AS Physics |
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Waves and Vibrations
Circular ripples.
Progressive Waves
Properties of Waves
Longitudinal and Transverse Waves
Superposition
Stationary Waves
Normal Modes of Vibration
Harmonics
Refraction
Snell's law
Refractive Index
Total Internal Reflection
Dispersion of White Light
Diffraction
Huygen's construction
Single-Slit Diffraction
Diffraction Gratings
Interference
Young's Double Slit Experiment
Fringe Spacing
Progressive Waves
Inaccurate AQA definition of waves = "Oscillation of the particles of the medium"
The above AQA definition is inaccurate because waves don't necessarily need particles or media (plural of 'medium') to propagate. In an EM wave (for example) an electric and magnetic (wave) field oscillate and don't need any medium to do so (i.e. it can propagate in a vacuum).
More accurate definition = "An oscillation accompanied by a transfer of energy, in time".
The above AQA definition is inaccurate because waves don't necessarily need particles or media (plural of 'medium') to propagate. In an EM wave (for example) an electric and magnetic (wave) field oscillate and don't need any medium to do so (i.e. it can propagate in a vacuum).
More accurate definition = "An oscillation accompanied by a transfer of energy, in time".
Properties of Waves
Amplitude = maximum displacement from equilibrium position (this is measured in different units depending on what type of wave it is e.g. if its a wave on a water it is measured in meters)
Wavelength, 'λ' (meters, m) = length of one complete wave
Displacement (meters, m) = distance a particle moves from its equilibrium position
Period, 'T' (seconds, s) = time to complete one wave
Frequency, 'f' (Hertz, Hz) = number of waves per second
Wave speed, 'c' or 'v' (m/s) = wavelength / period
Phase angle (degrees or radians) = the position along the wave. One complete cycle is 360 degrees (or 2π)
Wavelength, 'λ' (meters, m) = length of one complete wave
Displacement (meters, m) = distance a particle moves from its equilibrium position
Period, 'T' (seconds, s) = time to complete one wave
Frequency, 'f' (Hertz, Hz) = number of waves per second
Wave speed, 'c' or 'v' (m/s) = wavelength / period
Phase angle (degrees or radians) = the position along the wave. One complete cycle is 360 degrees (or 2π)
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Longitudinal and Transverse Waves
Longitudinal
Each particle oscillates parallel to the direction of propagation of the wave. The particle doesn't move perpendicularly to the wave propagation direction.
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Transverse
Each particle oscillates perpendicular to the direction of propagation of the wave. There particle doesn't move parallel to the wave propagation direction.
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Polarisation
Transverse waves exist in planes of polarisation. Polarised light consists of one plane of polarisation. When unpolarised light travels through a polaroid material the wave is 'polarised'. Polarisers that are at 90 degrees to one another (aka 'cross polarisers') will not let any wave through.
Unpolarised passing through a polarizer
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Polarizers that are placed at 90 degrees will not let light through.
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Stationary Waves
The combining effect of two (or more) waves is known as superposition.
Principal of superposition = when two waves meet, the total displacement at a point is equal to the sum of the individual displacements at that point.
Crest meets crest = ‘supercrest’
Trough meets trough = ‘supertrough’.
Crest meets trough = zero displacement
These are formed if two progressive waves continually cross each other. They appear to be standing still and not progressing. They combine at fixed points along the wave to form points of no displacement, or nodes (‘points that dont move’). At each node the two waves are always 180° (or π) out of phase, so they cancel each other out.
Principal of superposition = when two waves meet, the total displacement at a point is equal to the sum of the individual displacements at that point.
Crest meets crest = ‘supercrest’
Trough meets trough = ‘supertrough’.
Crest meets trough = zero displacement
These are formed if two progressive waves continually cross each other. They appear to be standing still and not progressing. They combine at fixed points along the wave to form points of no displacement, or nodes (‘points that dont move’). At each node the two waves are always 180° (or π) out of phase, so they cancel each other out.
Resulting wave
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Two waves crossing paths
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The image to the left shows a clear time series (t0.... t4) showing the principle of superposition. When waves overlap (in phase) they constructively interfere, when they are out of phase they destructively interfere.
The resulting wave is a stationary wave, whose nodes remain stationary. An 'anti-node' is a point of maximum amplitude - this oscillates from positive to negative. A 'node' is a stationary point that does not move. The distance between adjacent nodes is ½ λ. |
Normal Modes of Vibration
Fundamental mode at different fractions of the full cycle i.e. period, T
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Stationary waves occur at specific frequencies that depend on the length between the fixed ends; this series of discrete stationary wave frequencies are called 'normal modes' of oscillation. The fundamental mode is the lowest frequency mode (see image on the left) i.e. lowest frequency standing wave.
The higher modes are often called 'overtones'. After a full cycle the mode is back to its original position (this goes for any mode of vibration). There are nodes at either end of this string. |
Harmonics
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(images thanks to Wikipedia here)
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Wave representation (e.g. on a string)
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Molecular representation (e.g. in a wind instrument)
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Fundamental mode or 'first harmonic':
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First overtone or 'second harmonic':
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Second overtone or 'third harmonic':
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For first harmonic λ1 = ½L
For second harmonic λ2 = L For third harmonic λ3 = ⅔L Time taken for a wave to travel along the string and back is t = 2L/c. Time taken for to pass through a whole number of cycles = n/f (where n is a whole number). 
v = wave speed, μ = mass per unit length, L = length Derivation of frequency of the n-th mode |
Refraction
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Refraction occurs because the speed of light is different in different media. All angles are measured from the normal to the light beam. Normal = perpendicular line crossing the boundary between two media; it is drawn where the beam touches the interface. When light travels from dense to less dense medium = light bends away from the normal.
Less dense to dense = light bends towards the normal. 
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Wavefronts = lines that connect points in a wave that are in phase. The distance between wavefronts is the wavelength.
Prettttty wavefronts on water
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Wave moving from a less dense medium (white) to a dense medium (grey). Wavefronts are further apart (longer wavelength) in the white medium in comparison with the grey medium (shorter wavelength)
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The wavefronts slow down upon entering the denser medium and you can see why the ray changes direction (above). Note: the frequency of the wave does not change when refraction occurs (this is bizarre!). If the speed of the wave and its wavelength is reduced then the frequency remains the same (from c=fλ). So blue light in air is slightly different from blue light in glass!
Snell's Law
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The speed in medium 1 is c1.
YY’= c1t and XX’= c2t Use geometry and similar triangles to get i and r. Using trig: c1t=XY’sin(i) And c2t=XY’sin(r). Divide one by the other... |
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Refractive index, n
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The refractive index is the ratio of the velocity of light in a vacuum divided by the velocity of light in that medium. The refractive index is a ratio. It has no units ('dimensionless'). A high refractive index means that light travels slow in that medium.
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Another helpful form of Snell's law, using refractive indices
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Some values for n to give an idea of order of magnitude
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Total Internal Reflection
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Total internal reflection takes place if:
1. The incident substance has a larger refractive index than the other substance. 2. The angle of incidence exceeds the critical angle. Critical angle, θc, is the angle above which light is internal reflected at an interface: |
If you get n1 and n2 the wrong way round your calculated will explode!
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Dispersion of White Light
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White light is made out of many frequencies of light. A glass prism refracts light by different amounts depending on the wavelength of light. The shorter the wavelength the greater the angle of refraction. ‘Dispersive’ effect occurs because the speed of light in the glass depends on the wavelength. The refractive index for the different colours is will be different. |
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Diffraction
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This is the spreading out of waves as they pass through a narrow slit (not just light). For diffraction to occur the slit must be of comparable size to the wavelength of the wave passing through the slit.
To increase the amount of ‘spread’: • Decrease size of gap • Increase wavelength |
Small slit
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Large slit
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Huygen's Construction
Christiaan Huygens (1629 - 1695)
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Huygens said that each point on the single slit acts a source of secondary wavefronts, as seen in the diagram on the right. This is a useful model representation showing how waves spread out. |
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Single Slit Diffraction
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Single-slit diffraction is the phenomenon whereby a wave will 'spread' through a narrow slit that is roughly the same size (or smaller) than the wavelength of that wave. This applies for anything with a wavelength, even high energy electrons (see wave-particle duality). A diffraction pattern is the result. This consists of light and dark fringes at specific distances and angles, as will be derived below. |
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Imagine the first dark fringe occurs at a point P on a screen a large distance from the single slit. Consider light waves travelling from two points A1 and A2 a distance a/2 apart to the point P.
To produce the first minimum the light from A1 and A2 must interfere destructively, so: The path difference: A1P–A2P=λ/2 The path difference: A1Q=λ/2 Triangle A1A2Q is similar to triangle PA2O From triangle A1A2Q: sinθ=A1Q/A1A2 Using: A1Q=λ/2 and A1A2=a/2 Therefore: sinθ = (λ/2)/(a/2) Therefore, first dark fringe is at: sinθ=λ/a |
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Minima occur at different angle, θ:
Otherwise written as:
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Distance to minimum from central fringe, x:
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As the slit gets narrower, the spacing increases:
Diffraction Gratings
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A diffraction grating is a large number of regularly spaced, narrow slits. When illuminated by monochromatic coherent light a diffraction pattern is produced. This consists of a series of sharp rays of light, spreading out from the diffraction grating in well defined directions. The more slits there are the more interference you get. Eventually you only get constructive interference in very specific places. |
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In certain directions the secondary wavefronts line up and produce rays of light. In all other directions the waves of light interfere destructively. In the directions in which the wavefronts line up, the waves interfere constructively. The path difference between parallel rays of light from neighbouring slits is equal to a whole number of wavelengths of the light.
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Consider light from neighbouring slits P and Q a distance d apart. The wavelength of the light is λ. For constructive interference, the path difference between parallel rays of light from neighbouring slits is: QY=nλ
The separation between the slits: QP=d sinθ=QY/QP Therefore, the diffraction equation is: 
d = distance between slits.
n = nth order of light. θ = angle to order. The maximum number of orders is when θ=90° (sinθ=1). Calculate d/λ and round down to the nearest whole number for the maximum number of orders. Examples: |
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Red laser passing through diffraction grating
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White light passing through diffraction grating. The colours are separated according to wavelength.
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Comparing green light (top) and white light (bottom) going through a diffraction grating
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Interference
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Light sources need to be coherent if they are to interfere. This means that they must have the same frequency with a constant phase difference. Constructive interference = waves in phase. Destructive interference = waves out of phase. Coherent sources are ones that have a constant phase difference between them. In the double slit experiment this can be achieved by either using a monochromatic laser source or using an incoherent source with a single slit. In the double slit experiment a crest/trough passes through each slit at exactly the same time.
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Thomas Young (1773-1829)
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Young's Double Slit Experiment
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Excellent visualisation from here.
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Fringe Spacing
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Consider two slits S1 and S2, a distance s apart. The distance between the slits and the screen is D. Imagine a point P, a distance x away from the centre of the screen. The wavelength of the light is λ.
For bright spots to be produced at the point P, the path difference between the paths from the two slits S1P and S2P must be a whole number of wavelengths. In this case the waves interfere constructively. The path difference: S1P – S2P = nλ Image a point Q such that QP = S2P The path difference: S1Q = nλ |
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For constructive interference the path difference must be a whole number of wavelengths:
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Destructive interference must be a half number of wavelengths:
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For D >> x and D >> s: triangle S1S2Q is similar to triangle PMO ('small angle approximation')
From triangle S1S2Q: sinθ=nλ/s From triangle PMO (using OM ≈ PM = D): sinθ=x/D |
Therefore, distance of bright fringe from central fringe, w:
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n = fringe number
λ = wavelength s = slit separation D = distance from slits to screen. |
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Therefore, the distance between bright-bright and dark-dark fringe is just:
