Phil,
Preferring, as I do, discussion centering around ideas, facts and the implications of
these, I have only reluctantly quoted so called "authorities" in this debate on soundboard
behavior, and have done so, if memory serves, in two or three instances.
Now, since I have been accused, apparently, of dangerously disrupting the consciousness
of the entire world with error and recklessly spreading misconception on the Internet ( Oh
Gosh, the Horror of It All!), I have taken the time to compile a set of quotes pertinent to
the point of view advocated by myself and others. It would seem that it should be incumbent
upon the proponents of the contrary point of view to be aware of the points in these and
numerous other similar quotes, evidently, judging from the specious arguments they offer, they
are not. It is with some trepidity and sense of wasted effort then that I take time from a
demanding schedule to abstract from the literature that which they, were they efficient
students of the subject, would already know and give evidence of, and are, no doubt,
determined to deny the relevance for. However, as pianos are a subject dear to my heart, as
it must be for them, which is something I respect them for even when I think they are wrong
then I will hazard the effort.
Now quoting:
"MODERN PHYSICS" by Dull, Metcalfe and Williams, p.292, "Certain parts of a string
vibrating with a standing wave pattern never move from their equilibrium positions. These
points are called nodes. ..... When standing waves are produced in a stretched string,
neither END (caps mine, rh) of the string can move. Thus in such a vibrating string, both
ends must be nodes. ....In a standing wave, energy is not transferred past the nodes where the
particles are continuously at rest."
This is a rather rudimentary book. It should be noted that the context of the quote is
that of transverse wave behavior and the manifestations of its energy.
"THE PHYSICS OF SOUND" by Richard Berg and David Stork, p. 65, "The sum wave is
standing, that is, progressing neither to the right nor to the left, but continually
oscillating back and forth between configurations labeled (1) and (5). Its displacement
varies between zero (no visible wave) and twice the amplitude of either of the two component
waves. ....Points ....where there is no motion of the rope...are called nodes and labeled N
(remember NO displacement.)" ( the caps are mine, the book merely italicizes it). I do not
need to be reminded that a rope is not a piano wire. - rh.
p. 67, "this discussion has involved only transverse waves in a rope or similar
medium.....we shall see that such standing waves are basic to the production of sound in many
musical instruments."
p. 67, " ...Now we shall look at what happens to waves at the end of a rope and then
study the standing waves produced in a short section of such a medium. ....The "force" of the
pulse hitting the fixed end of the rope causes the pulse to change from positive to negative,
a 180 degree phase reversal. The wall pulls down on the end of the rope to ensure that there
is no displacement at the end. This downward force can be thought of as generating a negative
pulse traveling away from the end, the reflected pulse. No such phase reversal occurs when
the pulse reflects off a free end, because there is no downward force exerted on the end of
the rope."
p. 68, "...It is now apparent that to form a standing wave it is not necessary to have
two initial waves; a single wave reflecting off an end will suffice. There must be a node at
the fixed end, since that end is constrained to ZERO DISPLACEMENT AT ALL TIMES" (caps mine,
rh)
These two rudimentary books give a brief, only lightly mathematical treatment to wave
mechanics. It is incumbent to all that profess to be experts on the subject to comprehend
what these, elementary approaches themselves imply, notwithstanding casual observation and
that is that a standing wave NECESSARILY IMPLIES RIGID TERMINATIONS AND THE ASSOCIATED
REFLECTIONS ARISING THEREFROM. Otherwise, the standing wave itself cannot exist.
Additionally, a critical distinction must be drawn once again: These matters are
concerned with a local, traveling wave, its reflection and the consequent standing wave that
develops thereby. When the local pulse hits the bridge, the evidence of its reflection and
progressive superposition, is, as I have said before, the harmonic development on the string,
something that exists in a rich degree on a piano string. The soundboard supplies a force
pulling downward when the pulse pulls upward and vice versa, resulting is a phase reversal,
this is its work in this context. The string pulls approximately, up or down, the board
contrarily, resulting in reflection.
Incidentally, the speaking length of the fundamental of a piano string is but half the
wavelength. A wave requires 360 degrees of cycling to complete its period. The fundamental
standing wave on a piano is but 180 degrees and as such implies a node at its ends, as should
be evident from the quotes above. To emphasize, a node exists at the ends of the standing
wave, this point is motionless. That is not to say, however, that forces do not exist there -
another subject.
A far better book, but one substantially more mathematical is J.P.Den Hartog's
MECHANICAL VIBRATIONS. This lucid book has a much better treatment of the entire subject of
vibrations, free and forced and both damped and damped. It also includes analysis of another,
less encountered type known as self-excited vibrations. Of particular relevance to the
confused articulations of some on this list regarding transverse and longititudinal waves and
their capabilities is the chapter entitled "Many Degrees of Freedom", that is Chapter IV, in
which will be found on p. 135 art 4.4:
"Strings and Organ Pipes; Longitudinal and Torsional Vibrations of Uniform Bars. These
four types of problem will be treated together because their mathematical and physical
interpretations are IDENTICAL" (caps mine, rh). There follows then the treatment of these
subjects. As to whether longititudinal vibration can, as I have, said transmit energy in
"equal measure" with the transmissions of transverse vibrations, and could do so at the
bridge and this be transmitted through the soundboard, get the facts, guys. You can find them
in the chapter referenced above, among many others. .
In giving consideration to approximations of the possible shapes a stretched string may
take in its free vibrations, using Rayleigh's method, Hartog posits as one extreme a
parabola as being the shape of the fundamental but then through a series of reasoned
mathematical expressions shows that this cannot be completely the case. He then says on p.
144 "The spring effect driving a particle dx of the string back to equilibrium lies in the
curvature( this word was italicized which is not rendered here, rh), or d^2y/dx^2, of the
string. At the ENDS (caps mine, rh) the string particles do not MOVE so that there they have
obviously NEITHER INTERTIA FORCE NOR SPRING FORCE."(caps rh). I ask all, how do you suppose
the bridge to be made to rock then?
He then proceeds to contrast the parabola with a string lifted at its centerpoint to
produce a triangular shape and shows that this is equilavent to loading the string with a
central mass. Reasoning similarly he shows that the resonances or shapes of the free
vibration will be as before but that the kinetic energy will be greater; the frequency less,
and observes that that an impacted string must have a shape somewhere between these extremes.
With regard to the equivalence of effect of longitudinal and transverse vibrations note
p. 144, "...we shall solve the combination problem of a heavy string of total mass M, in the
middle of which is attached a single concentrated weight of the same mass M. this problem is
AGAIN EQUIVALENT (caps, rh) to that of the longitudinal (or torsional) vibrations of a bar
clamped at both ends and having a concentrated disk in the middle with a mass (or moment of
inertia) equal to that of the bar itself."
On p. 136, "The problem of longitudinal vibrations in a bar(italicized) is quite
similar to that of the string..." He then proceeds to derive the partial differential
equation of the bar and labels it (4.21); that of the string is labeled (4.20). On p. 137 when
speaking about the resulting equation for the bar, that is (4.21), one finds "...This is the
same differential equation as (4.20)."
Another, more mathematical but less general treatment of waves can be found in PHYSIC OF
WAVES by Elmore and Heald. With regard to longitudinal waves there can be found on p. 47
"...When a wave traveling on a string is reflected or absorbed in some manner, the momentum it
carries is either reversed in its direction of flow or is transferred from the wave to the
external bodies that serve to ABSORB (caps mine, rh) the energy of the wave. We therefore
expect, from Newton's second law, that in either event the wave must exert a longitudinal
force on its surroundings equal to the rate of change of momentum of the wave in its direction
of propagation. " Note that the energy of the wave is said to be absorbed; we are not
speaking of deflections.
In order to illustrate the nature of elastic action in a stretched wire in contrast to
that of static deflection and the implications arising thereby, the following quote is offered
from PHYSICS OF WAVES p. 46 "....we must replace ( partial differential equations are quoted
in which the difference of the two is their treatment of tension versus deflection, rh) in
order to bring into the calculation the fact that we are dealing with a transverse wave
obeying (the usual wave equation,rh) and not simply with a static state of sideways
displacement of the string brought about by some external system of forces applied to the
string in the y direction."
In my opinion it is critical to grasp the importance of this distinction, both as regards
the nature of the elastic action of the string itself, and, in particular, its effect upon
the bridge/ soundboard. I have attempted to make the point repeatedly that the
bridge/soundboard would be better viewed with these distinctions in mind. In this context I
would argue that the kinetic energy transferred to the string by the hammer has gone into the
elastic strain of the wire, which then pulses through the bridge/soundboard interface as an
elastic wave. The supposition of an infinite stiffness is necessary for this to occur.
In these and numerous other books rigid supports are assumed as the starting point for
the discussion of traveling waves on a stretched string. Their efficiency enables the
subsequent development of standing waves. Where one can find standing waves then by some
measure superposition is implicit. In the case of a stretched string this measure is the
reflection of the traveling pulse. The standing waves do not move and they therefore are not
reflected. They are, in fact, a new equilibrium in the wire itself, created by the change in
energy of the string imposed upon it by the impact of the hammer. To quote again from Den
Hartog p. 126, "This gives the shapes of the vibration, or the "normal modes" ....These are
the only ...configurations in which the system can be in equilibrium under the influence of
forces which are proportional to the displacements x (as the inertia forces are)."
Insofar as the inaccuracy of the one degree of freedom used by Ron N as illustrating a
supposed scientific approach to this matter, which judging from the terminology used appears
to be followed by others sharing his view, the following quote from COLLEGE PHYSICS by Sears
and Zamansky will be helpful, p. 381, " We can now see an important difference between a
spring-weight system and a vibrating string. The former has but one natural frequency while
the vibrating string has an infinite number of natural frequencies....If a weight suspended
from a spring is pulled down and released, only one frequency of vibration will ensue. If a
string is initially distorted so that its shape is the same as any one of the possible
harmonics, it will vibrate, when released, at the frequency of that particular harmonic. But
when a piano string is struck, not only the fundamental, but many of the overtones are present
in the resulting vibration." Incidentally, this is a most clearly expressed book as regard
wave mechanics. Inasmuch as the one degree model is woefully unsuited for describing a
uniform, isotropic string with its complexities, how much more so must be the case when
applied to the far, far more abundant complexities of the rim/bridge/soundboard system
Stipulating for the moment that the string acts as a force, which in fact in a qualified
manner, I have never disputed, in regard as to whether this would in fact result in actual
motion, another quote from Den Hartog is useful, p. "If an alternating force acts on a mass
of an n-degree-of-freedom system, there will be n-1 frequencies at which that mass will stand
still while the rest of the system vibrates. As the end of the string is a node, it is not
required therefore that forces operating there produce motion.
I have made no attempt to quote from books on statics and strength of materials, although
much critically, pertinent information is to be found there also. A few posts on this subject
have contained such quotes.
Contemplation of the fact of the node represented by the end points of the string and its
implications will be clarified by considering those on the string itself. At this point in
things, I am not sure whether the statement that these particular nodes do not move would
provoke argument or not, although, in fact, they do not, but it is certain they illustrate a
point in which forces are at work, superpose and cancel resulting in, essentially,
compression of the medium without bodily motion. With the illustration of the fact that this
can happen, we are, as you say, back to where we started.
Regards, Robin Hufford
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