On Wed, 23 Jan 2002 08:52:44 John Delacour wrote: >At the bridge termination, the situation is different. The point >where the string terminates is not rigid, the soundboard/bridge >system is to a degree flexible and mobile and is not so massy as the >rigid termination. As a result, this end of the string is able to >move in all planes, and so it does. The distance it moves in any >direction is, of course, infinitessimal in the case of a piano >string, and this movement is periodic. Upon how much it is free to >move will depend how much of the energy of the string is reflected >back into the string and how much is propagated through the system. >The stiffness and mass of the termination will determine these >proportions. Agreed, except I would say that how much it is free to move is not dependent on string energy but on soundboard and bridge properties. How much it actually does move depends on string energy. > >The soundboard with its bridge, glued at its perimeter to a >relatively massive and stiff structure is quite clearly similar to a >drum or a diaphragm and when you thump it, whether with strings on or >without, it will produce a fairly distinct note at a certain >frequency. This is the fundamental frequency at which it will >resonate. It will also resonate at higher frequencies, though the >relationship between these frequencies will not be the same as the >relationship between the partials of a vibrating string. Agreed. > >If we ignore the transient effect of the hammer blow for the moment, >and consider a string vibrating smoothly at say 220 c/s, what we have >is the non-rigid termination of the string exerting a periodic force >at the bridge in various directions. Sometimes it will be _trying_ >to move the bridge up and down, sometimes sideways, sometimes >diagonally, and always _trying_ at the same time to rock the bridge. > Agreed, except that I would not say that the termination of the string is exerting a force at the bridge but that the string itself is exerting a force at the bridge. >As a result of this movement, that point of the bridge in contact >with the string at the termination, will be forced to move with it. >If, for simplicity's sake, we consider only the up and down movement, >the molecules next to the string will be forced, 220 times per second >to move up and down. We can now consider a single ray or column of >molecules extending from the string termination at the top of the >bridge through the glue line to the underside of the soundboard. The >length of this column will be, say 40 mm. This column of molecules >is elastic and is one dimension of the elastic medium that comprises >the bridge. If I press on the unloaded soundboard (no strings) it is >clear that every molecule of this column will move downwards at >exactly the same speed and the distance between the molecules on the >column will not change; no force is being applied to alter their >relationship to one another. The column is moving as a body. This doesn't seem consistent with your stack of magnets analogy to me. Even in an unloaded board, the bridge has stiffness and the board has stiffness, so you're offering some resistance to the bottom magnet. If we're going to talk about the molecular level then when a force is applied (one time - from zero to full load - no cyclic load) to be consistent, a pressure wave has to start at the top of the bridge and work its way through the bridge and soundboard. So it seems to me that the molecule at the bottom of the bridge has to move slightly behind the molecule at the top of the bridge. It's not clear to me that there is a difference between this situation and one in which the force is applied every 1/220 second. I believe this point is one of the sources of disagreement - that the soundboard behaves differently under a static load that is does under a dynamic load. > >The case is quite different when we consider the periodic force >acting up and down at the top of the bridge when the string vibrates. >For reasons of stiffness and mass or inertia, the whole column of >molecules cannot move as a body; and yet the molecule at the top is >being forced by the string to move up and down in relation to the >bottom molecule. It therefore gets closer to the molecule below it >and this molecule opposes this approach by moving downwards and so on >through the column to the bottom. Thus a wave of pressure moves down >through the bridge and this takes time. The time taken for this wave >to travel down the 40 mm column of beech or maple molecules will be 7 >or 8 microseconds and at any instant during that time the >relationship between the molecules will be different; some will be >pressed closer together and others will be moving apart from each >other. When the string stops moving down and starts moving up again >(every 4545 microseconds), the top molecule will be stationary while >other molecules will still be moving either up or down. A few comments here. The top of the bridge starts to move before the bottom of the bridge. The bottom of the bridge moves 7 or 8 microseconds behind it but is then moving too. I would call this bodily movement of the bridge. If this is the reason for saying that the bridge does not move bodily and does not move the soundboard bodily then there would seem to be little disagreement about what's going on except over definition of terms. You say that some molecules of the bridge will be pressed together and some will be moving apart? Why is that? On the first downward cycle of the string it seems that all the molecules will be pressing together. On the upward cycle of the string the molecules would be pulling apart? So, as I see it, what you're saying is, on the initial downward cycle of the string, the string exerts a downward force on the top of the bridge, which causes the top of the bridge to start moving, the bottom of the bridge moves shortly thereafter, and the soundboard follows as the load or pressure wave propogates to the rim. This sounds like the string moving the bridge which moves the soundboard. On a macro scale we would see that as a result of the string moving down the top of the bridge moves down but lagging somewhat the downward movement of the string. The words that some people would use for this are that the string is moving the bridge and soundboard. This would also seem to be the behavior described by vibration theory for a one degree system with a mass and spring. I thought there was some disagreement about whether such a model was correct for this situation. > >If it were possible to isolate this phenomenon from everything else >that is going on (for example the transverse and flexural vibrations >of the system at its natural modal frequencies and others), as it >probably is, then two accelerometers or laser devices, one at the >string termination and one under the bridge would register different >voltages separated in time by 7 or 8 microseconds. And what does the output from the accelerometers mean? Does it indicate that those points are physically moving? I thought this idea was scoffed at before? >This mechanical energy is transferred >through the system to end up as pressure waves in the air, or audible >sound. Agreed. I think there has never been a dispute about this. Phil F
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