Sound waves(The behavior of soundboards)

[Robin Hufford]

John D.
     The point of the post was to contrast, a little, the differences
between the frequency of the standing waves on the string and the wave or
phase velocity of the traveling wave itself and thereby to suggest that
inasmuch as these are different  an implication exists that at a boundary
discontinuity the part of the small part of the  traveling wave that
passes this boundary cannot be the part driving the board as its frequency
is different with respect to that of the standing waves which we hear.
     This seems a subtle but significant ramification of the questions
arising from the bridge/string interaction as by it we should be able to
rule out  the passing part of the wave as being the driver of the board,
as it were.  In excluding  this component one is left with the action of
the standing wave, something the proponents of the bridge motion view of
course would claim they have intended to be the case, and something we
argue too,  as the principal mechanism of  moving the bridge.  The
mechanism of this influence is the point of contention.
     Since the passing wave has been ruled out we must consider the
possibilities of mechanical action to account for the bridge motion point
of view.   Considering  the string to be operating as a cable under
tension,which seems to be part of  view  the bridge motion proponents will
argue, and  which case implies a shortening of the string length,  a
fluctuating pitch and other great complexities, I find this to be most
difficult to accept for reasons stated previously.       Taking the view
instead  that now a force is being exerted on the bridge then we are
forced to think of a lever.  I am greatly perplexed to comtemplate the
efficiency arising from a thin, highly flexible lever, operating from
short to substantial lengths,  and essentially pivoting from the front
termination, to move the relatively massive, inflexible bridge soundboard
system up and down as is claimed.   The kinetic energy of the hammer, due
precisely to the elastic action and nature of the wire loads the system
in a way such that cyclic strain develops in certain configurations on the
string.
       As to stress and strain, these terms are used, as is that of
impedance, in a kind of colloquial form here and I merely wished to point
out that thinking of force and stress as synonymous can lead to
problems;   from my view, part of those problems are demonstrated when one
believes that a string which is undergoing cyclic stress or strain as it
were can, in the case of a piano soundboard, operate as a force and induce
the proposed  motion which is under analysis here.  I am attempting to
offer a different view in this regard and acoustics, at the moment, is not
central to this objective.  I still try to restrain my commentary from
passing on to other subjects, such as Acoustic Radiation,  as I believe
the question taken up last month has not been exhausted, notwithstanding,
I am sure, the frustration of some who inversely relate worthiness of
expression to worthiness of analysis.
Regards, Robin Hufford
John Delacour wrote:

> 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