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
This PTG archive page provided courtesy of Moy Piano Service, LLC