4. Oscillations & Waves

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)


Type of EMR Wavelength Range (m) Applications Other

RadioMicrowavesInfra-RedVisible LightUltra-VioletX-RaysGamma Rays


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

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


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



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


Potential EnergyThe potential energy of a mass moving in SHM is

given by:EP = ½ m w2 x2

EP = potential energy, J



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


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

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4. Oscillations & Waves