Waves

Waves

1. Superposition

   • When two or more waves overlap or superpose the resultant displacement at a given instant is the sum of the individual displacements at that instant and position
   • You can use an oscilliscope to measure the time period of a wave to calculate the frequency
   • Phase describes the stage in a wave cycle
   • When two points are at the same stage they are in phase
   • Phase difference
     • Two waves in phase have no phase difference
     • Two waves doing the exact opposite at the same moment are in antiphase
     • If neither in phase or in antiphase then they are said to be out of phase
     • Phase difference can be measured by a phase angle
     • We use a rotating phasor to show where a wave is in the cycle, 1 cycle = 2π = 360°
     • Angle 0 = 2πft
     • displacement = a sin 2π ft
     • We can calculate displacement when two waves superpose by adding the individual wave displacements together as a vector tip to tail
   • Standing waves
     • When two progressive waves of the same frequency are travelling in opposite directions the waves can appear to stop moving - when this happens a standing wave is formed
     • When a string attached at both ends is plucked waves:
       • Move along the string in opposite directions
       • Reflect at the end of strings
       • Superpose as they pass through one another
     • There will be a maximum amplitude of oscillation at points along the string where they meet in phase these are called antinodes
     • Where waves meet in antiphase there is zero amplitude - a node
     • Wave speed v (ms-1) = frequency f (Hz) x wavelength λ (m)
     • Standing waves in air
       • Basis of most wind instruments
       • 1) A sound travels along the tube
       • 2) The wave reflects at the end of the tube
       • 3) Waves travelling up and down the tube superpose eachother
       • Sound can be reflected from an open and a closed end of a column

2. Refraction

   • Waves change speed when they change medium - the change in speed causes light rays to bend and change direction
   • Refractive index
     • Is the ratio of speed in light in one medium to another
     • Refractive index = speed of light in vacuum / speed of light in material
     • Refractive index = sini/sinr

3. Hudgens wavelets

   • He imagined light spreading out in tiny wavelets
   • He pictured every point on a wavefront as a source of circular wavelets
   • Hudgen suggested that when the wavelets meet in phase they form a new wavefront
   • Everywhere else the wavelets from one part of the wave will be in antiphase with those from another part of the wave and will cancel out
   • Ratio of speeds of light = ratio of sin(i) and sin(r)

4. Path difference and phase difference

   • Interference of waves
     • Interference is used to describe the effect of superposition of waves
     • Where waves meet in antiphase the sound is quieter
     • Where waves meet in phase the sound is louder
   • Path difference
     • Waves meeting with 0 path difference will be in phase
     • In order to be in phase the waves path difference has to be an integer multiple of λ or nλ
     • Waves meet in antiphase when the path difference = (n+ 0.5)λ
   • Coherence
     • Stable superposition patterns only occur when there is a constant phase difference between waves from the two sources
     • Waves with a constant phase difference are coherent
   • Constructive interference occurs when waves meet in phase (maximum) - results in larger amplitude
   • Destructive interference is when waves meet in antiphase (minimum) - results in lower amplitude

5. Diffraction

   • When waves pass through a gap roughly the same width as their wavelength the waves spread out
   • The amount the wave spreads depends on the width of the gap compared to the wavelenght
   • Youngs double slit experiment
     • He passed light through two pinholes very close together and observed a pattern of dark and bright fringes
     • Where light from both slits meet at the screen in phase a bright fringe is observed
     • sin0 = λ/d
       • d = slit separation
       • 0 = angle made at the slits between one fringe and another
     • nλ = dsin(0n)
   • Order of maxima
     • The order of maximum shows the number of wavelengths path difference between light from 2 adjacent slits and the screen
     • Zeroth order, n=0 - path difference = 0
     • First order, n=1 - the maximum for a path difference between the slits and path difference of one wavelength
     • Second order, n=2 - maximum produced by a path difference of 2 wavelengths
     • λ = xd/L
       • x = screen separation
       • d = slit separation
       • L = distance between slits and screen
   • Diffraction grating diffraction
     • A multiple slit version of two slit system
     • nλ = dsin(0n) STILL APPLIES
     • Using a grating increases the brightness of the image as more light gets through
     • Diffraction grating spreads out light into its wavelengths (colors) because each wavelength will produce a maximum at a different angle since λ is changing
     • Diffraction gratings are described by the number of lines per mm
     • line separation (m) = 1/number of lines per m
   • Single slit diffraction
     • As a wave passes through a slit we imagine each point in the wavefront as a source of new wavelets
     • Where the phasor arrows are in the same direction we get a large resultant creating a maximum
     • At an angle 0 the path difference across the whole slit is λ
     • Each phasor will reach the screen a little out of phase with the phasor from an adjacent point
     • All the phasors will add together to give a zero resultant - this is a diffraction minimum which occurs wwhen sin0 = λ/b
       • b = slit size
     • Phasors add to 0 if they make a complete circle

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