At 9:57 AM -0600 1/30/02, Ron Nossaman wrote:
RH
> >I daresay I have done the same with similar use of physics and its
>principles.
>>It is a measure of the possible futility of this discussion that you seem to
>>unawares of this.
>
>I apparently am. I have read that if I had read anything about the more
>advanced aspects of vibration, I would understand your theory. I have, and
>I don't.
I will preserve an impartial stand on this matter, since I cannot
take sides with either you or Robin. I began this enquiry many weeks
ago with a vague idea of what sound is and have since learned a great
deal. In my episodic replies to Phil Ford, I have attempted to
present the picture of a tiny section of the bridge as I see it. In
all essentials I believe this simple picture not to be too
inaccurate. If we consider a steady periodic force (the string once
it has settled down) acting purely vertically at the string
termination, we will get a progressive compression of the bridge
travelling perpendicular to the soundboard, which will obviously
exert a normal force on the soundboard. The amplitude of this
_vibration_ as it reaches the soundboard and even its content may be
somewhat different from the amplitude and content at the string
termination, but that does not concern me here. There will also be a
lot else happening, a local flexure of the bridge in various planes,
the radiation and reflexion of compression waves in other directions
but the y axis, the setting up of bending waves in the bridge etc.
etc. but neglecting all these things we have a periodic force at the
frequencies of the string acting at right angles to the soundboard
directly below the string termination.
Rayleigh writes, at the beginning of Chapter IV:
"The displacements possible to a natural system are infinitely
various, and cannot be represented as made up of a finite number of
displacements of specified type. To the elementary parts of a solid
body any arbitrary displacements may be given, subject to conditions
of continuity. It is only by a process of abstraction so constantly
practised in Natural Philosophy, that solids are treated as rigid,
fluids as incompressible and other simplifications introduced so that
the position of a system comes to depend on a finite number of
coordinates"
Already this suggests we are not in an easy area. In fact the
mathematics required to go into any depth even in the consideration
of the vibration of the simplest thing, the string with no stiffness,
is not elementary. When we come to bars things get more complicated,
with membranes more so, and the most complicated of all is the
vibration of plates, let alone that of orthotropic or anisotropic
plates of curious shape such as the soundboard.
I have obtained a few books that deal with our subject but of all of
them, the ones that will, I know, be most informative to me, are Lord
Rayleigh's classic "Theory of Sound" and Philip Morse's "Vibration
and Sound". Both of these are very heavy works and include a lot of
very heavy mathematics, but they are written with the utmost clarity
and authority and it is therefore possible, with a great deal of
effort, to understand the principles without knowing all the
mathematics. As Philip Morse writes in his introductory chapter, "It
is important to realize that the mathematical solution to a set of
equations is not the answer to a physical problem; we must translate
the solution into physical statements before the problem is finished".
Now to return to the piano. Anders Askenfelt wrote:
"2. At the very onset of the tone there is a shock excitation of the
bridge as the first transversal wave on the string arives. This is a
general phenomenon which occurs in all systems when a signal is turned
on suddenly. This first wave excites all modes ("resonances") of the
instrument including all the soundboard resonances, and is heard as a
prominent "thump"."
What he is talking of here is the natural frequencies of the system,
and we have all read about the modal vibrations of the
plate/soundboard, seen the Chladni patterns etc. Most illustrations
show only the first few modes but there are theoretically an infinite
number of modes. The frequency of these modes depends on the
stiffness and mass of the system. None of these frequencies will,
except by pure chance, be in tune with any of the strings and we
don't actually want to hear these modal vibrations at all as such.
This initial impulsive force may be considerable, as when the string
is hit with a hammer and all sorts of resonances are excited, or it
may be quite slight, as when a tuning fork is brought into contact
with the bridge.
"This transient decays and after some time the partial frequencies of
the string dominates the driving of the soundboard and hence the
radiated sound."
We now come to the _forced_ vibrations introduced into the system by
the strings, as opposed to the natural vibrations of the system or
modal vibrations.
If we return to the bottom of my bridge, we have a vertical
displacement of particles acting at right angles to the board. The
board is to a degree flexible and consequently it flexes or bends,
and the periodic flexure or bending at this point gives rise to waves
of flexures or bends (the bending wave or flexural wave) which travel
outwards from the point, will be reflected, absorbed, dissipated etc.
Thus at its simplest, a progressive compression travelling vertically
down through the bridge at this point gives rise to a bending in the
system which travels away from the point as a ("circular") bending
wave. I am told that this means that
both surfaces of the board have the same displacement at any one time
due to such a wave. To this extent then, there is a similarity
between the vibrations of a string and that of a membrane and that of
a plate, except that the restoring force of a string is primarily
tension rather than compression and stretching, or bending. I could
hardly deny, of course, that a simple bending wave (if there is such
a thing as a simple one, which I doubt!) might be described as a
ripple!
What interests me now is to get a truer idea of the interaction of
the _forced_ bending waves with the _natural_ resonances of the
system. I'm some way along this road but not very far, but whatever
distance I have travelled, I have managed to do it without the
baggage of differential and integral calculus, which would probably
speed the journey but not increase the insight.
I'll leave you with an extract from a message I received from a man
working in the field of plate vibrations:
"I don't wish to discourage you, but if you start taking into account the
orthotropicity of the sound board then things very interesting! We can
no longer distingush between B, L and T waves but there may be
different types of wave which are "mixtures" of these! As an example of
the weirdness that can go on... Waves which are predominantly bending
have the interesting property that the direction in which the wave
travels and the direction in which the wave carries energy are not the
same thing. The energy of the wave may be carried in a completely
different direction to the physical wave disturbance."
Simple eh?!
JD
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