Superposition and Interference of Sound Waves: MDCAT Physics notes
Superposition and Interference of Sound Waves notes for MDCAT: principle of superposition, coherent sources, path difference conditions, beats and traps.
Principle of superposition
When two or more waves overlap at a point, the resultant displacement at that point is the algebraic (vector) sum of the displacements due to each wave:
$$y = y_1 + y_2 + y_3 + \dots$$
After overlapping, each wave carries on unchanged. Two troughs travelling towards each other on a string combine at the centre into a single, deeper trough of larger amplitude; a crest meeting an equal trough momentarily gives zero displacement.
Interference
Interference is the effect produced when two identical waves travelling in the same medium superpose, giving a pattern of reinforcement and cancellation. For a steady pattern the sources must be coherent: same frequency (hence same wavelength) and a constant phase difference. Two loudspeakers driven by the same signal generator are coherent.
Constructive interference
Waves arrive in phase (crest on crest). The amplitude is the sum of individual amplitudes; for equal amplitudes, the resultant amplitude is double, and the sound is loud.
$$\text{Path difference} = n\lambda, \quad n = 0, 1, 2, \dots$$
Destructive interference
Waves arrive out of phase by $\pi$ (crest on trough). For equal amplitudes they cancel, giving silence.
$$\text{Path difference} = \left(n + \tfrac12\right)\lambda$$
| Path difference | Phase difference | Result |
|---|---|---|
| $0, \lambda, 2\lambda$ | $0, 2\pi, 4\pi$ | Loud (constructive) |
| $\lambda/2, 3\lambda/2$ | $\pi, 3\pi$ | Silence (destructive) |
Example: $\lambda = 50\ \text{cm}$ and path difference $100\ \text{cm} = 2\lambda$, so the point is loud. A path difference of 25 cm ($\lambda/2$) would give silence.
Beats
When two sound waves of slightly different frequencies $f_1$ and $f_2$ superpose, the loudness rises and falls periodically. These periodic variations are called beats:
$$\text{beat frequency} = |f_1 - f_2|$$
Beats are interference in time. They are used to find an unknown frequency (e.g. of a tuning fork) by sounding it with a fork of known frequency, and to tune musical instruments: tuning stops when beats disappear. The ear can detect beats only if $|f_1 - f_2|$ is less than about 10 Hz.
Key formulas
- $y = y_1 + y_2$
- Constructive: $\Delta x = n\lambda$
- Destructive: $\Delta x = (n + \tfrac12)\lambda$
- Phase difference $= \dfrac{2\pi}{\lambda}\Delta x$
- Beats: $f_b = |f_1 - f_2|$
Common MDCAT traps
- Interference needs coherent sources, not sources of different frequencies.
- Resultant displacement is the sum of displacements, never the product.
- Maximum loudness at path difference $\lambda$ (or $0, 2\lambda$), not $\lambda/4$.
- Path difference of a whole number of wavelengths gives loudness, not silence.
- Beats measure frequency, not speed or wavelength.
Quick revision
- Waves in phase: amplitude adds.
- Equal waves in antiphase: cancel.
- Two troughs meeting: bigger trough.
- Beat frequency = difference of frequencies.