Acoustic resonance in a wooden rectangular box

[simon56modder]

I could have ask this on an other forum... but I thought it'd be cool to see how great the KSP community is

So I'm planning to build a celesta from scratch. For those who don't know what a celesta is :

Spoiler

it's an instrument that works and looks similar to a standard upright piano, but instead of strings it uses string plates over wooden resonator boxes.

So I want to draw the design but there is one thing I have to know before I start. I have to find out the size of the resonators. (note : each frequency/key on the instrument has it's own resonator)

I found this formula on the Acoustic resonance wikipedia page :

Quote

For a rectangular box, the resonant frequencies are given by

where v is the speed of sound, L_x and L_y and L_z are the dimensions of the box.  , , and are nonnegative integers that cannot all be zero.

but how do I apply it ? For instance if I want to know the size of the resonator needed for a 442Hz A (note : each frequency/key on the instrument has its own resonator), how do I proceed ?

Edit : One last thing : I don't know if it's important to know but the rectangular resonators are opened at the top - the side just below the metal sound plate.

Edited February 21, 2017 by simon56modder

[Vanamonde]

Looks like there's going to be some science in this thread. And so, moved to Science & (Not)Spaceflight. 

[Steel]

7 hours ago, simon56modder said:

I could have ask this on an other forum... but I thought it'd be cool to see how great the KSP community is

So I'm planning to build a celesta from scratch. For those who don't know what a celesta is :

  Hide contents

it's an instrument that works and looks similar to a standard upright piano, but instead of strings it uses string plates over wooden resonator boxes.

So I want to draw the design but there is one thing I have to know before I start. I have to find out the size of the resonators. (note : each frequency/key on the instrument has it's own resonator)

I found this formula on the Acoustic resonance wikipedia page :

but how do I apply it ? For instance if I want to know the size of the resonator needed for a 442Hz A (note : each frequency/key on the instrument has its own resonator), how do I proceed ?

Edit : One last thing : I don't know if it's important to know but the rectangular resonators are opened at the top - the side just below the metal sound plate.

It definitely will matter that the boxes aren't closed, but unfortunately I don't remember enough on resonances to tell you how it matters

[JayPee]

are l, m, and n the eigenvalues of the resonance or something?

[mikegarrison]

If one of the lengths is much larger than the other, then it will resonate as a so-called "organ pipe". Essentially the dominant tone will be a compression wave that moves up and down the pipe. If the three dimensions are closer to being equal to each other then you can get standing waves for each of the three dimensions.

[mikegarrison]

The equation you gave is for all the resonant frequencies of the box. The lower case l, m, and n are integers corresponding to different wave modes that fit in each of the three dimensions. The first mode looks like half a sine wave. The second mode looks like a full sine wave. The third mode looks like one and a half sine waves, etc.

The ends of the waves are determined by the boundary condition (whether there is a wall there or not). If there is a wall there, the pressure of the standing wave has to be at peak amplitude. If there is no wall there, the pressure has to be zero (ie. matching atmospheric).

Edited February 22, 2017 by mikegarrison

[simon56modder]

Thanks a lot @mikegarrison ! Considering all you said, don't you think it'd be easier to build organ-pipe-like resonators instead of wooden boxes ?

[cwaynemc]

@simon56modderreviving this as I have a similar project going.  How did you end up determining resonator size?

[K^2]

Usually, for something like Piano, mechanical resonances of walls, especially the soundboard, are more important than resonances of air inside the box, which is what the formula in OPs post refers to. So I don't know if any of this is going to be helpful.

I'm not aware of any shorthands. Historically it's been trial and error. Still mostly is, but these days, you can build an FEM simulation, and then, if you're looking for specific combination of acoustic and mechanical characteristics, you can run something like a genetic algorithm to select for the optimal shape. It's a lot of work on top of building an accurate model of the instrument, though. Not sure if it's worth it.

For something practical, my recommendation would be to find dimensions and materials of existing instrument of high quality and copy it. It's very difficult to do better than that.