PHYA2 - Waves

Waves the pass through a substance are vibrations

• Sound waves and seismic waves are examples of mechanical waves: vibrating particles cause nearby particles to vibrate and so on
• EM waves do not need a substance in order to vibrate

Londitudal Waves

• They vibrate parallel to the direction of propagation

Transverse Waves

• They vibrate perpendicular to the direction of propagation

Transverse waves are plane-polarised if the vibrations are only in one plane

• Longditudal waves cannot be polarised
• If unpolarised light passes through two filters and one filter is said to be crossed relative to the other then the light intensity is at a minimum
• Light coming through the first filter is blocked by the second because the second is at 90 degrees to the first

• Displacement of a vibrating particle - the distance and direction of the particle from its equilibrium position
• Amplitude - The MAX displacement of a vibrating particle (this can be positive or negative i.e crest/trough)
• Wavelength - the least distance between two adjacent vibrating point (e.g. trough-trough)
• Period - time for one complete wave to pass a fixed point

Period = 1/f

Wave speed, c = freq x wavelength

Phase difference = fraction of wavelength x 2pi

Reflection

• Straight waves directed at a certain angle to a surface will reflect off at the same angle
• Angle between the reflected wavefront and wall and incident wavefront and wall are equal

Refraction

• When a wave changes speed when it crosses a boundary, its wavelength also changes
• If wavefronts are at a non-zero angle to the boundary the direction of the wave will change as well

Diffraction

• Diffraction occurs when waves spread out after passing through a gap
• The narrower the gap the more the waves spread out
• The longer the wavelength the more the waves spread out

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

• Where a crest meets a crest reinforcement takes place and a supercrest if formed
• When a crest meets a trough the resultant displacement is zero

Stationary Waves

• When two people send waves along a rope from both ends the two sets of progressive combine to form a stationary wave
• On a stationary there a points of zero displacement called nodes (where the waves cancel each other out) and maximum displacement called antinodes
• At nodes the two waves are antiphase
• When waves pass through each other at a constant frequency and phase difference, cancellation and reinforcement occur at fixed points. This is interference
• Coherent sources produce an interference pattern due to the constant phase difference and frequency of the superimposing waves.

•  Stationary waves are formed in air by the superposition of two waves
• The stationary wave is formed in the region of the detector
• These are the wave travelling towards the wall and its reflection from the wall 
• minima and maxima are formed by destructive interference and constructive accordingly at points where the two waves are continuously in antiphase or in phase
• The 2 waves have the same frequency and travel in opposite directions

Stationary wave patterns are only produced at resonant frequencies. The is when an exact number of half-wavelengths fit into the string. Stationary waves don't transfer energy

• The simplest wave pattern is produced at the fundamental frequency
• There are nodes at either end of the wave and they have no amplitude
• At a node there is destructive interference and at a antinode there is constructive interference
• n^th overtone = n+1^th harmonic

• At the a^th harmonic, the number of antinodes = a, and the number of nodes = a+1
• At the a^th harmonic a/2 wavelengths fit into the string
• If you are given the fundamental frequency, the resonant frequency of the a^th harmonic = a x fundamental frequency

Light shone through a narrow slit will sometimes produce a diffraction pattern.

• The light source needs to be monochromatic (same wavelength) so that the waves diffract by the same amount resulting in a clear interference pattern
• The pattern has a bright central fringe and alternating bright and dark fringes
• Constructive interference has path difference n wavelengths, destructive interference has path difference (n+0.5) wavelengths

Double-slit interference

• At a bright fringe the light wave from each slit arrive in phase with each other
• At a dark fringe the light wave from each slit arrive antiphase (180 degrees out of phase) with each other
• distance from central maxima to first order maxima is the fringe spacing, w
• D is distance from the screen

Diffraction is the spreading of waves as they pass through a gap

Diffraction of a single slit

• Central fringe is produced and has the highest intensity
• Central fringe is twice as wide as the outer fringes
• Intensity of light decreasing further from the central fringe
• Each outer fringe is of equal length
• Greater wavenlength of light produces fringes that are further apart

Diffraction of double slits

• Slits need to be close enough so that the light from each slit overlaps
• Each slit must be narrow enough to allow for sufficient diffraction

The double slits are coherent sources because the light emitted is of constant phase difference.

In the double slight experiment the fringe seperation depends on the colour of light used

• White light consists of a continuous spectrum of colours: from with the range 650nm (red) - 350nm (blue)
• Each colour of light has it own wavelength
• Fringe seperation for red light is greater than that of blue light because the wavelength of red light is greater
• The longer the wavelength, the greater the fringe seperation
• Vapour lamps/discharge tubes produce light of a dominant colour
• Light from a filament lamp covers a continuous spectrum therefore a filter is required to lower the range of wavelengths
• Laser light is highly monochromatic. Double slits can be illuminated directly without the use of a single slit first

For white light:

• central fring is always white
• inner fringes are blue on the inner side and red on the outer

When monochromatic light is incident upon many close slits a giffraction grating is produced

• Light from a diffraction grating only travels in certain directions
• Light waves through adjacent slits reinforce each other in certain directions and cancel out in other directions
• Central beam of a diffraction grating is the zero order beam, which goes in the same direction as the incident beam
• Angle between zero order beams and following orders increases if light of a longer wavelength is used, grating with closer slits is used

• d = dist. between gratings (m), smaller d = greater theta
• theta = angle of diffraction
• max no. of orders = d/wavelength (sin90=1)