# Section 3 - Waves - complete

- Created by: scarlett
- Created on: 01-09-20 13:18

## Waves

- a progressive (moving) wave carries energy from one place to another without transferring any material

- a wave is caused by something making particles or fields oscillate (vibrate) at a source

- these oscillations pass through the medium (field) as the wave travels, carrying energy with it

- there are ways to tell that waves carry energy:

1 - electromagnetic waves cause things to heat up

2 - x-rays and gamma rays knocnk electrons out of their orbits, causing ionisation

3 - loud sounds cause large oscillations of air particles which can make things vibrate

4 - wave power can be used to generate electricity

5 - since waves carry energy away, the source of the wave loses energy

## Components of Waves

- cycle - one complete vibration of the wave

- displacement, x, metres - how far a point on the wave has moved from its undisturbed position

- amplitude, A, metres - maximum magnitude of displacement

- wavelength, lamda, metres - the length of one whole wave cycle, from crest to crest or trough to trough

- period, T, seconds - the time taken for a whole cycle (vibration) to complete, or to pass a given point

- frequency, f, hertz - the number of cycles per second passing a given point

- phase - a measurement of the position of a certain point along the wave cycle

- phase difference - the amount one wave lags behind another

- phase and phase difference are measured in angles (in degrees or radians) or as fractions of a cycle

## Reflection and Refraction of Waves

Reflection

- the wave is bounced back when it hits a boundary

- e.g. you can see the reflection of light in mirrors

- the reflection of water waves can be demonstrated in a ripple tank

Refraction

- the wave changes direction as it enters a different medium

- the change in direction is a result of the wave slowing down or speeding up

## Frequency Equation of Waves

- the frequency is the inverse of the period

- frequency = 1/period

- f = 1/T

- 1 Hz = 1s^{-1}

## Wave Equation

- wave speed (c) = distance travelled (d) / time taken (t)

- from this you can get to the wave equation:

- speed of wave (c) = wavelength x frequency

Example

A wave has a wavelength of 420m and travels at a speed of 125ms^{-1}

Find the frequency of this wave.

speed of wave = wavelength x frequency

frequency = speed of wave/wavelength

frequency = 125/420 = 0.30 Hz

- c is also the speed of light in a vacuum

- light is a type of electromagnetic wave

- all EM waves travel with a constant spped in a vacuum (3.00 x 10^{8} ms^{-1})

- the c used in the wave equation is the speed of the wave in question & it can take any value depending on the wave

## Transverse Waves

- all electromagnetic waves are transverse

- they travel as vibrating magnetic and electric fields - with vibrations perpendicular to the direction of energy transfer

- other examples of transverse waves are ripples on water or waves on strings

- there are two main ways of drawing transverse waves:

1 - they can be shown as graphs of displacement against distance along the path of the wave

2 - or they can be shown as graphs of displacement agains time for a point as the wave passes

- both sorts of graph often give the same shape so you have to check the label on the x-axis

- displacements upwards from the centre line are given a + sign and displacements downwards are given a - sign

## Longitudinal Waves

- the most common example of a longitudinal wave is sound

- a sound wave consists of alternate compressions and rarefactions of the medium its travelling through

- some times of earthquake shock waves are also longitudinal

- its hard to represent longitudinal waves graphically

- they are usually plotted as displacement against time

- however they can be confusing as they look like a transverse wave

## Polarised Waves

- if you shake a rpe the make a wave you can move your hand up and down or side to side or in a mixture of directions but it will still make a transverse wave

- if you try to pass waves in a rope through a vertical fence, the wave will only get through if the vibrations are vertical

- the fence filters out vibration in other directions

- this is called polarising the wave

- polarisation can only happen for transverse waves

- in 1808, Etienne-Louis Malus discovered that light was polarised by reflection

- physicists at the time thought that light spread like sound, as a longitudinal wave, so they struggled to explain polarisation

- in 1817, Young suggested that light was a transverse wave consisting of vibrating electric and magnetic fields at right angles to the transfer of energy

- this explained why light could be polarised

## Polarising Filters

- ordinary light waves are a mixture of different directions of vibration

- they can be polarised using a polarising filter

- if you have two polarising filters at right angles to eachother, then no light will get through

- light becomes partially polarised when reflected from some surfaces - some of it vibrates in the same direction

- if you view reflected partially polarised light through a polarising filter at the correct angle you can block out unwanted glare

- polarioid sunglasses make use of this affect

## TV and Radio Signals

- if you walk down the street and look up at TV aerials on houses, you'll see that the rods on them are all horizontal

- the reason for this is that TV signals are polarised by the orientation of the rods on the broadcasting aerial

- to receive a strong signal, you have to line up the rods on the receiving aerial with the rods of the transmitting aerial (if they aren't aligned, the signal strength will be lower)

- its the same with radio, if you try tuning a radio and then moving the aerial around, your signal will come and go as the transmitting and receiving aerials go in and out of alignment

## Superposition

- at the instant the waves cross, the displacements due to each wave combine

- then each wave goes on as usual

- you can see this if two pulses are sent simultaneously from each end of a rope

- the principle of superposition says that when two or more waves cross, the resultant displacement equals the vector sum of the individual displacements

- 'superposition' means 'one thing on top of another thing'

- you can use the same idea in reverse - a complex wave can be separated out mathematically into several simple sine waves of various sizes

## Interference

- a crest plus a crest gives a bigger crest

- a trough plus a trough gives a bigger trough

- these are both examples of constructive interference

- a crest plus a trough of equal size gives nothing

- the two displacements cancel each other out completely

- this is called destructive interference

- if the crest and the trough aren't the same size, then the destructive interference isn't total

- for the interference to be noticeable, the two amplitudes should be nearly equal

- graphically, you can superimpose waves by adding the individual displacements at each point along the x-axis, and then plotting them

## In Phase Waves

- two points on a wave are in phase if they are both at the same point in the wave cycle

- points in phase have the same displacement and velocity

- it's mathematically handy to show one complete cycle of a wave as an angle of 360^{o} (2 pi radians)

- two points with a phase difference of zero or a multiple of 360^{o} are in phase

- points with a phase difference of odd-number multiples f 180^{o} (pi radians, or a half cycle) are exactly out of phase

- you can also talk about two different waves being in phase

- in practive this happens because both waves come from the same oscillator

- in other situations there will nearly always be a phase difference between two waves

## Interference Patterns

- interference still happens when you're observing waves of different wavelength and frequency but it happens in a jumble

- in order to get clear interference patterns, the two or more sources must be coherent (and be in phase)

- two sources are coherent if they have the same wavelength and frequency and a fixed phase difference between them

## Path Difference

- whether you get constructive or destructive interference at a point depends on how much further one wave has travelled than the other wave to get to that point

- the amount by which the path travelled by one wave is longer than the path travelled by the other wave is called the path difference

- at any point an equal distance from two sources that are coherent and in phase, you will get constructive interference

- you also get constructive interference at any point where the path difference is a whole number of wavelengths

- at these points the two waves are in phase and reinforce each other

- but at points where the path difference is half a wavelength, one and a half wavelengths , two and a half wavelengths etc., the waves arrive out of phase and you get destructive interference

- constructive interference occurs when path difference = n(wavelengths) where (where n is an integer)

- destructive interference occurs when path difference = (2n +1)lamda/2 = (n + 1/2) lamda

## Reflected Progressive Waves

- a stationary (standing) wave is the superposition of two progressive waves with the same frequency (wavelength), moving in opposite directions

- unlike progressive waves, no energy is transmitted by a stationary wave

- you can demonstrate stationary waves by setting up a driving oscillator at one end of a stretches string with the other end fixed

- the wave generated by the oscillator is reflected back and forth

- for most frequencies the resultant pattern is a jumble

- however, if the oscillator happens to produce an exact number of waves in the time it takes for a wave to get to the end and back again, then the original and reflected waves reinforce each other

- at these "resonant frequencies" you get a stationary wave where the pattern doesn't move - it just sits there, bobbing up and down

## Stationary Waves

- each particle vibrates at right angles to the string

- nodes are where the amplitude of the vibration is zero

- antinodes are points of maximum amplitude

- at resonant frequencies, an exact number of half wavelengths fits onto the string

First Harmonic

- this stationary wave is vibrating at the lowest possible resonant frequency

- it has one "loop" with a node at each end

Second Harmonic

- it is twice the frequency of the first harmonic

- there are two "loops" with a node in the middle and one at each end

Third Harmonic

- the third harmonic is three times the frequency of the first harmonic

- 1 1/2 wavelengths fit on the string

## Stationary Microwave & Sound Waves

- microwave stationary wave apparatus:

- a probe connected to a meter or loudspeaker is placed between a metal plate and microwave transmitter

- you can find the nodes and antinodes by moving the probe between the transmitter and reflecting plate

- stationary sound waves are produced in the glass tube

- the lycopodium powder laid along the bottom of the tube is shaken away from the antinodes but left undistrubed at the nodes

## Investigating Factors Affecting the Resonant Frequ

1 - start by measuring the mass and length of strings of different types using a mass balance and a ruler

- then find the mass per unit length (µ) of each string using: µ = M/L

2- set up the apparatus (signal generator connected to a vibration transducer. string fixed to the transduced and then masses connected to the other end over a pulley)

- record µ, measure and record the length and work out the tension (T) using:

T = mg (where m is the total mass of the masses in kg)

3 - turn on the signal generator and vary the frequency until you find the first harmonic - i.e. a stationary wave that has a node at each end and a single antinode

- this is the frequency of the first harmonic, f

Then investigate how the length, tension or mass per unit length of the string affects the resonant frequency by:

- keeping the string type and the tension in it the same and altering the length

- do this by moving the vibration transducer towards or away from the pulley

- find the first harmonic again, and record f against l

- keeping the string type and lenght the same and adding or removing masses to change the tension. find the first harmonic again and record f against t

- keeping the length and tension the same, but using a different sting to vary µ, find the first harmonic again and record f against µ

## Investigation Conclusions

- the longer the string, the lower the resonant frequency because the half wavelength at the resonant frequency is longer

- the heavier (i.e. the more mass per unit length) the string, the lower the resonant frequency - because waves travel more slowly down the string

- for a given length a lower wave speed, c, makes a lower frequency, f

- the looser the string the lower the resonant frequency because waves travel more slowly down a loose string

- the frequency of the first harmonic, f, is: f = 1/2l x square root of T/µ

- where l is the string length in m , T is the tension in the string and µ is the mass per unit length of the string

## Corners and Gaps

- the way that waves spread out as they come through a narrow gap or go round obstacles is called diffraction

- all waves diffract, but it's not always easy to observe

- the amount of diffraction depends on the wavelength of the wave compared with the size of the gap

1) when the gap is a lot bigger than the wavelength, diffraction is unnoticeable

2) you get noticeable diffraction through a gap several wavelengths wide

3) you get the most diffraction when the gap is the same size as the wavelength

4) if the gap is smaller than the wavelength, the waves are mostly just reflected back

- when sound passes through a doorway, the size of gap and the wavelength are usually roughly equal, so a lot of diffraction occurs

- thats why you have no trouble hearing someone through an open door to the next room, even if the other person is out of your line of sight

- the reason that you cant see themis that when light passes through the doorway, it is passing though a gap around a hundred million times bigger than its wavelength so the amount of diffraction is tiny

- so to get a noticeable diffraction with light, you must shine it through a very narrow slit

## Light Diffraction Pattern

- to observe a clear diffraction pattern for light you need to use a monochromatic, coherent light source

- monochromatic just means all the light has the same wavelength (and frequency) and so is the same colour

- lasers are a monochromatic and coherent light source

Demonstrating Light Diffraction Patterns with a Laser

- if the wavelength of light is about the same size as the aperture, you get a diffraction pattern

- you'll see a central bright fringe (central maximum), with dark and bright fringes alternating on either side

- the dark and bright fringes are caused by destructive and constructive interference of light waves

## Diffracted White Light

- white light is actually a mixture of different colours, each with different wavelengths

- when white light is shone through a single narrow slit, all of the different wavelengths are diffracted by different amounts

- this means that instead of getting clear fringes (as you would with a monochromatic light source) you get a spectra of colours

## Intensity of Light

- the central maximum in a single slit light diffraction pattern is the brighest part of the pattern

- this is because the intensity of light is highest in the centre

- intensity is the power per unit area

- for monochromatic light, all photons have the same energy, so an increase in the intensity means an increase in the number of photons per second

- so there are more photons per unit area hitting the central maximum per second than the other bright fringes

## Width of the Central Maximum

- when light is shone through a single slit, there are two things which affect the width of the central maximum

1) increasing the slit width decreases the amount of diffraction

- this means the central maximum is narrower, and the intensity of the central maximum is higher

2) increasing the wavelength increases the amount of diffraction

- this means the central maximum is wider, and the intensity of the central maximum is lower

## Two-Source Interference in Water & Sound

- its easy to demonstrate two-sound interference for either sound or water because tbey've got wavelengths of a handy size that you can measure

- you need coherent sources, which means the wavelength and frequency have to be the same

- the trick is to use the same oscillator to drive both sources

- for warwe, one ******** drives two dippers

- for sound, one oscillator is connected to two loudspeakers

## Young's Double-Slit Experiment

1) to see two-source interference with light, you can either use two separate, coherent light sources or you can shine a laser through two slits

- laser light is coherent and monochromatic

2) Young's double-slit experiment shines a laser through two slits onto a screen

3) the slits have to be about the same size as the wavelength of the laser light so that it is diffracted, then the light from the slits acts like two coherent point sources

4) you get a pattern of light and dark fringes, depending on whether constructive or destructive interference is taking place

- thomas young (the first person to do this is experiment with a lamp) came up with an equation to work out the wavelength of the light from this experiment

- working with lasers can be dangerous because laser light is focused into a very direct, powerful beam of monochromatic light

- if you looked at a laser beam directly, you eye's lens would focus it onto your retina, which would be permanently damaged

- you should: never shine the laser towards a person, wear laser safety goggles, avoid shining the laser beam at a reflective surface

- have a warning sign on display

- turn the laser off when its not needed

## Similar Experiment with Microwaves

- to see interference patters with microwaves, you can replace the laser and slits with two microwave transmitter cones attached to the same signal generator

- you also need to replace the screen with a microwave receiver probe

- if you move the probe along the path of the waves, you'll get an alternating pattern of strong and weak signals - just like the light and dark fringes on the screen

## Young's Double-Slit Formula

- the fringe spacing (w), wavelength (lamda), spacing between slits (s) and the distance from slits to screen (D) are all related by Young's Double-Slit formula

fringe spacing = (wavelength x distance) / spacing

- since the wavelength of light is so small you can see from the formula that a high ratio of D/s is needed to make the fringe spacing big enough to see

- rearranging, you can use wavelength = (fringe spacing x spacing) / distance to calculate the wavelength of light

- the fringes are so tiny that it's very hard to get an accurate value of w

- its easier to measure across several fringes then divide by the number of fringe widths between them

## Evidence for the Wave Nature of EM Radiation

- towards the end of the 17th century, two important theories of light were published - one by Isaac Newton and the other by a man called Huygens

- Newtons theory suggested that light was made up of tiny particles, which he called "corpuscles"

- Huygens theory suggested they were made up of waves

- the corpuscular theory could explain reflection and refraction, but diffraction and interference are both uniquely wave properties

- Young's double-slit experiment provided the necessary evidence that wave could also behave like a wave

- it showed that light could both diffract and interfere

## Sharp Interference Patterns

- you can repeat Young's double-slit experiment with more than two equally spaced slits

- you get the same shaped pattern as for two slits but the bright bands are brighter and narrower and the dark areas between are darker

- when monochromatic light is passed through a grating with hundreds of slits per mm, the interference pattern is really sharp because there are so many beams reinforcing the pattern

- sharper fringes make for more accurate measurements

## Monochromatic Light on a Diffraction Grating

- for monochromatic light, all the maxima are sharp lines

- there's a line of maximum brightness at the centre called the 0 order line

- the lines just either side of the central one are called first order lines

- the next pair out are called second order lines and so on

- for a grating with slits a distance (d) apart, the angle between the incident beam and the nth order maximum is given by:

d sin θ = nλ

- so by observing d, θ and n you can calculate the wavelength of the light

## Deriving the Monochromatic Light Equation

1) at each slit, the incoming waves are diffracted

- these diffracted waves then interfere with each other to produce an interference pattern

2) consider the first order maximum

- this happens at the angle when the waves from one slit lline up with the waves from the next slit that are exactly one wavelength behind

3) call the angle between the first order maximum and the incoming light θ

4) the angle is θ, d is the slit spacing and the path difference is λ

5) using trig, d sinθ = λ

6) the other maxima occur when the path difference is 2λ, 3λ, 4λ etc

- so to make the equation general, just replace λ with nλ, where n is an integer (the order of the maximum)

## Conclusions from dsinθ = λ

- if λ is bigger, sinθ is bigger, and so θ is bigger

- this means that the larger the wavelength, the more the pattern will spread out

- if d is bigger, sinθ is smaller

- this means that the coarser the grating, the less the pattern will spread out

- values of sinθ greater than 1 are impossibles

- so if for a certain n you get a result of more than 1 for sinθ you know that that order doesn't exist

## Diffraction Gratings and Identifying Elements

- white light is really a mixture of colours

- if you diffract white light through a grating then the patterns due to different wavelengths within the white light are spread out by different amounts

- each order in the pattern becomes a spectrum, with red on the outside and violet on the inside

- the zero order maximum stays white because all the wavelengths just pass straight through

- astronomers and chemists often need to study spectra to help identify elecments

- they use diffraction gratings rather than prisms because they're more accurate

- the wavelength of x-rays is of a similar scale to the spacing between atoms in crystalline solids

- this means that x-rays will form a diffraction pattern when directed at a thin crystal

- the crystal acts like a diffraction grating and the spacing between atoms (slit width) can be found from the diffraction patterm

- this is called x-ray crystallography & it was used to discover the structure of DNA

## Refractive Index

- light travels fast in a vacuum

- it slows down in other materials because it interacts with the particles in them

- the more optically dense a material is, the more light slows down when it enters it

- the absolute refractive index of a material, n, is a measure of optical density

- it is found from the ratio between the speed of light in a vacuum, c, and the speed of light in that material, c_{s}

- the relative refractive index between two materials, _{1}n_{2}, is the ratio of the speed of light in material 1 to the speed of light in material 2

- combining the equations gives you : _{1}n_{2 }= n_{2} / n_{1}

- the absolute refractive index of a material is a property of that material only

- a relative refractive index is a property of the interface between two materials & its different for every possible pair

- because you can assume n_{air} = 1, you can assume the refractive index for an air to glass boundary equals the absolute refractive index of the glass

## Snell's Law

- the angle the incoming light makes to the normal is called the angle of incidence

- the angle the refracted ray makes with the normal is the angle of refraction

- the light is crossing a boundary, going from a medium with refractive index n_{1} to a medium with a refactive index n_{2}

- when light enters an optically denser medium it is refracted towards the normal

- n, θ_{1 }and θ_{2} are related by Snell's Law: n_{1}sinθ_{1} = n_{2}sinθ_{2}

## Refraction Away from the Normal

- when light goes from an optically denser material into an optically less dense material (e.g. glass to air), interesting things can happen

- shine a ray of light at a boundary going from refractive index n_{1} to n_{2}, then gradually increase the angle of incidence

- the light is refracted away from the normal, so as you increase the angle of incidence, the angle of refraction gets closer and closer to 90^{o}

- eventually θ_{1} reaches a critical angle θ_{c} for which θ_{2} = 90^{o}

- the light is refracted along the boundary

- as sin 90^{o} = 1, Snell's law n_{1 }sinθ = n_{2}sinθ_{2} becomes n_{1}sinθ_{c} = n_{2} x 1 so: sinθ_{c} = n_{2}/n_{1} = _{1}n_{2}

- at θ_{1} greater than the critical angle, refraction is impossible

- all the light is reflected back into the material (total internal reflection)

## Optical Fibres

- an optical fibre is a very thin flexible tube of glass or plastic fibre that can carry light signals over long distances and round corners

1) step-index optical fibres themselves have a high refractive index but are surrounded by cladding with a lower refractive index to allow total internal reflection

- cladding also protects the fibre from scratches which could let light escape

2) light is shone in at one end of the fibre

- the fibre is so narrow that the light always hits the boundary between the fibre and cladding at an angle bigger than the critical angle

3) so all the light is totally internally reflected from boundary to boundary until it reaches the other end

## Absorption and Dispersion

Absorption

- as the signal travels, some of its energy is lost through absorption by the material the fibre is made from

- this energy loss results in the amplitude of the signal being reduced

Dispersion

Modal Dispersion - light rays enter the fibre at different angles, and so take different paths

- the rays which take a longer path take longer to reach the other end than those that travel down the middle of the fibre

- a single-mode fibre only lets light take one path, so it stops modal dispersion

Material Dispersion - light consists of different wavelengths that travel at different speeds in the fibre

- this causes some light wavelengths to reach the end of the fibre faster than others

- using monochromatic light can stop material dispersion

- both types of dispersion lead to pulse broadening

- the signal sent down the fibre is broader at the other end

- broadened pulses can overlap each other and confuse the signal

- an optical fibre repeater can be used to boost and regenerate the signal every so often, which can reduce signal degradation caused by both absorption and dispersion

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