Sound waves(The behavior of soundboards)

[John Delacour]

At 7:34 PM +0000 1/17/02, Phillip L Ford wrote:

>John,
>I don't have a problem with any of this.  Please proceed.

OK, to continue -- though I've gathered quite a bit on new and very 
interesting stuff in the meantime.

At 7:44 PM +0000 1/15/02, John Delacour wrote:
>.....If we move the top magnet a certain distance and maintain the 
>pressure, the magnets will (very fast but _not_instantaneously_) 
>take up different positions in respect to each other.  Each magnet, 
>or particle will have been displaced a different amount and the ray 
>will be in a state of compression.

I don't know how other people's minds work, but I'm one who needs to 
visualize models and more than that actually make things and 
experiment during designing things.  I very rarely begin with the 
drawing board and an idea; the drawing board comes second and the 
maths comes in when its needed.  Anyway, with this soundboard thing, 
I found that the only way to get a proper mental picture of things 
was to see the whole system as particles subject to forces; that way 
I can magnify the picture in my mind or on paper and see whats 
happening.  Things like the animated gifs at

<http://www.kettering.edu/~drussell/Demos/waves/wavemotion.html>

which Stephen mentioned a few weeks ago are most useful to me because 
they cut through all the calculus and present a very simple picture 
of what's going on in certain types of wave.  It's a pity he doesn't 
go a bit further.

So...I have so far given an analogy of a ray of magnets/particles 
held in equilibrium in a perspex or glass tube by a repulsive 
magnetic force.  We press on the top magnet and one by one down the 
tube the other magnets are displaced downwards, except for the bottom 
one which comes up against a rigid obstacle at the moment.  Before we 
leave this analogy, let's immerse the tube in a tank of syrup and 
press down the same amount and the same distance on the top particle. 
The magnets will rearrange themselves exactly as before but this time 
very much more slowly.  It will now be possible to observe the wave 
of pressure reaching each particle in turn as it travels downwards.

PARTICLES IN AN ISOTROPIC MEDIUM

o o o o o o o o o o o o o o o o o o o o o o o o o o o
  o o o o o o o o o o o o o o o o o o o o o o o o o o o
o o o o o o o o o o o o o o o o o o o o o o o o o o o
  o o o o o o o o o o o o o o o o o o o o o o o o o o o

In order to get a more useful picture of an isotropic elastic medium 
and to allow us to consider it in two and three dimensions, let's now 
get a tank of transparent liquid silicone rubber and suspend lots of 
ball bearings equally spaced in the fluid until it sets.  I'll now 
refer to the balls as particles.  The silicone rubber is, of course, 
the forces that keep them in equilibrium, equally spaced one from 
another. The distance separating the particles will be infinitessimal 
in comparison with the ascii picture drawn above.

The density of the material and Young's Modulus of elasticity for the 
material will determine how it behaves and how fast waves will move 
through it.  A disturbance of any kind to one particle will upset the 
equilibrium existing between it and its neighbours, and this 
disturbance will be passed on through the medium as a wave. at the 
instance of any stress, the particles will tend to restore themselves 
to a position of equilibrium, so if you imagin the picture above is a 
12 mm thick sheet of our stuff and you curl it round a hammer-head as 
you would hammer felt, then the particles at the bottom will be 
forced closer together and those at the top pulled further apart. 
When the force is removed, the internal forces between the particles 
will resore the sheet to its flat state.

If you whack the left end of the sheet, the particles at the end will 
push the neighbouring "column" of particles to the right and bounce 
back and this wave of column-pushing will proceed along the sheet at 
a definite speed.  See the Dan Russell animation of this.

Soundboard wood behaves very differently along the grain and across 
the grain and would be considered roughly 'orthotropic' as opposed to 
isotropic (same in all directions) and anisotropic (different in all 
directions).  As a result there are varying values of Young's Modulus 
for spruce and the speed of longitudinal sound waves in the material 
will be far greater along the grain than across it.  This is why I 
find it useful to think rather of a homogeneous, isotropic system 
first and introduce the complications of the real wood later, since 
the priciples are the same.

However, the speed of sound (longitudinal wave speed) along the grain 
of a plate of spruce (as oppused to a bar or rod) is maybe 5,000 
metres per second and is related to its elasticity and density as 
follows

CL = sqrt( E / (rho * (1 - mu^2))

CL = Wave speed
E = Young's modulus for the material (effectively, it's stiffness)
rho = material density (0.33 for Sitka spruce)
mu = Poisson's ratio (depends on the material but say 0.3)

The speed of a Bending Wave (or Flexural Wave) is directly related to 
the longitudinal wave speed, and consequently to stiffness; but also 
to its frequency.  I will come to bending waves later on, when I've 
got a better picture not so much of how they look as how they are set 
up.

I will repeat that I find it useful to see all waves as what they 
are, namely phenomena that happen to particles of a medium in reponse 
to forces.

Let me know if all that makes sense.  In the meantime here are a 
couple of URLs of limited interest, but which deal with some of the 
quantities I've mentioned.

<http://www.fpl.fs.fed.us/documnts/pdf1998/ross98d.pdf>
<http://www.fpl.fs.fed.us/DOCUMNTS/pdf1998/liu98a.pdf>
<http://www.fpl.fs.fed.us/documnts/pdf2000/liu00c.pdf>
<http://www.ndt.net/article/apcndt01/papers/988/988.htm>

JD