Interference & Reflection of Sound
Two sounds can add to silence, if they stay in step.
Lesson at a glanceShow the key points
- Add sound waves as pressure changes (scalars), not displacement vectors.
- Phase difference from a path difference Δx: δ = 2πΔx/λ (16.11); add any initial phase difference of the sources.
- Constructive: Δx = nλ, p₀ = p₀₁ + p₀₂; destructive: Δx = (n + ½)λ, p₀ = |p₀₁ − p₀₂| (16.12, 16.13). Equal waves: four times the intensity, or silence.
- Sources must be coherent (constant phase difference); incoherent sources just give the sum of intensities.
- Quincke's tube splits sound along two paths of adjustable length and recombines it; the intensity rises and falls as one path is lengthened.
- Reflection from a rigid wall: the pressure wave keeps its phase (compression returns as compression). From an open end: phase change π.
Two loudspeakers playing the same note don’t simply give twice the sound everywhere. Walk across the room and the note swells and fades: the two waves are adding in some places and cancelling in others.
Adding sound waves
Sound waves obey the principle of superposition. Since detectors respond to pressure, we add the excess pressures, which are ordinary numbers, not vectors. Suppose two waves of the same frequency arrive at a point:
If they travel different distances from their sources, they arrive with a phase difference. One extra wavelength of path means one full cycle (2π) of phase, so:
The resultant is again a sinusoidal wave of the same frequency, with amplitude
δ = 2nπ, i.e. Δx = nλ. Amplitudes add: p₀ = p₀₁ + p₀₂ (16.12). Two equal waves give twice the amplitude, four times the intensity.
δ = (2n + 1)π, i.e. Δx = (n + ½)λ. Amplitudes subtract: p₀ = |p₀₁ − p₀₂| (16.13). Two equal waves cancel: silence.
Two speakers emit the same 680 Hz note in phase. The speed of sound is 340 m/s. At point A the path difference is 1.25 m; at B it is 1.0 m. What does a listener notice at each?
Coherent and incoherent sources
A steady interference pattern needs the phase difference between the sources to stay constant. Such sources are coherent, like two speakers driven by the same amplifier. Two separate violinists are incoherent: their relative phase wanders randomly, so the cos δ term averages to zero and the intensities simply add.
Quincke’s tube
A neat demonstration: sound enters a tube at one end and splits into two curved branches, which rejoin at the other end, where the listener is. One branch slides like a trombone, so its length can be increased. Pulling it out by d adds 2d to that path.
- When 2d is a whole number of wavelengths, the two parts arrive in phase: the sound is loud.
- Pull it a further λ/4 (adding λ/2 of path) and they arrive out of phase: the sound becomes faint.
Measuring the slide’s movement between successive minima gives λ/2, and so the wavelength.
Reflection of sound
When sound meets a boundary, part is reflected. What happens to the pressure wave depends on the boundary:
- Rigid wall (or a closed end of a tube): the air can’t move there, so a compression comes back as a compression. The pressure wave is reflected without a phase change.
- Open end of a tube: the pressure there is held at the outside atmospheric value, so a compression comes back as a rarefaction. The pressure wave is reflected with a phase change of π.
- 1Add sound waves as excess pressures.
- 2Path difference Δx gives phase difference 2πΔx/λ.
- 3Δx = nλ: constructive; Δx = (n + ½)λ: destructive.
- 4Steady interference needs coherent sources; incoherent sources just add intensities.
- 5Quincke's tube shows interference by changing one path's length.
- 6A rigid wall reflects pressure without phase change; an open end reflects it with a phase change of π.
Check yourself
Keep exploring
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