Richard and Ron,
A stretched piano string when struck by a hammer acquires a momentary deflection
upward which is dependent upon the hammer, the wire, the striking point, etc. When a
rock is dropped into a pond a similar momentary disturbance exists and can be seen at
the point of impact as a series of depressions of the surface with consequent
rebounding and decreasing amplitudes which disturbs the water and results on the
surface in a visible wavetrain departing from the region of impact. The analogous
event on a string in a piano is invisible, but nevertheless a momentary disturbance
comprised, due to the elasticity of the material and its tension, of among other
things, a local deformation and local tension occurs. In both cases, in string and
water, subsequently a series of pulses will now propagate away from the disturbed and
flexing point of impact and can, of course, be seen as ripples radiating outward in
every increasing circles on the surface of the pond. In the case of the water, this
is, of course, familiar enough; neglecting complications of the hammer and shank, a
similar event happens locally at the point of interaction between string and hammer,
although not visible to the eye as are the ripples on the surface of the pond.
Richard has clearly illustrated the three dimensional, compressional aspect of the
wave in the pond and accounted for its appearance on the surface as a ripple due to
the absence of reflection. He has clearly shown the future of such a wave - it must
propagate outward in three dimensions as the surface of a sphere and, unless reflected,
is doomed to dissipation as its energy is progressively distributed over the surface of
a sphere of ever increasing size. He has emphasized the fact that the wave is, in
fact, what it is, that is the energy of motion on a molecular level, moving through
the medium, not the medium itself, and that the particles of the medium itself have an
oscillatory motion, which is to say they oscillate around their neutral positions. In
so doing they transmit the motion of the wave itself, while they themselves, return for
the most part, to their neutral position.
In the case of the piano string, with the obvious caveat that a string is not three
dimensional as is the pond, similar, analogous events occur to a point. At this point
a new phenomenon is introduced as a result of reflection. As has been noted the
impact of the hammer causes, due to the elastic nature of the wire a local transverse
deformation and local change of tension which then propagates away in a series of
traveling waves; one moving toward the bridge and the other toward the front
termination, be it agraffe or Capo'd'Astro. This aspect is, in fact, similar to the
analogy of the water disturbance. The tension in the wires ensures that this process
is invisible to the eye. Absent reflection, on a sufficiently long string, the two
traveling wave trains would be doomed, just as in the case of the water waves, to an
inevitable dissipation.
On a piano, however, boundary discontinuities exist, imposing in a mechanical
fashion effective reflection on the traveling wave trains at each termination. Given a
rigid termination(and man, is this the rub!) this change of velocity will be a change
of direction, not speed and will be accompanied by an attendant change of phase of 180
degrees. That is, an incoming pulse will be inverted upon reflection. Were a pulse
reflected by the free end of a wire, which incidentally, would necessarily require a
stiffer wire than the music wire found in pianos to occur, a similar change in
velocity will be noted but the pulse, however, will not be inverted as occurs upon
reflection from a clamped end but will remain oriented upon reflection just as it was
upon incidence, that is there will no change of phase. In both cases, free and
clamped, velocity is reversed. It is evident that a measure of energy must be
required to reverse the direction of the traveling wave; in the case of the clamped end
occuring in the piano, this is supplied through the forces acting at the
bridge/soundboard termination and by the agraffe or capo and plate as the case may be;
in the case of the free end this energy is supplied by the elastic action of the medium
itself.
As the wave trains are reflected at the terminations and travel back towards the
midpoint of the wire, they pass through outgoing, incident waves still emanating from
the original, attenuating local disturbance comprised of local deformation and local
tension. They then encounter one another, pass through, reflect and continue as before
many times. By this mechanism of recurrent superposition of wave trains constructive
and destructive interferences occur as the phasing and phase angle relationships of the
original and weakening pulses cycle through augmentation, intermediary effects and
cancellation, resulting in a new but entirely second order phenomenon, one nevertheless
highly significant to us all. We are all aware, of course, of what this is and it is
what we can see, hear, tune, and are ordinarily concerned with; that is, the transverse
standing waves on the string. Other terms applied, in this context, to these waves are
resononces, modes of vibration or free vibrations, however one wishes to characterize
them.
These configurations develop on the string completely secondarily to the original
transversly oriented but longitudinally traveling wave. As they represent cyclical,
recurrent, superpositions of the original traveling waves they are the result and not
the cause of reflection at the terminations and are not reflected themselves as they do
not move along the string or one-dimensional waveguide. Their existence and
periodicity, as such, is a measure of the efficiency of the terminations at which
reflections of the traveling waves occur and the regularity of the medium. Were these
terminations unstable or moving to any appreciable degree they must degrade this
process and hinder the cycling of the harmonic structure on the string. While no piano
is perfect, nor is there any such thing as a perfectly rigid support, any appreciable
motion here would be highly deleterious to the entire process.
Incidentally, the wave equation for a vibrating string, which is a partial
differential equation of the second order, is the same as the equation for that of a
longitudinally vibrating bar struck on one end which transmits a longitudinal pulse or
compression wave to the other end of the bar; the one being expressed a rate of change
of transverse displacements relative to the unperturbed string, which is taken to be
the x axis; the other being the rate of change of density fluctuations moving along
this axis. As to the capability of a longitudinal compression wave to transport energy
in equal measure with that of a transverse wave there should be no doubt. The
transverse structure of the standing waves pulses, at the terminations compression
waves, as I have said before, into the bridge and thence they travel to the
soundboard. The wave propagation through the bridge and board creates a similar
reflection and recurrency which, in turn, causes sufficient motion of the board to move
the air in contact with it. This process is very similar, except now occuring on a
two-dimensional wave guide or membrane, as it were.
In the context, solely of the transfer of energy from string to bridge, I believe
the ideal must necessarily by such degree of rigidity that mechanically enhances the
development of the harmonic structure on the string, although this may be in fact, and
I believe, is, at odds to the requirements of the soundboard when it is viewed as an
acoustically radiating structure, where flexibility is critically necessary, another
subject in and of itself. Implicit in this rigidity is the requirement that the string
tension be maintained at a stable value, otherwise the wave velocity and subsequent
frequencies of the harmonic system on the string will be altered. Even though the
frequency stability of a string can be easily shown to vary with an ETD, I believe
this arises from other causes than rocking or flexing motion of the bridge, and this is
also another subject.
As the string continues past the bridge and bridge pin on one end, and on through
the Capo or agraffe on the other, no actual, rigid termination of the wire exists at
either point. Rather, the speaking length is terminated and as the wire continues on
through the agraffe, bridge or Capo, a boundary discontinuity is what, in fact, has
been encountered. This situation, on pianos, is essentially similar to that above.
Reflection occurs with somewhat reduced efficiency as some very small part of the
traveling wave may pass the discontinuity. This wave possesses some small fraction of
the energy of the original, incident and essentially reflected, traveling wave, which
itself is of a small magnitude relative to the mass of bridge and soundboard and is
not capable of physically, flexing or rocking the bridge or soundboard, nor is it the
mechanism of energy transfer as should be apparent from consideration of the fact that
its frequency is that of the original wave velocity and no any of those of the
harmonic activity on the string. The transverse standing waves generated on the string
are transduced by the stress transduction method I have described, apparently ad
nauseum and infinitum to some, and generate in essence localized, periodic stress
concentrations and dissipations at the string/bridge interface, that is alternate
compressions and rarefactions that then propagate in accordance with the normal
distributions inherent in the inhomogenity of the wood.
Regards, Robin Hufford.
Richard Brekne wrote:
> Ron Nossaman wrote:
>
> > >> So how could a compression wave traveling through the bridge possibly move
> > >> the soundboard, and what's the physics behind it?
> > >
> > >Actually, that bothered me a while as well, but it was this rock in pond
> > >example
> > >that your camp through out that got me on to it. And I think I understand
> > where
> > >that reasoning comes from now. Lets take another look at the pond analogy.
> >
> > And once again, you're talking about what happens in the soundboard after
> > something has physically moved it, not what that something was, and what
> > that something was is the entire point.
> >
>
>
>
>
> >
> > >I still don't really see how any of this is really so totally incompatible
> > >with the
> > >diaphragm idea. As I have said all along I would suspect the real truth to
> > >all this
> > >lies in some combination of these two rationale and probably some other
> > >things we
> > >lay folk and for that matter real physicists as well haven't grasped or
> > >thought of
> > >yet.
> >
> > At this point it has nothing to do with the diaphragm idea or anything else
> > besides the cause and effect between an applied force and a resultant
> > movement. We have not gotten one iota away from the initial assumption and
> > statement that an internal compression wave initially moves the soundboard,
> > which then moves the bridge. I'd like to know how that is supposed to work.
>
>
>
> >
> >
>
>
> --
> Richard Brekne
> RPT, N.P.T.F.
> Bergen, Norway
> mailto:rbrekne@broadpark.no
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