Ron N,
I must say, in all of this heavy weighty argumentation that when I encountered
your comment about salmon, I had a most pleasant visceral laugh, not at your
argument but rather the appearance, or should I say the return, of the salmon so
unexpectedly. Was it oscillating or had it actually traveled? Now, putting levity
aside, to continue.
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; 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.
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 and, in order to avoid these complexities stress, which is force per
unit area can be replaced with total force which is force times the area. In this
way ordinary vectors may be employed. I point this out to emphasize that stress is
dependent upon the area of a section of a body and is not simply a force operating
on a body. The importance of this is that stress can become concentrated, or
lessened in localized areas, and can be distribute in a kind of variable way
through a medium even though we ordinarily think of a force as acting on or through
a body in a kind of consistent, uniform way. The distribution of stress is of real
importance for vibrating bodies.
In the case of a piano string the cut section may be perfectly transverse,
that is at ninety degrees to the length of the wire, or it may take any other
configuration as long as it sections the wire. If oblique or perfectly transverse,
the imaginary cut section, which obviously in real life does not exist, evidently,
as the wire is not actually cut into two parts, is in equilibrium. The set of
forces per unit area, placed upon this or any cut section, along with the angle
of the section if it is not 90 degrees to the line of action of the forces, which
maintains equilibrium across the cut section are the stresses.
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. Furthermore, as the transverse flexing areas of the
soundboard, radiate sound away through the air a similar thing happens now in the
room when the sound, again reflects upon itself. None of this is, particularly,
original with me but, I think, is well known in wave mechanics. Your blanket
analogy was insufficient, in my mind, as it ignored the extremely important
consequences of reflection.
Regards, Robin Hufford
>
>
> "the blanket analogy. Since the soundboard doesn't have the depth of a pond,
> and the bottom side tends to follow the top side at any given point, the
> internal compression waves are a much smaller part of the overall motion
> than the transverse, and are of considerably less consequence as a result."
>
>
> I say they are the reason for the transverse, modal behavior or resonances as I
> have described above.
>
>
>
>
>
>
.
>
>
> "Yes, you have said this repeatedly. What you haven't explained is how this
> compression wave moves the board rather than the periodic forces (cyclic
> load) of the transverse string vibrations moving the bridge, which moves
> the board. I say the vibrating string pushing and pulling on the bridge
> moves it and the board just like anything is moved by applied force, you
> say it does not, but the compression wave resulting from the pushing and
> pulling passes through the unmoved bridge and moves the soundboard. This is
> the original point of contention. How is this possible?"
>
By a method that is, essentially the same as happens on the string itselfs and
which also happens in the air.
>
> Regards, Robin Hufford
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