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 > > >
This PTG archive page provided courtesy of Moy Piano Service, LLC