Oscillations and Waves

Wave Characteristics

Waves

A wave is a means of transferring energy and momentum from one point to another without there being any transfer of matter between the two points.

This illustrates that there is no net transfer of the medium through which the wave travels, only energy moves from place to place.

In many examples, the wave carrying medium will oscillate with simple harmonic motion (i.e. a -x).

Progressive Waves

Any wave that moves through or across a medium (e.g. water or even a vacuum) carrying energy away from its source is a progressive (travelling) wave.

E.g. A duck on water:

As the wave passes the duck, the water (and duck) only oscillate vertically.

Wave direction Duc

k os

cilla

tion

Describing Waves1. Mechanical or Electromagnetic

Mechanical waves are made up of particles vibrating. e.g. sound – air molecules; water – water moleculesAll these waves require a substance for transmission and so none of them can travel through a vacuum.

Electromagnetic waves are made up of oscillating electric and magnetic fields.e.g. light and radioThese waves do not require a substance for transmission and so all of them can travel through a vacuum.

Describing Waves2. Progressive or Stationary

Progressive waves are waves where there is a net transfer of energy and momentum from one point to another. e.g. sound produced by a person speaking; light from a lamp

Stationary waves are waves where there is a NO net transfer of energy and momentum from one point to another. e.g. the wave on a guitar string

Describing Waves3. Longitudinal or Transverse

Longitudinal waves are waves where the direction of vibration of the particles is parallel to the direction in which the wave travels. e.g. sound

wave direction

vibrations

LONGITUDINAL WAVE

Describing WavesTransverse waves are waves where the direction of vibration of the particles or fields is perpendicular to the direction in which the wave travels. e.g. water and all electromagnetic waves

Test for a transverse wave:Only TRANVERSE waves undergo polarisation.

wave direction

vibrations

TRANSVERSE WAVE

Polarisation

The oscillations within a transverse wave and the direction of travel of the wave define a plane. If the wave only occupies one plane the wave is said to be plane polarised.

Polarisation

Light from a lamp is unpolarised. However, with a polarising filter it can be plane polarised.

Polarisation

If two ‘crossed’ filters are used then no light will be transmitted.

Aerial alignmentRadio waves (and microwaves) are transmitted as plane polarised waves. In the case of satellite television, two separate channels can be transmitted on the same frequency but with horizontal and vertical planes of polarisation.

In order to receive these transmissions the aerial has to be aligned with the plane occupied by the electric field component of the electromagnetic wave.

The picture shows an aerial aligned to receive horizontally polarised waves.

Measuring wavesDisplacement, xThis is the distance of an oscillating particle from its undisturbed or equilibrium position.

Amplitude, aThis is the maximum displacement of an oscillating particle from its equilibrium position. It is equal to the height of a peak or the depth of a trough.

amplitude a

undisturbed or equilibriumposition

Measuring wavesPhase, φThis is the point that a particle is at within an oscillation.

Examples: ‘top of peak’, ‘bottom of trough’

Phase is sometimes expressed in terms of an angle up to 360°. If the top of a peak is 0° then the bottom of a trough will be 180°.

Measuring wavesPhase difference, ΔφThis is the fraction of a cycle between two particles within one or two waves. Example: the top of a peak has a phase difference of half of one cycle compared with the bottom of a trough.

Phase difference is often expressed as an angle difference. So in the above case the phase difference is 180°. Also with phase difference, angles are usually measured in radians. Where: 360° = 2π radian; 180° = π rad; 90° = π/2 rad

Measuring wavesWavelength, λThis is the distance between two consecutive particles at the same phase. Example: top-of-a-peak to the next top-of-a-peakunit – metre, m

wavelength λ

Measuring wavesPeriod, TThis is equal to the time taken for one complete oscillation in of a particle in a wave.unit – second, s Frequency, fThis is equal to the number of complete oscillations in one second performed by a particle in a wave.unit – hertz, HzNOTE: f = 1 / T

The wave equationFor all waves:

speed = frequency x wavelength

c = f λ

where speed is in ms-1 provided frequency is in hertz and wavelength in metres

CompleteWave speed Frequency Wavelength Period

600 m s-1 100 Hz 6 m 0.01 s

10 m s-1 2 kHz 0.5 cm 0.5 ms

340 ms-1 170 Hz 2 m 5.88 ms

3 x 108 ms-1 200 kHz 1500 m 5 x 10-6 s

Each bright line in this diagram represents a crest and can be regarded as a WAVEFRONT.

Wavefronts and rays. 

 

A RAY can be thought of as a locus of one point on a wavefront showing the direction in which energy is travelling.

Download from http://phet.colorado.edu/simulations/index.php?cat=Sound_and_Waves

In the late 19th century physicists had been working extensively with electricity and magnetic fields. A great many discoveries in these fields were being made. At the same time it became universally accepted that the best model for light was the wave model.

James Clerk Maxwell summarised, synthesised and unified these ideas. He came up with the idea that all of these phenomena, including light, were simply different forms of

ELECTROMAGNETIC RADIATION

Electromagnetic waves are created by accelerating charges which result in a rapidly changing magnetic field and electric field travelling at right angles to each other and to their direction of travel.

EM Wave applet: http://micro.magnet.fsu.edu/primer/java/scienceopticsu/electromagnetic/index.html

Although the previous image showed only two transverse waves, EM waves don’t look like these in reality. There are actually many planes of electric-magnetic field oscillations.

Source: http://sol.sci.uop.edu/~jfalward/physics17/chapter11/chapter11.html

Source: http://outreach.atnf.csiro.au/education/senior/astrophysics/images/em_spectrumextended.jpg

It is the FREQUENCY of the waves that determines the type of electromagnetic wave and the different frequencies make up the ELECTROMAGNETIC SPECTRUM

Note that VISIBLE LIGHT only makes up a small part of the spectrum

Source: http://imgs.xkcd.com/comics/electromagnetic_spectrum_small.png

All electromagnetic waves travel with the same speed in free space. It is worthwhile to recall the orders of magnitude of the wavelengths of the principal radiations in the electromagnetic spectrum, such as the following

Wave Pulses

• A pulse wave is a sudden distortion or disturbance that travels through a material or medium

Reflection from fixed or free end

Reflection of Pulses

String 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 180º• The shape of the pulse stays the same, except that it is

inverted and travelling in the opposite direction• The 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 Pulses

String with a free end• If 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

SuperpositionSuperposition is seen when two waves of the same type cross. It is defined as “the vector sum of the two displacements of each wave”:

Superposition of wavesThis is the process that occurs when two waves of the same type meet.

The principle of superposition When two waves meet, the total displacement at a point is equal to the sum of the individual displacements at that point

reinforcement

cancellation

Superposition

Law of Superposition (interference)

Whenever two waves of the same type meet at the same point, the total amplitude (displacement) at that point equals the sum of the amplitudes (displacements) of the individual waves.

(You tube link1 and link2)

For constructive interference at any point, wavefronts must be ‘in phase’ and their path difference must be a whole number of wavelengths:

path difference = nλFor destructive interference at any point, wavefronts are ‘π out of phase’ and their path difference is given by:

path difference = (n + ½) λ

Superposition of Sound Waves

Constructive interference i.e. Loud or bright. Waves are in phase

Destructive interference i.e. dark or quiet. Waves are π rads out of phase.

Path Difference

S1

S2

O

Q

P

Q’

P’

At O : Zero path difference

At P and P’ path difference = 1λ

At Q and Q’ Path difference = 2λ

2nd subsidiary maximum

1st subsidiary maximum

Central maximum

1st subsidiary maximum

2nd subsidiary maximum

So in the above example…

S2Q – S1Q = 2λ

For constructive interference at any point, wavefronts must be ‘in phase’ and their path difference must be a whole number of wavelengths:

path difference = nλ

For destructive interference at any point, wavefronts are ‘π out of phase’ and their path difference is given by:

path difference = (n + ½) λ

Task: On your interference diagram…

i. Draw in lines of constructive and destructive interference

ii. Indicate the lines that join points…

a. in phase

b. 2π out of phase (path difference = λ)

c. 4π out of phase (path difference = 2λ)

d. 3π out of phase (path difference = 1.5λ)

Coherent waves

A stable pattern of interference is only obtained if the two wave sources are coherent.Two coherent wave sources…

i. have a constant phase difference,ii. thus produce waves with equal frequency.

InterferenceInterference occurs when two waves of the same type (e.g. both water, sound, light, microwaves etc.) occupy the same space.Wave superposition results in the formation of an interference pattern made up of regions of reinforcement and cancellation.

Double slit interference with light

This was first demonstrated by Thomas Young in 1801. The fact that light showed interference effects supported the theory that light was a wave-like radiation.

Thomas Young 1773 - 1829

Experimental detailsLight source: This needs to be monochromatic (one colour or frequency). This can be achieved by using a colour filter with a white light. Alternatives include using monochromatic light sources such as a sodium lamp or a laser.

Single slit:Used to obtain a coherent light source. This is not needed if a laser is used.

Double slits:Typical width 0.1mm; typical separation 0.5mm.

Double slit to fringe distance:With a screen typically 1.0m.The distance can be shorter if a microscope is used to observe the fringes.

Interference fringesInterference fringes are formed where the two diffracted light beams from the double slit overlap.

A bright fringe is formed where the light from one slit reinforces the light from the other slit. At a bright fringe the light from both slits will be in phase. They will have path differences equal to a whole number of wavelengths: 0, 1λ, 2λ, 3λ etc…

A dark fringe is formed due to cancelation where the light from the slits is 180° out of phase.They will have path differences of: 1/2λ, 3/2 λ , 5/2 λ etc..

Young’s slits equation

fringe spacing, w = λ D / s

where:s is the slit separationD is the distance from the slits to the

screenλ is the wavelength of the light

w

Question 1Calculate the fringe spacing obtained from a double slit experiment if the double slits are separated by 0.50mm and the distance from the slits to a screen is 1.5m with (a) red light (wavelength 650nm and (b) blue light (wavelength 450nm).

fringe spacing w = λ D / s

Question 2Calculate the wavelength of the green light that produces 10 fringes over a distance of 1.0cm if the double slits are separated by 0.40mm and the distance from the slits to the screen is 80cm

1.0 cm

Demonstrating interference with a laser

0.5m to 2m

A laser (Light Amplification by Stimulated Emission of Radiation) is a source of coherent monochromatic light.

Oscillations and Waves

Wave Properties

Wave diagrams1) Reflection

4) Diffraction3) Refraction

2) Refraction

Refraction

• Refraction is when waves bend as they travel from one medium to another

• When a wave travels into a different medium: – the wave speed changes– the wavelength changes– the frequency stays the same– If the wave hits the new medium at an angle, the

wave direction will change

RefractionRefraction occurs when a wave passes across a boundary at which the wave speed changes.

The change of speed usually, but not always, results in the direction of travel of the wave changing.

A wave slowing down on crossing a media boundary

Refraction of light(a) Less to more optical dense transition (e.g. air to glass)

angle of incidence

normal

angle of refraction

AIR GLASS

Light bends TOWARDS the normal.

The angle of refraction is LESS than the angle of incidence

(b) More to less optical dense transition (e.g. water to air)

angle of incidence

normal

angle of refraction

WATER AIR

Light bends AWAY FROM the normal.

The angle of refraction is GREATER than the angle of incidence

Refractive index (n)This is equal to the ratio of the wave speeds.

refractive index, ns = c / cs

ns = refractive index of the second medium relative to the firstc = speed in the first region of mediumcs = speed in the second region of medium

Question 1When light passes from air to glass its speed falls from 3.0 x 108 ms-1 to 2.0 x 108 ms-1.Calculate the refractive index of glass.

ns = c / cs

= 3.0 x 108 ms-1 / 2.0 x 108 ms-1

refractive index of glass = 1.5

Question 2

The refractive index of water is 1.33. Calculate the speed of light in water.

ns = c / cs → cs = c / ns

= 3.0 x 108 ms-1 / 1.33

speed of light in water = 2.25 x 108 ms-1

Examples of refractive indexExamples of ns for light (measured with respect to a vacuum as the first medium)

vacuum = 1.0 (by definition)air = 1.000293 (air is usually taken to be = 1.0)ice = 1.31water = 1.33alcohol = 1.36glass = 1.5 (varies for different types of glass)diamond = 2.4

The law of refractionWhen a light ray passes from a medium of refractive index n1 to another of refractive index n2 then:

n1 sin θ1 = n2 sin θ2

where:θ1 is the angle of incidence in the first mediumθ2 is the angle of refraction in the second medium

n2

Medium of refractive index, n1

θ2

θ1

In the data booklet the angles of incidence and refraction are called θ1 and θ2.

It can further be shown that…

Note that this is written in the data booklet as…

1n2 = sinθ1 = v1 = n2 sinθ2 v2 n1

sinθ2 = v2 = n1 sinθ1 v1 n2

E.g. 1

A wave travelling at 12cms-1 is incident upon a surface at an angle of 55° from the normal.

a. If the angle of refraction is 40°, determine the speed of the wave in the second medium.

b. If the initial wavelength is 6cm determine the frequency of the wave in the second medium.

sinθ2 = v2

sinθ1 v1

sin 40 = v2

sin 55 12v2 = 9.4 cms-1

In first medium: v = fλ f = v/λ = 0.12 / 0.06 = 2.0 Hz

Frequency does not change during refraction f = 2.0 Hz

E.g. 2

For light travelling from water into glass, r=20°.

If nw = 1.33 and ng = 1.50, determine i (θ1).

sinθ2 = n1 sinθ1 n2

sin20 = 1.33 sinθ1 1.50

sinθ1 = 0.34 / 0.89

= 0.38

θ1 = sin-1 0.38

= 22.5°

QuestionCalculate the angle of refraction when light passes from air to glass if the angle of incidence is 30°.n1 sin θ1 = n2 sin θ2 → 1.0 x sin 30° = 1.5 x sin θ2 1.0 x 0.5 = 1.5 x sin θ2

→ sin θ2 = 0.5 / 1.5 = 0.333

→ angle of refraction, θ2 = 19.5°

Complete:medium

1n1 θ1 medium

2n2 θ2

air 1.00 50o water 1.33 35.2o

glass 1.50 30o air 1.00 48.6o

water 1.33 59.8o glass 1.50 50o

air 1.00 50o diamond 2.4 18.6o

air 1.00 50o unknown 1.53 30o

Answers

Total internal reflectionTotal internal reflection (TIR) occurs when light is incident on a boundary where the refractive index DECREASES.And the angle of incidence is greater than the critical angle, c for the interface.

n2 ( < n1 )

n1

θ1θ1 > c

Critical angle (c)This is the angle of incidence, θ1 that will result in an angle of refraction, θ2 of 90 degrees.

n1 sin θ1 = n2 sin θ2 becomes in this case:n1 sin c = n2 sin 90° n1 sin c = n2 (sin 90° = 1)

Therefore: sin c = n2 / n1

n2 (<n1)

n1

θ1θ1 = c

θ 2 = 90o

Finding the Critical Angle…1) Ray gets refracted

4) Ray gets internally reflected3) Ray still gets refracted (just!)

2) Ray still gets refracted

THE CRITICAL ANGLE

Question 1Calculate the critical angle of glass to air. (nglass = 1.5; nair =1)

sin c = n2 / n1

→ sin c = 1.0 / 1.5= 0.667

→ critical angle, c = 41.8°

Question 2Calculate the maximum refractive index of a medium if light is to escape from it into water (nwater = 1.33) at all angles below 30°.

sin c = n2 / n1

→ sin 30° = 1.33 / n1

→ 0.5 = 1.33 / n1

→ n1 = 1.33 / 0.5

→ refractive index, n1 = 2.66

Optical fibresOptical fibres are an application of total internal reflection.

Step-index optical fibre consists of two concentric layers of transparent material, core and cladding.

The core has a higher refractive index than the surrounding cladding layer.

core cladding

Total internal reflection takes place at the core - cladding boundary.

The cladding layer is used to prevent light crossing from one part of the fibre to another in situations where two fibres come into contact.

Such crossover would mean that signals would not be secure, as they would reach the wrong destination.

QuestionA step-index fibre consists of a core of refractive index 1.55 surrounded by cladding of index 1.40. Calculate the critical angle for light in the core.

sin c = n2 / n1 → sin c = 1.40 / 1.55= 0.9032

→ critical angle, c = 64.6°

Optical fibres in communicationA communication optical fibre allows pulses of light to enter at one end, from a transmitter, to reach a receiver at the other end. The fastest broadband systems use optical fibre links.

The core must be very narrow to prevent multipath dispersion. This occurs in a wide core because light travelling along the axis of the core travels a shorter distance per metre of fibre than light that repeatedly undergoes total internal reflection. Such dispersion would cause an initial short pulse to lengthen as it travelled along the fibre.

input pulse

output pulse

Multipath dispersion causing pulse broadening

The EndoscopeThe medical endoscope contains two bundles of fibres. One set of fibres transmits light into a body cavity and the other is used to return an image for observation.

Optical fibres

Uses of Total Internal Reflection

Optical fibres:

An optical fibre is a long, thin, _______ rod made of glass or plastic. Light is _______ reflected from one end to the other, making it possible to send ____ chunks of information

Optical fibres can be used for _________ by sending electrical signals through the cable. The main advantage of this is a reduced ______ loss.

Words – communications, internally, large, transparent, signal

DiffractionDiffraction occurs when waves spread out after passing through a gap or round an obstacle.

Sea wave diffraction

Diffraction becomes more significant when the size of the gap or obstacle is reduced compared with the wavelength of the wave.

i) Diffraction by a "large" object

  ii) Diffraction at a "large" aperture

                              

                         

                                                         

iii) Diffraction by a "small" object

  iv) Diffraction by a "narrow" aperture

                                                            

            

 

                                                   

InterferenceInterference occurs when two waves of the same type (e.g. both water, sound, light, microwaves etc.) occupy the same space.Wave superposition results in the formation of an interference pattern made up of regions of reinforcement and cancellation.

CoherenceFor an interference pattern to be observable the two overlapping waves must be coherent.

This means they will have:1. the same frequency2. a constant phase difference

If the two waves are incoherent the pattern will continually change usually too quickly for observations to be made.

Two coherent waves can be produced from a single

wave by the use of a double slit.

Path differencePath difference is the difference in distance travelled by two waves.

Path difference is often measured in ‘wavelengths’ rather than metres.

Example: Two waves travel from A to B along different routes. If they both have a wavelength of 2m and the two routes differ in length by 8m then their path difference can be stated as ‘4 wavelengths’ or ‘4 λ’

Double slit interference with light

This was first demonstrated by Thomas Young in 1801. The fact that light showed interference effects supported the theory that light was a wave-like radiation.

Thomas Young 1773 - 1829

Experimental detailsLight source: This needs to be monochromatic (one colour or frequency). This can be achieved by using a colour filter with a white light. Alternatives include using monochromatic light sources such as a sodium lamp or a laser.

Single slit:Used to obtain a coherent light source. This is not needed if a laser is used.

Double slits:Typical width 0.1mm; typical separation 0.5mm.

Double slit to fringe distance:With a screen typically 1.0m.The distance can be shorter if a microscope is used to observe the fringes.

Interference fringesInterference fringes are formed where the two diffracted light beams from the double slit overlap.

A bright fringe is formed where the light from one slit reinforces the light from the other slit. At a bright fringe the light from both slits will be in phase. They will have path differences equal to a whole number of wavelengths: 0, 1λ, 2λ, 3λ etc…

A dark fringe is formed due to cancelation where the light from the slits is 180° out of phase.They will have path differences of: 1/2λ, 3/2 λ , 5/2 λ etc..

Fundamentals of Physics 99jw

Standing Waves & ResonanceA standing wave is created from two traveling waves, having the same frequency and the same amplitude and traveling in opposite directions in the same medium.

Using superposition, the net displacement of the medium is the sum of the two waves.

When 180° out-of-phase with each other, they cancel (destructive interference).When in-phase with each other, they add together (constructive interference).