# What happens to the pressure time graph when two musical instruments play at the same time?

I'm learning acoustics for the first time and I'm having trouble picturing what the pressure time graph would look like when two instruments play together.

Here are two graphs for the C4 note of a clarinet and a flute. When they both play together, will the graph just be a superposition of both their individual graphs?

First of all the link to the figures you posted do not show pressure versus time, they show the spectrum of the instrument in the frequency domain. Since an FFT is a linear operation I do suppose that two instruments would have their spectra added.

But there is more to the story that that. Pressure waves travel away from the source and their phase will depend on distance from the instrument to a listener's ear or microphone.

To correctly predict the combination of the two at a point in space you need to have a full description of the wave...

p = sin(wt - kr + phi)/r

where w and k are the angular frequency and wave number and phi is a phase shift based on initial conditions. Adding the pressure wave is called a coherent sum and accounts for the details in the two sources assuming they are knowable. This will generate an interference pattern in space.

What typically happens in an orchestra is there are small variations in phi, and this is considered a random variable. So one cannot completely specific the wave and we do an incoherent sum where we add the power values (or intensities). This is not a trivial difference as the two result in a different estimate for the amplitude of the sound wave. A coherent sum over N sources in phase will create an amplitude of N*p0, and a power of ~N^2*P0, whereas the incoherent sum will produce a power of N*P0, and an amplitude of sqrt(N)*p0.

The amplitude of the spectrum will likely not change by the relative phases in the modes will for each case.

• That's right, but the amplitude spectrum may change depending on the relative phases. If the two instruments happen to play the same note, at least the fundamental, and maybe some of the "harmonics" (excuse the slight abuse of the term) will have the same frequency, so where they meet in space you get linear superposition. This means that the summed magnitude will be between 0 and A1 + A2 (where those are the respective amplitudes at this point in space). This will happen for all frequencies that are common to the two spectra. Jan 31, 2020 at 23:04

Yes, the resulting pressure graph would be the sum of the two.