At 5:52 AM +0000 1/23/02, Phillip L Ford wrote: >Robin, >[...]As you and Den Hartog point out the end of the string is a node >and by theoretical definition can have no motion. I don't have an >explanation of how this is consistent with the idea that the string >is physically moving the bridge. However, as someone else pointed >out (and I think you agreed) if the end of the string did have a >small amount of movement it could still act effectively as a node. >You say that the motion of the string does not necessarily imply >motion of the bridge. Others say that the motion of the bridge must >occur for the system to work. Can you propose an experimental means >of verifying directly on a piano that motion does not occur or that >if it does it has nothing to do with sound production? Phil, I'll leave Robin to answer in his own way, but I'd say the explanation is quite simple to make, so I'll have a go. At the string's termination at the stud (agraffe) the wire is prevented from moving up and down (or sideways) by the very massive and very stiff nature of the string plate. As a result only an extremely small amount of mechanical energy is transferred into the metal structure and this is essentially rigid. Almost all the energy is thus retained the string. If the string is stretched between two such massive and stiff terminations, it will continue to vibrate for far longer than in a piano and its energy will slowly be dissipated in heat through internal friction of the wire and through the displacement of some air. A string of no thickness between two perfectly rigid terminations in a vacuum and not subject to gravity, would presumably vibrate for ever. 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. 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. 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. 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. 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. 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. This is only a tiny part of the story, of course, viewed in one dimension. Some people have imagined I have ascribed all acoustic phenomena in the piano to compression waves because I stated repeatedly that sound waves are pressure waves or longitudinal waves, as indeed they always are. That is a problem for them, and doesn't concern me. In fact I have never done so. What happens at the string termination is a periodic pressure change in varying directions resulting in a periodic change in velocity of the particles at that point. According to the properties of the woods, the moments of forces etc., this mechanical energy is transferred through the system to end up as pressure waves in the air, or audible sound. Depending on the direction of the forces, different sorts of internal stresses will be set up in the wood. What begins as a periodic disturbance of molecules of wood and steel at the top of the bridge ends up as the periodic disturbance of millions of molecules at the surfaces of the soundboard.
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