Contents: 7 sections
Syllabus points
- State the principle of superposition and apply it to interference and to stationary waves.
- Explain the formation of a stationary wave and describe nodes and antinodes.
- Describe diffraction and its dependence on the size of the gap relative to the wavelength.
- Understand the conditions for two-source interference and the need for coherence.
- Use λ = ax/D for double-slit interference and d sin θ = nλ for the diffraction grating.
The principle of superposition
When two or more waves meet at a point, the resultant displacement is the vector sum of the individual displacements.
Everything in this chapter is that one sentence applied to different situations. Note that it is displacements that add, not intensities or amplitudes, and that displacement has a sign, so waves can cancel.
Stationary waves
A stationary wave forms when two waves of the same frequency and amplitude travel in opposite directions and superpose. In practice this is a wave and its own reflection.
- Nodes are points of permanently zero displacement, where the two waves always arrive in antiphase.
- Antinodes are points of maximum displacement, where they always arrive in phase.
The distance between adjacent nodes is λ/2, so the wavelength is twice the node spacing. Reading the node spacing as the wavelength halves every answer that follows, and it is the most common error in the topic.
Stationary against progressive
| Progressive | Stationary | |
|---|---|---|
| Energy | Transferred along the wave | Stored, not transferred |
| Amplitude | Same for all particles | Varies from zero at a node to a maximum at an antinode |
| Phase | Changes continuously along the wave | Particles between two nodes are all in phase; particles either side of a node are in antiphase |
| Waveform | Moves along | Does not move |
The phase row is worth learning as written. Between two nodes every particle reaches its maximum at the same instant, so they are in phase despite having different amplitudes, and everything on the far side of a node is exactly half a cycle behind.
Harmonics on a string
A string fixed at both ends has a node at each end.
- Fundamental (first harmonic): one antinode, length L = λ/2, so λ = 2L.
- Second harmonic: two antinodes, L = λ, so λ = L.
- nth harmonic: λ = 2L/n, and f_n = n f₁.
For a pipe closed at one end, there is a node at the closed end and an antinode at the open end, so the fundamental has L = λ/4 and only odd harmonics exist.
Diffraction
Diffraction is the spreading of a wave as it passes through a gap or around an obstacle.
The amount of spreading depends on the ratio of the wavelength to the gap width. The spreading is greatest when the gap is comparable to the wavelength. A gap much wider than the wavelength produces almost no spreading.
This is why sound diffracts around a doorway but light does not: the doorway is comparable to the metre-scale wavelength of sound and enormous compared with the 10⁻⁷ m wavelength of light.
Diffraction changes neither the wavelength nor the frequency nor the speed. Only the direction of travel and the amplitude change.
Two-source interference
For a stable interference pattern the sources must be coherent: they must have a constant phase difference, which requires the same frequency. In practice this is achieved by using a single source and splitting it, which is what the double slit does.
Similar amplitudes are needed for good contrast, so that the destructive minima are close to zero.
The conditions at a point:
- Constructive interference where the path difference is a whole number of wavelengths, nλ.
- Destructive interference where it is an odd number of half wavelengths, (n + ½)λ.
The double slit
λ = ax / D
where a is the slit separation, x is the fringe spacing, and D is the slit-to-screen distance.
Read the symbols carefully: a is the separation of the slits and x is the separation of the fringes, and swapping them is a common slip because both are called "the spacing" in conversation.
Rearranged as x = λD / a, the equation says what happens when something is changed:
- Longer wavelength, so red rather than blue: wider fringes.
- Slits moved further apart, larger a: narrower fringes.
- Screen moved further away, larger D: wider fringes.
Worked example. Light of wavelength 600 nm falls on slits 0.50 mm apart. The screen is 2.0 m away. What is the fringe spacing?
x = λD / a = (600 × 10⁻⁹ × 2.0) / (0.50 × 10⁻³) = 2.4 × 10⁻³ m, or 2.4 mm.
The diffraction grating
d sin θ = nλ
where d is the grating spacing, the distance between adjacent lines, and n is the order.
If a grating is quoted as having N lines per metre, then d = 1/N. For 500 lines per mm, d = 1 / (500 × 10³) = 2.0 × 10⁻⁶ m. Missing the conversion from lines per millimetre to lines per metre is the standard error.
Because sin θ can never exceed 1, the maximum order visible is the largest integer n for which nλ/d ≤ 1. Questions ask how many orders are seen, and the answer is 2n + 1 counting both sides and the central maximum.
A grating gives sharper and brighter maxima than a double slit, because many slits contribute, which is why it is used for measuring wavelengths.
With white light the central maximum is white, because all wavelengths have zero path difference there. Every other order is spread into a spectrum, with red deviated most because it has the longest wavelength. That is the opposite of a prism, where red is deviated least, and the pair is worth remembering together.
Common mistakes
- Taking the node spacing as the wavelength rather than as half of it.
- Saying particles between adjacent nodes are out of phase because their amplitudes differ.
- Saying diffraction changes the wavelength.
- Defining coherent as "in phase" rather than "constant phase difference".
- Swapping a and x in λ = ax/D.
- Forgetting that d = 1/N for a grating, or leaving N in lines per millimetre.
- Saying the central maximum of a grating with white light is a spectrum.
- Saying red is deviated least by a grating, which is true of a prism instead.