Sound waves(The behavior of soundboards)

[John Delacour]

Robin,

Thanks for your latest two post, which seem to be leading towards a 
proper explanation of the way things work.  I am still awaiting 
delivery of Philip Morse's "Vibrations and Sound" from bn.com and am 
looking forward to getting deeper into this.


At 12:33 AM -0800 1/7/02, Robin Hufford wrote:

>The compression wave generated by the standing waves pulses periodic 
>energy into the bridge as a cyclic, local rate of strain that 
>propagates through the medium.  These are alternating compressions 
>and rarefactions.  In so doing reflection and stress concentration 
>then occur just as they did with the transverse pulse on the string, 
>that is, they occur through the medium of superposition of the 
>traveling, now longitudinal, and periodic waves.  Incidentally, the 
>wave velocity of the transverse wave on a stretched string is the 
>square root of the quotient of the tension and mass density;

I must say I find this terminology a little unsure.  The 'mass' is 
the product of the volume and the specific gravity (relative 
density), so "mass density" to me is tautologous and basically 
meaningless.  What you term "wave velocity" is what normally is 
termed 'frequency'.  The formula also involves the length, giving the 
Frequency as the reciprocal of twice the Length multiplied by the 
square root of the Tension divided my the Mass

F = 1/(2 * L) * SQRT(T/M)

>  that of a compression wave in a solid medium is the half of the 
>square root of the quotient of the Modulus of Elasticity and the 
>mass density.

Here you are speaking of the natural frequency of longitudinal 
vibration of a stretched string, and here I would say the velocity is 
the square root of (Youngs Modulus E divided by the specific gravity 
rho)

V = SQRT(E/rho)

and that the Frequency is given by the formula

F = 1/(2 * L) * SQRT(E/rho)

This mode of vibration is peripheral to our present discussions and 
not really under consideration, but to pick up on an earlier thread 
where we were talking of this kind of vibration, it might be 
interesting at some point to consider the way sound at audible 
frequencies is imperfectly reflected (as opposed to ultra-sound) and 
this might explain a problem Stephen Birkett had with calculations 
involving the longitudinal mode...but that's by the by and not really 
relevant at the moment.  The longitudinal vibrations we are talking 
about now are at arbitrary superimposed frequencies and Young's 
Modulus for fir and beech are presumably nowhere in the picture :-)


>In these discussions a clear agreement as to what in fact stress 
>actually is should be had by all and requires some imagination. 
>Stress is not simply a force and as such does not obey the laws of 
>vector addition.  Stress requires both the idea of a force and a 
>plane visualized as cutting a section of a body to be correctly 
>understood and as such it is, in fact, force per unit area and 
>dependent upon the arbitrary angle of the plane chosen to cut the 
>body. Equilibrium has to be maintained through the imaginary cut 
>section by placing parallel forces operating across it.  The forces 
>operating through the cut section, will have moments if the cut 
>section is oblique; the effect of these forces cannot be specified 
>without taking into account the angle of the cut section relative to 
>the body in order to comprehend the effect of the moments.  This 
>distinguishes stress from a force and requires more complicated 
>methods to be expressed mathematically.  These methods are tensors...

One day I really must get to grips with the Calculus!  However 
Einstein successfully managed to explain his Theory of Relativity in 
a very readable book for the man devoid of calculus, so I guess there 
are also nice easy ways of getting the idea of stress and strain 
across.  None of us needs to be "blinded by science".


>The compression waves pulsing into the bridge travel preferentially 
>according to the characteristics of the wood.  Traveling through the 
>bridge, ribs and board they are distributed and reflected whenever 
>the reach the end of the board, whether free or attached.
>
>   During this process the inhomogeneous and obstructive nature of 
>wood causes stress concentrations and localizations particular to 
>the particular soundboard assembly under consideration.  The 
>subsequent superposition of these traveling longitudinal waves 
>creates, in a manner analogous to that of the transverse wave on the 
>string, standing areas, as it were, or resonances, free vibrations, 
>modes, etc.

Yes, and this is where things get interesting as we approach the 
concept of Acoustic Radiation; but between the bridge pin and the ear 
there's a lot of complicated stuff happening and it's the stuff in 
between that I think we can slowly work towards understanding. 
Whether rightly or wrongly, I see the notion of the propagation of 
the sound in the bridge as _relatively_ simple to grasp, though a 
full understanding of what happens would obviously be less simple. 
On the one hand, as Del has said, the whole system should be regarded 
as a unit, and so it is, but it's a very heterogeneous unit.  On the 
other hand, if too much consideration is given to its heterogeneity, 
it might take a lot longer to get a grasp of the main principles. 
I'm wondering if it might be useful to imagine, for the purposes of 
clarity, a bridge/soundboard/rim system or 'unit' composed of an 
unspecified homogenous elastic material, eliminating from discussion 
such things as ribs, different acoustic velocities etc. so as to have 
an idealized (certainly not ideal!) model to consider.  This would 
enable us to look first at the behavior of the pressure waves, which 
is to say the oscillatory movements of the particles of the material, 
at different points in the system.  At the same time I think it would 
simplify things if detailed discussion of acoustic impedance were 
eliminated.

To put it another way, what is happening at the string/bridge 
interface (our starting point) and what is happening, who can say 
exactly where, at the surface of the soundboard as regards acoustic 
radiation, are two different things.  What interests me is to trace 
the path between the two phenomena and, having established a clearish 
picture of the main phenomena, then to introduce variables to colour 
it in.

JD