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Notes by Category University Engineering

Mechanics & Stress Analysis*
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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.

Transverse and longitudinal waves diagram. EngineeringNotes

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 diagram, Refraction diagram, Diffraction diagram, Polarisation diagram. Wave properties diagram. EngineeringNotes


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 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 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 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

The Electromagnetic Spectrum, EM Spectrum, Electromagnetic Radiation, Visible Light spectrum. EngineeringNotes

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
Total Internal Reflection Diagram, critical angle diagram. EngineeringNotes

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