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Further Maths

  • A-Level Physics

Waves

Updated: Aug 22

There are two forms of waves - progressive and stationary. Progressive waves transfer energy and can either be transverse, where the vibrations are perpendicular to the direction of travel such as EM waves, or longitudinal, where the vibrations are parallel to the direction of travel such as sound waves.

Wave Motion

  • Displacement, x, is how far the wave has moved from its undisturbed position

  • Amplitude, A, is the maximum magnitude of displacement

  • Wavelength, λ, is the length of one whole wave cycle, e.g. from peak to peak

  • Wave speed, v, is the speed at which the wave moves – the distance it travels per second

  • Period, T, is the time taken for the whole cycle to complete

  • Frequency, f, is the number of cycles passing a given point per second

  • Phase is a measurement of the position of a certain point along the wave

  • Phase difference is the amount one wave lags behind another. It is measured in degrees or radians, where one wavelength represents 360° or 2π


Determining Frequency

Frequency can be determined using a cathode ray oscilloscope to measure voltage. It displays waves from a signal generator as a function of voltage (y-axis) and time (x-axis), from which the time period can be calculated, and this used to work out frequency.


The units of frequency are Hertz, Hz, or /s


The wave equation:

λ = 1/T Frequency = 1 / Time Period

If you know the frequency and the wavelength, you can use the wave equation to work out speed:

v = fλ wave speed = frequency × wavelength

Intensity is the measure of how much energy a wave is carrying. For example, the ‘brightness’ or ‘loudness’ of light and sound waves are just their intensity.

I = P/A Intesnisty = Power / Area

It is proportional to amplitude squared:

Intensity ∝ Amplitude Squared


Reflection, Refraction, Diffraction & Polarisation

Reflection

Waves are reflected when they change direction at a boundary between two different media but stay in the same medium. This means the wavelength and frequency stay the same, so the angle of incidence equals the angle of reflection.


Refraction

Refraction occurs when a wave moves from one medium into another, and it is the way in which the wave changes direction when this happens. This change in direction is a result of the wave speeding up when entering a less dense medium and bending away from the normal, or slowing down when entering a more dense medium and bending towards the normal. This happens because the frequency remains constant but the wavelength changes.


Diffraction

Diffraction occurs when waves pass through narrow openings – the effect is most noticeable when the gap is a wavelength or less across. This can be demonstrated by a ripple tank or shining monochromatic light through a slit, onto a screen.


Polarisation

Polarisation leaves waves vibrating in only one direction, e.g. 2D not 3D. Only transverse waves can be polarised, and two perpendicular polarising filters allow no waves through. This can be demonstrated using polarising filters for visible light, or microwave transmitters and receivers along with a metal grille for microwaves. Microwave transmitters transmit vertically polarised waves, so you only need one grille.



Electromagnetic Waves

Electromagnetic waves are a unique subset of transverse waves, because they can travel and transfer energy in no medium - they can travel through a vacuum. This is known as self-propagating, meaning they can sustain themselves due to their electric and magnetic fields, acting at right angles to one another.


All electromagnetic waves travel at the same speed in a vacuum, but at different frequencies and wavelengths.

The speed of light in a vacuum is 3.0E8 m/s

E.M. waves, like all transverse waves, can be reflected, refracted, diffracted and polarised.


The Electromagnetic Spectrum

Refraction of Light

Different materials slow down light by different amounts. We express this property of the material as the refractive index:

n = c/v refractive index = speed of light in a vacuum / speed of light in the material

Similarly, Snell’s law works this out using angles:

Total Internal Reflection

As well as a refractive index, materials also have a critical angle.

sin C = 1/n sin of the critical angle = 1 / refractive index

θ < C

When θ is less than the critical angle, some of the light is internally reflected, but most emerges and speeds up, bending away from the normal.


θ = C

When θ is the same as the critical angle, the light nor emerges nor reflects - it refracts along the edge between the two media.


θ > C

When θ is greater than the critical angle, all the light is internally reflected at the same angle, θ, from the normal. This is

called total internal reflection (TIR)



Superposition

Superposition occurs when two or more waves meet and pass through each other. When the meeting happens, the two waves instantaneously combine to form a resultant wave of the sum of the individual amplitudes. Then, the two waves separate and continue unchanged.


This moment of interference can be either constructive or destructive:

  • constructive interference is when a peak and a peak give a bigger peak

  • destructive interference is when a peak and a trough cancel out to give a smaller peak or no peak at all (if the two waves have the same amplitude)

Waves must be coherent to interfere – have the same wavelength and frequency, and be a fixed phase difference apart.


Destructive interference is use in noise-cancelling headphones. A microphone detects the sound waves from the surroundings, and the speaker generates the exact same waves but in antiphase.


Path & Phase Difference

Path difference is how much further one wave has travelled than the other to get to the point of interference.

  • If the path difference is a whole number of wavelengths, constructive interference occurs. This is known as being in phase and is commonly described as a multiple of 360˚or 2π radians.

  • If the path difference is a half number of wavelengths, destructive interference happens. This is known as being in antiphase, when the waves are 180˚or π radians apart.


Investigating Interference

Interference can be demonstrated with two parallel speakers connected to the same signal generator. When passing some distance in front of and parallel to them, the volumes will vary from loud to quiet depending on whether the path difference is a whole or half number of wavelengths.


Exactly the same experiment can be carried out with a microwave generator, a double slit, and a microwave receiver.



Young's Double Slit

A good way of demonstrating interference using visible light is with laser and a double slit. Laser light is, by definition, coherent, and if the slits are about the same length as a wavelength, the light is diffracted.


The diffraction patterns from each slit cross over one another, interfering when the waves meet. This gives a pattern like this:

Young’s experiment was the first evidence for the wave nature of light. Newton believed light to be a series of particles, like air, but Huygens believed it to be a wave.


The experiment can be used to calculate the wavelength of the light source by measuring the slit separation, a, the fringe spacing, x, and the distance between the slit and the screen, D.

λ = ax / D wavelength = slit seperation × fringe spacing / slit to screen separation

Slit separation is sometimes labelled a, sometimes d

A diffraction grating can be used to give a sharper image on the screen than a double slit, as when there are hundreds of slits per mm more beams reinforce the pattern. This means the bright spots are brighter and narrower and the dark spots are darker and wider. Measuring the fringe width from the zero order to the nth order allows you to calculate the angle of diffraction. This can be used to work out the wavelength more accurately:

d sinθ = nλ slit separation × sin(angle of diffraction) = order × wavelength


Stationary Waves

Stationary Waves occur when two identical waves meet from opposite directions (identical in speed, frequency and length). The two waves superpose to form a series of nodes (no amplitude) and antinodes (maximum amplitude). This happens all the time when waves reflect on themselves.

Phase difference on stationary waves is different:

  • in between adjacent nodes, particles in a stationary wave are in phase, yet their amplitudes are different

  • on different sides of a node, particles are in antiphase, as they reach the negative of the amplitude at the same time.

Unlike a progressive wave, there is no net energy transfer by a stationary wave as the two waves go in opposite directions.


The wavelength of a progressive wave is the distance between two adjacent nodes, whereas on a stationary wave the wavelength is the distance between two nodes/antinodes.



Forming Stationary Waves

Stationary waves can be formed with a microwave generator. When the waves are reflected off a metal sheet a stationary wave forms (if the distance is correct). A receiver will read a series of minimum intensities at the nodes, and maximum intensities at the antinodes. 



Harmonics

When a string is taught, there is a node at each end. If this is plucked, or caused to vibrate, it will form a note – the first harmonic – at its fundamental frequency. However, at the same time stationary waves of half the wavelength and double the frequency will form, creating the second, third, fourth harmonic etc.

The same occurs inside a hollow tube. If the tube is open at only one end, a node forms on the closed end and an antinode at the opening. This means only odd multiples of the fundamental frequency are possible, as each harmonic is ¼ of a wavelength, rather than a half.


A tube that is open at both ends, however, can form any integer multiple of the fundamental frequency, as the first harmonic is also half a wavelength. 

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