What is the equation used to determine the acoustic resonant frequency of a hallow sphere, and a string?

T

For strings, the wikipedia article is adequate, no idea about spheres though

Acoustic Resonance Wiki

For air in a sphere (is this your question?), the solution is the same as for a cone. You may find more easily the solution for a cone as it is known for music instruments like the saxophone, but beware that you'll have a speed node at the sphere's walls and a pressure node at the intrument's wide end.

The solution includes some Bessel functions.

@ Enthalpy:

It surprises me that the sphere and cone would have the same structure of resonances. The extreme case of a very long cone with a very small taper should produce Bessel functions, because these arise in cylinders. On the other hand, the natural modes in a sphere are given by the spherical harmonics, which are associated Legendre polynomials longitudinally, cosines in latitude, and powers of r in the radial direction. These seem somewhat different from the Bessel functions. This has got me interested; do you have a reference where I could find out more? (I tried Google, but it just gives me either Milo Wolff stuff or technical papers on inertial plasma containment; perhaps I haven't found the right combination of search terms yet....)


@ gre:

Short answer: strings are easy, cylinders are easy, spheres are harder. For both strings and cylinders, you have either f = nv/2L (when conditions at both ends are the same) or f = (2n+1)v/4L (when conditions at both ends are of opposite type). For strings, v = wavespeed = sqrt(tension*string_length/string_mass); for cylinders, v = speed of sound in air is approximately (331 + 0.6*T)m/s, where T is the Celsius temperature. Both strings and cylinders have sets of frequencies that are simple multiples of the fundamental frequency, which is why they make musically pleasing notes. Other systems (such as drums) make more complex mixtures of frequencies, which sound "noisier" or more "dissonant" to the ear. Flutes and violins can have a very low harmonic complexity, which makes them sound "sweet," while brass instruments have intermediate levels of harmonic complexity, because their shape includes both tubes and cones. A sphere would be more complex still, and would probably sound similar to a drum, a muffled sort of sound.

Hope that helps!

--Stuart Anderson

For a sphere. Would v/(2*pi*r) work for the fundamental tone?

Sphere and cone: I was only considering the radial modes, which aren't by far the only ones, so the similarity in only partial. Assemble many cones (with the appropriate sections...) at their apex, you can build a sphere of them. Now, as the speed is radial, you don't need the walls any more: remove them, you have a sphere.

I have a good book (and a few more) called "The Physics of Musical Instruments".
http://www.amazon.com/Physics-Musical-Inst...r/dp/0387983740
Complete solution of air cones, metal plates etc. Maybe 400 pages, but still concentrating on spectra, which helps little understanding what qualities a sound will have - though the authors recognize musical sounds aren't necessarily periodic and hence of harmonic spectra.

This book helped a friend of mine to build the Tubax, Soprillo, Contraforte, so it is a useful book.
http://www.eppelsheim.com
don't miss the recorded sound samples!

And if you like music instruments, a few addresses...
http://www.oddmusic.com/gallery/index.html
http://www.instrumentenweb.com/index.html
http://www.contrabass.com

I'd be very cautious about any link between spectrum contents and human perception. A bell for instance is not harmonic, and building it as harmonic as possible doesn't improve its sound. Also, a sinewave sound isn't any sweet, and a violin's spectrum is about as rich as an oboe.

Maybe the least bad discovery in the last 20 years is that aperiodicity is central to sound quality - and this kind of aperiodicity is barely visible on a spectrum. Researcher (at Rennes university maybe) randomized the start time of each period of a sawtooth waveform and got a more reasonable imitation of a violin - something a synthesis by the harmonic spectrum radically fails to do.

Having written a program that creates a sound from an arbitrary harmonic contents, I can tell that
- A clarinet or an oboe sound can be more or less suggested
- A violin, a saxophone, a flute, a bassoon not at all
- Phase is not perceived
And anyway, one just hears that a saxophone or bassoon sound vibrates.

v/(2*pi*r) : no, except maybe in some very artificial cases. Could you tell if you're interested in the resonance of air in a sphere, or in a full solid sphere, or in the resonance of the walls of a hollow sphere...?

Thanks for that information Enthalpy.

I'm looking for the resonance equation for both air sphere and solid sphere.