Adjacent in-phase sound sources compound power as the square of their number

Evidence

We can jump-start our understanding by considering two adjacent small pistons, which may as well be touching. By arguments very similar to those we encountered in the method of images, we can see that the power radiated from two such pistons is not twice, but rather four times the power from one alone. … It is useful to review two different perspectives that cause us to reach this conclusion. The first is that the amplitudes ξ from each of the pistons are equal at some distance from them. Since the power increases as amplitude squared, power α (ξ + ξ) = 4ξ2, whereas the power from one piston was just ξ2. … The second point of view is to ask why each piston produces twice the power it would if it were alone (so that two of them produce four times the power). The explanation is very simple: in addition to the pressure in front of the piston caused by its own acceleration, each piston feels the very nearly equal pressure from the adjacent piston. The pressure is very nearly equal because the wavelength is long compared to the size of the pistons and the spacing between them. Even though this pressure is mostly out of phase with the velocity, whatever small fraction that was in phase is now doubled, so twice the work must be done by each piston, or four times the work of one piston. So we get the factor of four by either argument. The reader will have noted that these arguments are a recapitulation of the proximity resonance phenomenon—that is, in-phase addition of closely proximate amplitudes. A closely similar situation applies to one lone and two adjacent Helmholtz resonators, as depicted in figure 13.7 (left and center). — Eric J. Heller, Why You Hear What You Hear- An…, 651