The principle of superposition, coherence and path difference, two-source and double-slit interference, diffraction, and stationary waves on strings and in pipes.

What this dot point is asking

CCEA wants you to state the principle of superposition, explain coherence and path difference, describe two-source and double-slit interference and the diffraction grating, describe diffraction, and explain how stationary waves form on strings and in pipes. Calculations on the fringe equation, the grating equation and resonant frequencies appear regularly, alongside the sketch of a standing-wave pattern.

Superposition and interference

For a stable, observable interference pattern the sources must be coherent: they keep a constant phase difference and have the same frequency (and ideally the same amplitude). A single source split into two, as in Young's double slit, guarantees coherence because any phase fluctuation affects both slits equally.

The grating gives much sharper, brighter maxima than two slits because thousands of slits reinforce only at the exact angles satisfying the equation, making it the instrument of choice for measuring wavelength accurately.

Diffraction

Diffraction is the spreading of a wave as it passes through a gap or around an obstacle. It is most pronounced when the gap is about the same size as the wavelength: a gap much wider than $\lambda$ barely spreads the wave, while a gap comparable to $\lambda$ spreads it through a wide angle. This is why sound (long wavelength) diffracts around a doorway easily but light (very short wavelength) does not, and why a single narrow slit produces a broad central bright band with dimmer fringes either side.

Stationary waves

A string fixed at both ends has a node at each end. The fundamental fits half a wavelength into the length, $L = \dfrac{\lambda}{2}$, giving $f_1 = \dfrac{v}{2L}$; the harmonics are integer multiples of this. A pipe closed at one end has a node at the closed end and an antinode at the open end, so the fundamental fits a quarter wavelength, $L = \dfrac{\lambda}{4}$, and only the odd harmonics are present. An open pipe has antinodes at both ends and behaves like the string for its harmonic series. Resonance occurs only at these specific frequencies.

Worked example: harmonics of a closed pipe

Examples in context

Example 1. Noise-cancelling headphones. A microphone samples the incoming sound and the electronics generate an inverted wave, half a wavelength out of step, so that superposition gives destructive interference at the ear. The unwanted hum is cancelled because the resultant displacement is the sum of two nearly opposite waves, a direct everyday use of the superposition principle.

Example 2. A guitar string. Plucking a string sets up a stationary wave fixed at both ends. The fundamental gives the note's pitch, while the harmonics add the timbre. Pressing a fret shortens the vibrating length $L$, which raises the fundamental $f_1 = \dfrac{v}{2L}$, exactly as the standing-wave condition predicts.

Try this

Q1. In a double-slit experiment the slit separation is $0.50\ \text{mm}$, the screen is $2.0\ \text{m}$ away and the fringe spacing is $2.4\ \text{mm}$. Find the wavelength. [3 marks]

• Cue. $\lambda = \dfrac{ws}{D} = \dfrac{(2.4 \times 10^{-3})(0.50 \times 10^{-3})}{2.0} = 6.0 \times 10^{-7}\ \text{m}$.

Q2. State the path difference condition for constructive interference. [1 mark]

• Cue. A path difference of a whole number of wavelengths, $n\lambda$.

Q3. A diffraction grating has $300$ lines per millimetre. Find the angle of the first-order maximum for light of wavelength $5.9 \times 10^{-7}\ \text{m}$. [3 marks]

• Cue. $d = \dfrac{1 \times 10^{-3}}{300} = 3.33 \times 10^{-6}\ \text{m}$; $\sin\theta = \dfrac{n\lambda}{d} = \dfrac{5.9 \times 10^{-7}}{3.33 \times 10^{-6}} = 0.177$, so $\theta = 10.2^{\circ}$.