Question: What is the resonance frequency of the anal sphincter?
Short answer: 161 Hz.
Long answer: According to various urban legends, there exists a sound frequency which can disrupt the function of the external anal sphincter, causing fecal incontinence. Some countries have supposedly explored the possibility of deploying this “brown note” as a weapon. Nothing has ever come of it.
Nevertheless, the acoustics of the sphincter are of flatological interest, because they must influence the sound of farts. Although we lack the technical facilities for measuring sphincter acoustics directly, we can infer their properties from our existing database and some simple calculations. The calculations are based on a model of speech production, which we briefly review here.
The articulation of speech involves a steady flow of air from the lungs through the windpipe. The vocal folds in the larynx then vibrate in such a way to create sound waves of different frequencies. Other features of speech are created by a complex interplay of resonances in the lips, tongue, mouth, and nasal cavity.
Similarly, farts are produced by air moving through the rectum, which produces its own acoustic resonance that is further influenced by the vibration of the external anal sphincter:
In previous work, we have estimated the contribution of acoustic resonances in the rectum to fart sounds. Given a reasonable estimate of the length of the human rectum, we predicted a peak frequency of around 340 Hz, which is not quite accurate, as the observed frequencies are closer to 270 Hz. This suggests that the external anal sphincter is responsible for important characteristics of fart sounds.
To estimate this contribution, we built on the strong similarities between speech production and fart production. Studies of speech typically make use of a source-filter model, in which air forced through a tube (e.g., the trachea or the rectum) has a broad-band spectral characteristic. Resonances within the system (e.g., the larynx or sphincter) amplify certain sound frequencies, while suppressing others. Mathematically, we assume that these are stable over short periods of time, yielding a linear, time-invariant system. For fart production, we can express the resulting system as:
Here f is the fart sound, x is the sound produced by air flow, r is the resonance of the rectum, and s is the resonance of the sphincter. The asterisks mean that the final output is produced by convolution of the source x and the filters r and s. The filters are related to what speech scientists call formants.
According to the convolution theorem, the above equation can equivalently be expressed as:
Here the fart, the filters, and the source are all described in the Fourier domain as a function of sound frequency ω.
where the typical length L of the rectum is 23 cm, n is an odd number, and the diameter d is a maximum of 6 cm; v is the speed of sound. The full model, with estimates for X, R, and F is shown here:
It follows that the contribution of the sphincter is given by:
Taking F from the observed data, R from rectal acoustic resonances, and assuming X to be broadband noise, we arrive at the following frequency spectrum for S:
The average resonance frequency of the sphincter is therefore 161 Hz. Of course, the anal sphincter is under voluntary control, being comprised of striated muscle, and so the actual frequency can be tuned by the farter in various ways, probably within the range of about 100 – 200 Hz. Interestingly, the vocal folds produce fundamental frequencies in the same range.
Note that this range is quite different from previous claims about the brown note, which for reasons unknown is reported to have a frequency below 20 Hz. Whether the typical sphincter frequency of 161 Hz functions as a brown note can be determined by playing the following video file:
Thus far, the military applications of this finding seem rather limited.