3, 11.44.2 Energy Changes in SHM 1h 11.24.3 Forced Oscillations and Damping 3h 11.5, 11.6
4.4 Wave Characteristics
“The impetus is much quicker than water, for it often happens that the wave flees the place of its creation, while the water does not; like waves made in a field of grain by the wind, where we see the waves running across the field while the grain remains in place.” Leonardo da Vinci
Wave TermsA pulse is a single disturbance that transfers
energy over a distanceAn oscillation is a periodic (regularly repeating)
disturbanceA wave is a periodic disturbance that transfers
energy over a distance without any net transfer of the medium
Mechanical waves (e.g. sound, water waves) travel through a medium (e.g. air, water)
Electromagnetic waves (e.g. light, X-rays, UV rays) do not require a medium to propagate
Transverse WavesIn transverse waves, the particles oscillate at
right angles to the direction of travelExamples of transverse waves include: water
waves, electromagnetic wavesThe point of maximum amplitude is called a crest,
the point of minimum amplitude is called a trough Although mechanical transverse waves can travel
along the surface of a liquid, they do not travel through liquids or gasses
Longitudinal WavesIn longitudinal waves, the particles oscillate along
the direction of travelExamples of longitudinal waves include: sound
waves, the primary waves of an earthquake The point where the particles are closest together is
called a compression, the point where the particles are farthest apart is called a rarefaction
Representing WavesWaves can be represented by:
an arrow (ray) showing the direction that the wave is propagating in
lines showing the wave crests (wavefronts) looking down on the waves from above
Wave Terms The displacement (x, m) of a particle is how far it has moved from its rest
position The wavelength (l, m) is the distance between two consecutive particles
that have the same displacement, i.e. to consecutive crests/compressions or troughs/rarefactions
The velocity (v, m.s-1) of a wave is the speed and direction in which it is travelling
The frequency (f, s-1 or Hz) of a wave is the number of wavelengths that pass a given point every second
The period (T, s) of a wave is the amount of time taken for one wavelength to pass a given point
The amplitude (A, m) is the maximum displacement of a particle from its rest position
The intensity (I, J.m-2.s-1) is the energy that a wave transports per unit time across a unit area of the medium. The energy transported is proportional to the square of the amplitude, I a A2
Frequency and PeriodFrequency is the inverse of period:
f = 1/TPeriod is the inverse of frequency:
T = 1/ff = frequency, Hz (or s-1)T = period, s
Wave VelocityThe velocity of a wave in a given medium is
calculated by: v = fl
v = velocity, m.s-1
f = frequency, Hzl = wavelength, mWorksheet: Wave Velocity Questions
Displacement-Position Graph
A displacement-position graph shows a ‘snapshot’ of a wave at a given time
Displacement-Time Graph
A displacement-time graph shows how the displacement of one particle in the medium varies over time
The following graph shows how the displacement of point P (previous graph) varies with time
Electromagnetic WavesElectromagnetic waves consist of a varying
electric field and a varying magnetic fieldThese two fields travel in the same direction and
are at right angles to each otherElectromagnetic waves are transverse waves and
they can propagate through a vacuum, i.e. they do not require a medium to propagate
The speed of electromagnetic radiation (EMR) in a vacuum (c) is 3.00 x 108 m.s-1
Electromagnetic waves are produced when a charge is accelerated
Electromagnetic Waves
Electromagnetic Spectrum
High energy EMR has a high frequency (and short wavelength)
Low energy EMR has a low frequency (and long wavelength)
Electromagnetic Spectrum
Type of EMR Wavelength Range (m) Applications Other
RadioMicrowavesInfra-RedVisible LightUltra-VioletX-RaysGamma Rays
Electromagnetic Spectrum
Type of EMR Wavelength Range (m) Applications Other
Radio <102 AM, FM, TV and radar signals
Microwaves 102-10-3 Cooking, communications
Infra-Red 10-3-10-6 Heat
Visible Light 700 nm-400 nm Vision
Ultra-Violet 10-7-10-8
Sterilisation from microbes, production of vitamin D in skin
X-Rays 10-8-10-10 Imaging bones, cancer treatment
Gamma Rays >10-8 Cancer treatment
4.5 Wave Properties
Reflection of PulsesString with a fixed end If a pulse travels along a string that is fixed to a rigid
support, the pulse is reflected with a phase change of 180ºThe shape of the pulse stays the same, except that it is
inverted and travelling in the opposite directionThe amplitude of the pulse is slightly less as some energy
is absorbed at the fixed end When the (upward) pulse reaches the fixed end, it exerts
an upward force on the support, the support then exerts and equal and opposite downward force on the string (reaction force), causing the inverted pulse to travel back along the string
http://rt210.sl.psu.edu/phys_anim/waves/indexer_waves.html
Reflection of PulsesString with a free endIf a pulse travels along a string that is tethered
to a pole but free to move, the pulse is reflected with no phase change
The shape of the pulse stays the same, except that it is travelling in the opposite direction
Reflection of Wavefronts
Reflection is when waves bounce off a surfaceThe first law of reflection is that the angle of
incidence equals the angle of reflection
qi = qr
The second law of reflection is that the incident ray, normal line and reflected ray all lie on the same plane
Reflection at a Plane Boundary
RefractionRefraction is when waves bend as they travel
from one medium to anotherWhen a wave travels into a different medium:
the wave speed changesthe wavelength changesthe frequency stays the same If the wave hits the new medium at an angle, the
wave direction will change
RefractionRefraction is when waves bend as they travel from
one medium into anotherFor example, light waves travelling from air to water;
ocean waves travelling from deep water to shallow water
Refraction occurs because the speed of the wave changes:when waves speed up, they bend away from the normalwhen waves slow down, they bend towards the normal
Water waves travel faster in deep water, light rays travel faster in less dense media
Refraction
Refractive IndexThe refractive index (n) of a medium relates to
how fast waves travel through that mediumRefractive index is a relative scale and does not
have units As light travels into a denser medium:
refractive index increaseswave velocity decreaseswavelength decreasesfrequency stays the same
Refractive IndexSome common refractive indices are:
nvacuum = 1 nwater = 1.33 nglass = 1.5
nrock salt = 1.5 nruby = 1.76 ndiamond = 2.4
n1 = refractive index of medium 1
n2 = refractive index of medium 2
v = velocity, m.s-1
l = wavelength, m
Snell’s LawSnell’s Law relates the angle of incidence, the
angle of refraction and the refractive indices of two media
n1sinq1 = n2sinq2
n1 = refractive index of medium 1
n2 = refractive index of medium 2
q1 = angle of incidence
q2 = angle of refraction
Huygen’s PrincipleHuygens’ Principle: Every point on a wave
front can be considered as a source of tiny wavelets. These wavelets spread out in the forward direction at the speed of the wave. The new wave front is given by the tangent of all of these wavelets.
DiffractionDiffraction is when waves bend slightly when
travelling past an obstacle or through an aperture
Diffraction is noticeable when the size of the obstacle is similar to the size of the wave
For example, ocean waves bending around a headland, shortwave radio bending around the corner of a building, AM radio waves bending over a hill
Superposition of WavesWhen two waves of the same nature travel past
each other, the displacement of the resultant wave is the sum of the displacements of the individual waves at that point
Constructive Interference
Constructive interference is when displacements of the individual waves are in the same direction, and the resultant wave has a greater amplitude
Destructive Interference
Destructive interference is when the displacements of the individual waves are in opposite directions, and the resultant wave has a lesser amplitude
Two-Source Interference
If two point sources produce coherent waves (i.e. same wavelength, same phase), a pattern of constructive and destructive interference occurs around them
Two-Source Interference
Where two crests (or two troughs) meet, constructive interference occurs resulting in a greater displacement; this is called an antinode
Where a crest and a trough meet, destructive interference occurs, resulting in no displacement; this is called a node
When viewed from above, lines of antinodes and nodes radiate outward from the point between the sources
If the sources are in phase, there will always be a central antinode (i.e. maxima or line of constructive interference)
Path Difference & Phase Difference
We can predict where nodes and anti-nodes will occur in two-source interference
This is done by measuring the path length from one source (S1) to a given point (A), and the path length from the other source (S2) to that same point (A)
If the path difference (S1A – S2A) is a whole number of wavelengths, the waves from the two sources will arrive in phase and constructive interference will occur; this will cause an antinode
path difference = nl for an anti-node If the path difference is a half wavelength greater, the waves
from the two sources will arrive 180º out of phase and destructive interference will occur; this will cause a node
path difference = (n + ½)l for a node
Interference of LightWhen monochromatic (one coloured) light
shines through a thin double slit (creating two sources of the same light), an interference pattern is observed
This experiment demonstrates the wave properties of light and was first performed by English scientist Thomas Young in 1801
Beat FrequencyIf two waves of the same nature but slightly
different frequency are allowed to interfere, the waves will drift in phase (causing constructive interference and greater amplitude) and then out of phase again (causing destructive interference and lesser amplitude)
This periodic oscillation in the amplitude of the combined waveform is called the beat frequency
The most obvious example is when two musical notes that are slightly out of tune are played together, the resulting sound has a regular increase and decrease in volume
Beat Frequency
4.1 Simple Harmonic Motion
Simple Harmonic Motion
Simple Harmonic Motion
An object is moving with Simple Harmonic Motion (SHM) if: the object experiences a net force (and therefore an
acceleration) towards the equilibrium position the magnitude of the net force (and therefore
acceleration) is proportional to the distance of the object from the equilibrium position
it is moving with a regular, repeating motion with constant period and (theoretically) constant amplitude
the motion of the object can be modelled by a ‘reference circle’
Examples of SHM include: a pendulum swinging, a mass bouncing on a spring, a guitar string that has been plucked, the tides
Terms in SHM The displacement (x, m or q, rad) of an object moving in SHM is the
distance of the object from its equilibrium position The amplitude (x0 or q0) is the maximum displacement of the object The period (T, s) is the time taken for one complete oscillation The frequency (f, Hz) is the number of oscillations per second; f =
1/T The angular frequency (w, rad.s-1) is equivalent to the angular
speed of an object moving in uniform circular motion
w = angular frequency, rad.s-1
f = frequency, Hz
T = period, s
The Reference CircleThe reference circle is used to understand SHM We consider the motion of an object moving with
uniform circular motion, and project this motion into one dimension
The amplitude, x0, of the SHM is given by the radius of the circle
The angular frequency, w, of SHM is given by the angular velocity of the object in the reference circle
Displacement in SHMThe displacement of an object in SHM is given by
the vertical component of the displacement of an object moving in the reference circle
x = x0cosq [or x = x0sinq (starting at 3 o’clock)]
Since w = q/t or q = wtx = x0cos(wt) [or x = x0sin(wt)]
x0 = amplitude of SHM, m
w = angular velocity, rad.s-1
x = displacement of the object in SHM, m
q = angle from starting position, rad
Velocity in SHMThe velocity of an object moving in SHM is given
by the vertical component of the velocity of an object moving in the reference circle
v = -x0wsin(wt) [or v = x0wcos(wt)]These equations can also be written
v = -v0sin(wt) [or v = v0cos(wt)]
Where v0 = wx0 and v0 is the maximum velocity
Another Equation for Velocity
The velocity of an object moving with SHM is given by v = -x0wsin(wt)
Since sin2q + cos2q = 1 or sinq = √(1 - cos2q)We can say v = -x0w√(1 - cos2(wt))
Putting the x0 inside the square root sign we get
v = -w√(x02 – x0
2cos2(wt))
Since x = x0 cos(wt), another equation for velocity in SHM is:
v = -w√(x02 – x2)
x0 = amplitude of SHM, m x = displacement of the object in SHM, m
w = angular velocity, rad.s-1 t = time, s
v = velocity of the object in SHM, m.s-1
Acceleration in SHMThe acceleration of an object in SHM is given by
the vertical component of the centripetal acceleration of the object in the reference circle
a = -x0w2cos(wt) [or a = -x0w2sin(wt)]
Since x = x0cos(wt)
a = -w2xThis is the mathematical definition of SHM:
acceleration is proportional to displacement but directed in the opposite direction to displacement (hence the negative sign). The acceleration is always directed towards the equilibrium position.
Phasor DiagramsA phasor diagram shows the phase angle of
the displacement, velocity or acceleration in SHM
We say that: velocity leads displacement by 90º (or p/2 radians)acceleration leads velocity by 90º (or p/2 radians)acceleration leads displacement by 180º (or p
radians)
Force in SHMFor an object moving in SHM, the force and
acceleration are always directed towards the equilibrium position
We can use F = ma to calculate the net force acting on an object moving in SHM
SHM of a PendulumThe period of motion for a pendulum is given by
T = period, sl = length of string, mg = acceleration due to gravity, m.s-2
SHM of a SpringFrom Hooke’s Law, the tension force in a spring is Ft = -kx
The restoring force in SHM is Fr = ma = -mw2xSince the tension force in the spring provides the restoring
force -kx = -mw2x or k = mw2
Since T = 2p/w
k = spring constant, N.m-1 x = displacement, m
Fr = restoring force, N m = mass, kg
w = angular frequency, rad.s-1 T = period, s
Questions in SHMQuestions in SHM can be solved mathematically
or graphically
4.2 Energy Changes in Simple Harmonic Motion
Energy Changes in SHMAn object moving in SHM has purely potential
energy at either end of its motion and purely kinetic energy at the mid-point in its motion
At other stages of its motion, the object has a mixture of potential and kinetic energy
The total energy remains the same throughout the cycle (assuming no friction)
Kinetic EnergyThe kinetic energy of an object is given by
EK = ½ mv2
Since the velocity of an object moving in SHM is v = -w√(x0
2 – x2)
EK = ½ m w2 (x02 – x2)
EK = kinetic energy, J x0 = amplitude of SHM, m
w = angular velocity, rad.s-1 m = mass, kg
x = displacement of the object in SHM, m
Potential EnergyThe potential energy of a mass moving in SHM is
given by:EP = ½ m w2 x2
EP = potential energy, J
x = displacement of the object in SHM, m
w = angular velocity, rad.s-1
m = mass, kg
Total EnergyThe total energy of a mass moving in SHM isET = EK + EP
ET = ½ m w2 (x02 – x2) + ½ m w2 x2
ET = ½ m w2 (x02 – x2 + x2)
ET = ½ m w2 x02
EP = potential energy, J
x0 = amplitude of SHM, m
w = angular velocity, rad.s-1
m = mass, kg
4.3 Forced Oscillations and
Resonance
DampingTheoretically, when an object is moving in SHM
the amplitude of oscillations stays the sameIn reality, friction forces oppose the movement
of the object and energy is dissipated as heatAs a result, the amplitude of oscillations
decreases This is called dampingExamples of damped oscillations include: a
swing that gradually swings lower and lower, a plucked guitar string that gradually stops vibrating, and the suspension of a car
DampingA lightly damped system is one in which the
oscillations decay very gradually, e.g. a well oiled pendulum
A heavily damped system is one in which the oscillations decay rapidly, e.g. a pendulum swinging underwater (causing a high amount of friction)
A critically damped system is one that is damped so heavily that the object comes to rest at the equilibrium position without oscillating at all
Critically damped systems are used in bridge and building designs where it is important that any oscillations are reduced rapidly
Natural Frequency If an external force is applied to an oscillating system, it
will oscillate at the natural frequency of the system, e.g. a child swinging on a swing after a single push
If an external driver repeatedly applies a force to an oscillating system, forced oscillations will occur, e.g. giving a child on a swing repeated pushes
If the forced oscillations are applied at the same frequency as the natural frequency, resonance occurs and the amplitude of the oscillations increases
Overall, the amplitude of the oscillations of an object moving in SHM will depend on the driving frequency, the natural frequency, the phase difference between these, the amplitude of the driving force, and the amount of damping
Resonance CurveIn a resonance curve, the amplitude of
oscillations (y-axis) is plotted against the driver frequency (x-axis)
When the driver frequency is equal to the natural frequency (or resonant frequency), the amplitude of oscillations will reach a peak
If the system is lightly damped, there will be a high, steep peak at the resonant frequency, giving a clearly defined peak
If the system is heavily damped, the resonance peak will be smaller and less steep
Resonance Curve
Examples of ResonanceThere are situations in which resonance can be harmful or
unhelpful, for example: when wind gusts blow a bridge at the natural frequency of the
bridge when the waves of an earthquake match the natural frequency
of a tall building when a car goes over a bumpy road at the natural frequency
of the car’s suspension when spinning machinery operates at the resonant frequency
of the structures supporting itResonant properties are also used by humans, for example:
sound resonates and is amplified in the cavity of an acoustic guitar or violin
electrical circuits can be tuned to the frequency of radio or TV signals resulting in reception of a radio or TV channel