posted 24 June 2003 11:08            

Some while ago, I submitted sounds that pretend to sound like struck bars and plates. These sounds used wavetables to compute the complicated Bessel functions needed for finding the relative amplitudes at the first few normal modes of vibration.

When I submitted those sounds, I blithely assumed that the sound emanating into a room would be given by approximately the square of the amplitudes of the respective normal modes of vibration.

Now, on second thinking, I see that I failed to take account of the relative coupling efficiency of these vibration frequencies to the air medium of the room.

When a vibrating object has small size (or wavelength) compared to the wavelength in air of its frequency, the amount of sound radiated into the room diminishes considerably. Metal, and other stiff materials, have much shorter wavelengths of vibration than the corresponding wavelengths in air. What we care about for our purposes is not the absolute relationship between these mismatches, but rather, the relative degree of mismatch at higher frequencies of vibration compared to the lowest mode vibration frequency (or fundamental).

This is an exceedingly difficult problem to solve in detail. But a back-of-the-envelope calculation can demonstrate the essential character. Assume that within one half-wavelength in air, we encompass N wavelengths of the same pitch in the stiff object. Go ahead and assume sinusoidal envelopes for both of these wavelengths, and simply integrate up the stiff sinusoids that fit within one half-wavelength of the air sinusoid.

That yields a Sinc(1/x) function, of sorts, for stiff-wavelength x. As x grows smaller this function oscillates with an amplitude proportional to x. So if the wavelength in metal of some frequency is smaller by a factor of 10 than the corresponding wavelength in air, then the coupling efficiency between the object and the room will be approximately 1/10.

Now metal rods and plates are dispersive mediums, meaning that the speed of sound of some vibration depends on the frequency of that vibration. It goes something like the square root of the frequency for a metal (or stiff) rod. The wavelength of an oscillation is inversely proportional to the speed of sound at that frequency. Hence, as the mode of oscillation increases above the fundamental, one should expect that the coupling efficiency of these higher frequency vibrations would be heard with much lower amplitudes, going something like 1/Sqrt(f).

Hence, my Sounds tended to exaggerate the higher (anharmonic) partials of the sounds of struck bars and plates. The results ought to sound a bit darker, but still have anharmonic character.

While not exact, I wouldn't be too surprised to find something very nearly 1/Sqrt(f) for the relative loudness of higher partials.

[An old professor of mine taught me many years ago, to first try to solve difficult problems on a napkin or the back of an envelope. If you can't get the general sense of the solution in that small amount of space, then you don't understand the problem well enought to warrant a massive detailed model... It was pretty wise advise.

... but that doesn't mean that I analysed this situation correctly... anyone else care to critique this answer? ]

Cheers,