**Question:** Why do many simultaneous farts sound like a single fart?

**Short answer:** Math.

**Long answer**: Our Fart Maker technology produces an infinite variety of farts. While the actual technology is proprietary, we can say that it works primarily by additively recombining real farts from our existing database. It has always been surprising to us that the sum of many farts sounds similar to an actual fart.

Recently, we pointed out a mathematical fact, namely that two farts with the same peak frequency but different phases would combine to produce a sound with the same frequency. In other words, if two people produced similar farts at slightly different times, the result would sound like a third fart. However, our mathematical demonstration was limited to the unrealistic case in which the two farts had the same amplitude and consisted of a single frequency. Here we show that this result applies to the more general case as well.

*Theorem*: The frequency content of the sum of many farts approaches the mean frequency spectrum of the individual farts.

*Proof*: Consider a composite signal comprised of *n* farts:

where *g _{k}* is the

*k*

^{th}fart.

An important contribution to the mathematics of flatology came from Joseph Fourier, who discovered the famous Fourier Transform.

Writing the Transform in amplitude-phase form, each *g _{k}* becomes:

The component of *g _{k}* at frequency ω

*for fart*

_{h}*k*is therefore:

where *A _{h}* and

*φ*correspond to the amplitude and phase for a single fart. The sum across farts for frequency

_{h}*ω*is then:

The righthand side of the equation follows from the Harmonic Addition Theorem. Moreover, it can easily be shown that:

The expected value of *A ^{2}* is then:

The sum on the right-hand side vanishes for *i* ≠ *j*, because *φ_{i} – φ_{j}* is uniformly distributed with a mean of zero. For

*i*=

*j*, cos(

*) = 1, so that*

*φ**φ*_{i}–_{j}The power at each frequency in the composite fart is therefore proportional to the average power in the individual farts. This completes the proof.

Note that for small *n*, it is possible for farts to be in antiphase, so as to annihilate each other through destructive interference, while for large *n*, constructive interference creates a coherent composite from those farts that happen to be in phase. This suggests that the most realistic composite fart will be obtained when Fart Maker settings are chosen so as to maximize the *richness* of the sound. Here is an example: