At a school function, two large speakers stand on either side of the stage, both playing the same song. Walk slowly across the ground in front of the stage. A long, steady note seems to swell in some places and fade in others, although you are no farther from the stage. Stand still, and the note stays the same. Move a step, and it changes again.
The principle of superposition: when worlds (or waves) collide
What happens when two sound waves meet at the same point in space? Do they crash into each other? Do they bounce off each other? The reality is much simpler, and more elegant. Sound waves pass right through one another. Each wave travels on as if the other were not there.
At any point where the waves overlap, the air feels both at once. The chapter Propagation of Sound Waves described a sound by its pressure variation \(\Delta P\), the change of pressure above or below the still air. If the first wave alone would give \(\Delta P_1\) at a point, and the second alone \(\Delta P_2\), then together they give \[ \Delta P = \Delta P_1 + \Delta P_2. \] The sum is algebraic: a compression, \(\Delta P > 0\), and a rarefaction, \(\Delta P < 0\), partly or wholly cancel. This is the principle of superposition, the same rule the chapter Superposition and Reflection used for displacements on a string.
When two or more sound waves overlap in air, the pressure variation at each point and instant is their algebraic sum. Each wave adds the pressure variation it would produce alone.
This simple act of addition gives rise to fascinating, and sometimes complex, patterns of sound. One consequence is interference. Two sources of the same pure tone can produce a stable pattern of loud and quiet zones in space. The rest of this lesson maps that pattern.
Superposition adds the pressure variations, not the loudness. Two equal waves that arrive together, crest on crest, give twice the pressure amplitude and so four times the intensity. Two that arrive crest on trough give silence, not half the loudness.
At one point in a room, the sound from one speaker alone would raise the pressure by \(0.03\,\text{Pa}\). At the same instant, the sound from a second speaker alone would lower it by \(0.01\,\text{Pa}\). What is the pressure variation there?
Interference: a spatial map of sound
Imagine two speakers, \(S_1\) and \(S_2\), perfectly synchronised and emitting the same pure tone. As the waves spread out from each source, they create a stable interference pattern throughout the room. Whether a particular place is loud or quiet depends entirely on how the two waves line up when they arrive there.
"Perfectly synchronised" has a precise meaning. The two speakers are wired to one amplifier, so they push the air out together and pull it back together. They emit waves of the same frequency with no phase difference between them. Such sources are coherent sources. If one speaker drifted in frequency, the loud and quiet places would wander, and no steady map would form.
The key factor is the path difference \[ \Delta L = |L_2 - L_1|, \] the extra distance that one wave has to travel to reach the listener at P. The two waves leave their speakers in step. The wave with the longer path arrives later, so it is behind in its cycle. The path difference therefore decides the phase difference between the two waves when they arrive.
How much phase does one metre of path cost? A wave repeats itself after one wavelength \(\lambda\), and its phase grows by \(2\pi\) over that distance. So a path difference \(\Delta L\) is \(\Delta L/\lambda\) of a cycle, and the phase difference is \[ \Delta\varphi = \dfrac{2\pi}{\lambda}\,\Delta L. \] This is the same rule that the chapter Superposition and Reflection found for waves on a string.
Two speakers in phase emit a tone of wavelength \(1.0\,\text{m}\). A listener is \(4.00\,\text{m}\) from one speaker and \(4.25\,\text{m}\) from the other. What is the phase difference of the two waves at the listener?
Loud spots and quiet spots
Two cases matter most. They are the two extremes, and every other path difference gives something in between.
Constructive interference, a loud spot. Suppose the path difference is a whole number of wavelengths, \(n\lambda\). Then the phase difference is \(2\pi n\), and the waves arrive perfectly in phase, crest meeting crest. They reinforce each other. This makes a point of maximum amplitude, a loud spot. This is constructive interference.
Destructive interference, a quiet spot. Suppose instead that the path difference is a half-integer number of wavelengths, \((n + \tfrac12)\lambda\). Then the phase difference is an odd multiple of \(\pi\), and the waves arrive perfectly out of phase, crest meeting trough. They cancel each other. This makes a point of minimum amplitude, or silence. This is destructive interference.
For a listener at a point P, with two coherent sources in phase:
Two speakers emit identical sound waves. At a point where the path difference is exactly one full wavelength, what kind of interference occurs?
Fig 1.2 shows what the listener at P receives in the two cases. Each trace is the pressure variation at P against time. Choose a path difference with the chips.
Look along a dashed line in Fig 1.2. It passes through a crest of \(\Delta P_1\). At \(\Delta L = \lambda\) and \(2\lambda\), the same line passes through a crest of \(\Delta P_2\), and the resultant has crests twice as tall. At \(\Delta L = \lambda/2\) and \(3\lambda/2\), the line passes through a trough of \(\Delta P_2\), and the resultant stays flat. Adding a whole wavelength of path changes nothing.
How loud is a place in between?
The chapter Superposition and Reflection found the resultant of two equal waves with a phase difference. The same algebra holds for pressure. If each wave alone has pressure amplitude \(\Delta P_0\) at P, the resultant has amplitude \[ \Delta P_R = 2\Delta P_0\left|\cos\dfrac{\Delta\varphi}{2}\right| = 2\Delta P_0\left|\cos\dfrac{\pi\,\Delta L}{\lambda}\right|. \] Intensity grows as the square of the pressure amplitude, so the intensity is \(I = 4I_0\cos^2(\pi\Delta L/\lambda)\), where \(I_0\) is the intensity of one wave alone.
Check the two extremes. At \(\Delta L = \lambda\), \(\cos\pi = -1\), so the amplitude is \(2\Delta P_0\) and the intensity \(4I_0\). At \(\Delta L = \lambda/2\), \(\cos(\pi/2) = 0\), so both are zero. Halfway, at \(\Delta L = \lambda/4\), the amplitude is \(2\Delta P_0\cos 45° = 1.41\,\Delta P_0\) and the intensity \(2I_0\), the same as the two waves added without interference.
Interference does not create or destroy energy. It moves the energy around. The loud spots get \(4I_0\) and the quiet spots get nothing. Averaged over the room, the intensity is \(2I_0\), the sum of the two speakers. The quiet spots lose exactly what the loud spots gain.
Two speakers in phase play a \(686\,\text{Hz}\) tone. Take the speed of sound as \(343\,\text{m/s}\). At a point P the path difference is \(1.25\,\text{m}\). What does a listener at P hear?
Two speakers in phase send waves to a point P, each with a pressure amplitude of \(0.02\,\text{Pa}\) there. The path difference is exactly one wavelength. What is the pressure amplitude of the sound at P, in Pa?
At a quiet spot the two waves cancel. Has the sound energy of the two speakers been destroyed there? Explain.
No. Interference only redistributes the energy in space. At the quiet spots the intensity is zero, but at the loud spots it is four times that of one speaker, not twice. Averaged over the pattern, the intensity equals the sum of the two speakers, so no energy is lost.
Now return to the walk in front of the stage. The two speakers play the same note in phase, so they are coherent sources. At each step your path difference to the two speakers changes. Where it is a whole number of wavelengths, the note swells. Where it is a half-integer number, the note fades. Standing still keeps the path difference fixed, so the loudness stays fixed too.
- Adding the loudness, or the intensities, of two coherent waves.Add the pressure variations first, then square. In phase, two equal waves give four times the intensity of one; out of phase they give zero.
- Taking the path difference as \(L_1 + L_2\), or as the distance between the speakers.The path difference is \(|L_2 - L_1|\), the extra distance one wave travels to reach the listener.
- Swapping the two conditions.A whole number of wavelengths puts crest on crest, which is loud. A half-integer number puts crest on trough, which is quiet.