Sound waves(The behavior of soundboards)

[Phillip L Ford]

John,
I haven't had much time to spend with this for a few days.  I've read this
over.  I looked at the wave illustrations.  Just glanced at the wood references.
I'll read them when I have more time.  As far as what's written here I'm still
following you, so feel free to carry on.

Phil F

On Fri, 18 Jan 2002 22:28:42  
 John Delacour wrote:

>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
>
>
>