At 2:17 AM -0600 2/2/02, Richard Moody wrote: >| Dear Mr. Brekne, in this case the simplest process -- transverse motion >| of the soundboard -- is also the dominant one. ....... I think you can >| understand it better as the bridge giving rise to standing waves in the >| soundboard which are, in fact, what are called modes of vibration of the >| soundboard. >| >- GW > > I take transverse motion to be the simple up and down movement of the >soundboard rather like the back and forth movement of a loud speaker cone >or headphone diaphram. Well, your use of the word "simple" suggests to me that you have a mental picture of these things that is false, which is not surprising, since their modal vibrations are invisible. From the point of view of its natural vibrations, the soundboard is classed as a plate. For the first few modes of a circular plate, you can see three little movies at <http://webphysics.davidson.edu/Alumni/AnAntonelli/work/math/Chladni/ModeDemo.html> which will begin to change this mental picture. If you then consider that the plate is vibrating in an indefinite number of natural modes more and more complicated than these, you will get a picture of a surface with all the modes happening at once and superimposed on each other and being reflected from the rim in different directions etc. etc. > I wish "standing waves" would be better defined, but the interesting >concept is that the soundboard "vibrates in modes". Modes I take to be >what acousticians call the segments of vibrating string that we call >"partials" to differentiate them from "harmonics".... harmonics being a >tone resulting from a fundamental with "overtones" (harmonics) of perfect >ratios. Because the string is stiff, we are told, the partials are sharp >from perfect and this sharpness is called inharmonicity. The frequency of the modes of a vibrating string are (in the absence of any stiffness) in small integer ratios with the fundamental. 1:2, 2:3 etc. Since there is a small degree of stiffness or bar-likeness in any string, the upper partial frequencies will be slightly "stretched", but it is tension rather than stiffness which restores the string to equilibrium > Now if the soundboard is vibrating in modes and it is conceivably >stiff therefore shouldn't it have its own inharmonicity? Inharmonicity is not applicable or relevant to the natural vibration frequencies of the plate or soundboard. There is no exact ratio of the frequency of one mode to another, and there doesn't have to be. The exact frequencies of the modes does not concern us because these natural frequencies are not responsible for radiating the sound of the strings. If you tap the soundboard or disturb it in any way it will vibrate in all its natural frequencies, just as the skin of a banjo will vibrate as a drum, but both these vibrating things are simply "receptacles" for the ordered and harmonic sound that comes from the strings. > So, does the SB somehow reproduce the frequency of the string's >partials (inharmonicity), or does the SB vibrating in modes with its >own stiffness determine (more or less) the inharmonicity of the >piano? No, neither. The natural vibration of the soundboard is one thing. We then introduce "forced" vibrations from the strings which travel through the bridge and cause the soundboard to vibrate in a different way and at frequencies not its own. As you know, when you tap the soundboard, the sound of the tap dies away very quickly. The same thing happens when the soundboard is first disturbed by the vibrations from the struck string - all the modes or resonances of the board are excited, but any sound they produce is transient. The string, however, continues to deliver its harmonious messages to the soundboard and this causes travelling waves of various frequencies and speeds to emanate from the area of excitation and spread all through the board, reflecting from the rim etc. Since the periodic force from the string is mainly at a right angle to the soundboard, it bends the system locally and this periodic bending produces a wave, called a bending wave. It is these waves that compress and rarefy the air in contact with the surfaces of the board and thus cuase the board to radiate the sound of the strings. The two types of vibration, i.e. the natural and the forced, are not completely divorced from one another. The frequencies at which the board vibrates in its natural modes must cover the range of frequencies that we want to hear. Here is how somone recently described it to me >The soundboard must be flexible enough for it to have many natural >frequencies over the frequency range of the piano. If it is too >stiff then the natural frequencies will be too high. You don't have >to vibrate a structure at exactly its natural frequency for it to >resonate. If you are slightly to one side of the natural frequency >then resonance can still take place. The further you move away from >the natural frequency, the less resonance you will get. If the >structure is highly damped then you do not have to be so close to >the true natural frequency to get resonance. If the soundboard is >big enough and flexible enough and the damping is sufficient then >the modes (resonances) overlap and almost any frequency will >resonate. I posted a demonstration of this a couple of weeks ago. If you thump the soundboard of a strung piano, you will hear a recognisable tone, which might be the fundamental of the second F in the bass (as it is on the piano beside me here). If you then play this F and listen carefully, you will find it has a depth or resonance that you will not find in the C below it or the C above it, because its frequency coincides almost exactly with one of the natural frequencies of the system. One of my tuners said to me the other day how strange he found it that the second C# on a grand so often sounds especially good. I have not had a chance to see many grands since, but I would not be surprised if that C# turns out to be the frequency of a natural mode on many grand pianos. JD
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