Resonance, acoustical resonant frequency of objects

[sheldon]

I have been trying to figure out the acoustical resonant frequency of objects. I think their are engineering formulas that could be used. For example if you have a 1x1x1inch block square of iron, at what acousticaly resonant frequency would it be tuned to? I imagine the shape would have something to do with it. For example if the iron block were a 1x1x1inch sphere, at what acoustical resonant frequency would it be tuned to? A tuning fork was designed to resonate at specific frequency of sound via the inherent physical properties its design contains. You can slap it and the tuning fork will produce the frequency of sound. Let's say the sound at the same frequency was produced via other method and directed at the tuned fork. Would it vibrate because the freq. is resonant to its own shape?

 

[arcnets]

sheldon,
here's what I believe to know about resonant frequencies, maybe you can use it.
0-dimensional: A pendulum has one definite resonance frequency. I think the tuning fork can roughly be treated as a (double) pendulum, and thus has similar properties.
1-dimensional: An ideal string has a basic resonance frequency. Plus the harmonics 2f, 3f, and so on.
2-dimensional: Drum skins, cymbals, bells, have a basic resonance frequency. Plus higher ones which are not harmonic. The design of a proper church-bell, for instance, is AFAIK a sort of secret art, passed down from master to apprentice...
3-dimensional: When it comes to solid bodies, resonant frequencies will be packed tightly. In analogy to molecular spectra, we must talk of 'bands' rather than 'lines'. I think it makes no sense talking about 'The resonant frequency' of a massive cube or sphere.

However, it's an interesting problem.

 

[Tyger]

It's an interesting topic

but really pretty complicated. In solid bodies sound has two speeds, not directly related so they have to be obtained from experiment or tables. The transverse speed and the longitudinal speed.

And you have to use Fourier Analysis and eigenvalue equations to analyze the different resonant modes of the shapes.

There is a Dover paperback by Bierley that has the solutions to many of these problems. It's an early introduction to the use of eigenvalues and quite an interesting little book.

 

[damgo]

Naa, you can find 3-d acosutic resonances. The technique is basically the same as you would use to find the resonant modes for electromagnetic waves in a cavity: you solve the wave equation

del^2 F = (1/v^2) * d^2/dt^2 F

with the proper boundary conditions, typically via separation of variables. For a homogenous isotropic cube, you will have frequencies of v0*sqrt(i^2+j^2+k^2) where i,j,k are integers and v0 is the frequency of a 2d plane wave with wavelength double the length of the cube.

For cylindrically-symmetric shapes you can use Bessel functions, and spherical harmonics for spherical shapes. For complex ones, you're prob best off using numerical methods to solve the diffeq/boundary value problem.

 

[sheldon]

http://mathworld.wolfram.com/Eigenvalue.html
http://farside.ph.utexas.edu/research/rfphtml/node45.html
http://www.phys.unsw.edu.au/~kryw/clectures/acoustic/Acoustic1279.html

Thanks everybody, I utilized your key words to look up a few sites. I really am way over my head and thought there would be a more simple equation. Or maybe I need it described in a different way. I will start with a question. If you figured out the resonant frequency of eather a sphere or cube of iron, would it be one specific frequency or a multiple level of frequencies and a different shaped sound wave to induce vibration into the object? I have heard of resonant frequencies of large structure to be devistating and engineers need to figure out the resonance of there designed structure in order to install buffers in optimal locations to keep the structure from vibrating apart?

 

[Tyger]

Let's take some examples

A cube: It will have basically two sets of transverse modes, which will be represented as standing waves, ones with integer and half integer wavenumbers. The will be simple harmnics, 1, 2, 3 etc..

A cylinder: It will have modes that are symmetrical and represented by Bessel Functions. You can just look them up in a table of Bessel Functions.

But these shapes also have many modes that aren't symmetrical or simple to describe. Also modes vary greatly in their ease of excitation, so the modes most frequently met with are those that are easily excited.