At 6:38 PM -0800 2/4/02, Robin Hufford wrote: > I am surprised to see DenHartog is still alive. He must be >quite old. In point of fact, he was Professor of Mechanical >Engineering and not a Professor of Physics. I will attempt to >contact him. At 8:36 PM -0800 1/13/02, Robin Hufford wrote: > 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: Robin, I haven't yet got Den Hartog's book, since Rayleigh and Morse are plenty to be getting on with. When it comes to dispersion in bending waves in bars and plates, both these authors deal with the question, though Morse seems to rely very much on Rayleigh. Rayleigh recommends Love's Mathematical Treatise on Elasticity (another Dover classic) for further reading and I have ordered this. One book which would certainly go into more detail and extend the treatment to orthotropic plates etc. would be Lothar Cremer's "Structure-Borne Sound", but this book is $175! A borrowing job. There seems to be a problem with presenting a simplified explanation of dispersion in physical terms, though the mathematical principles are not in dispute. Dan Russell's site gives a simple demonstration of dispersion in a bar (which is the same as in a plate, except that the thickness of the plate needs to be factored in) but to actually quantify the wave speeds in a piano soundboard at different frequencies would probably be impossible at present. I don't really see how it would benefit us to know in any case. The principle is clear and well documented for over 100 years and is much used in practical work involving ultrasound - more so probably than in the consideration of audio frequencies. At 10:50 PM +0100 2/3/02, Richard Brekne wrote: >This would seem to present a situation where essentially the higher >the string partial, the quicker the corresponding soundboard wave >runs through the SB system... which my unschooled mind wants to >jump at wondering why I dont hear the high frequencies before I hear >the fundementals.... to put a point on it. Benade does not deal extensively with dispersion, but writes (p.348) "...as we go from one part of the scale to the other, the predominantly excited components have different frequencies and therefore different 'spreading times' across the soundboard." Since the waves are travelling at several kilometres per second, we're not going to notice the high notes arriving first, though they will decay sooner for this reason and others. Dispersion is also observable in water waves and a visual demonstration could probably be set up, though the nature of the waves is completely different from flexural waves. At 1:38 AM -0800 2/3/02, Robin Hufford wrote: >This incidence device...clearly shows substantial motion of the >bridge when the string has been struck and at the same time shows >plainly that deflecting the string mechanically by applying pressure >and causing it to be displaced statically, does not have an effect >of any similarity to that when the string is struck and vibrates >harmonically. Absolutely. And this relationship is dealt with at great length in all books dealing with vibrations. Rayleigh writes "That the excursion should be at its maximum in one direction while the generating force is at its maximum in the opposite direction ... is sometimes considered a paradox..." The only way to resolve the paradox is to read and study how such a situation can arise, and simple examples exist as an introduction to this study. The relationship between the phase and amplitude of the vibrations at the string termination and those at a point on the soundboard just below has no similarity to the static "equivalent", and that is basic to the whole consideration of vibrations. When in addition we factor in all the other vibrations both natural and forced to which the point is subject, it is seen to be absurd to consider that the excursions of the soundboard, even at the point in question, in any way follow the excursions of the string point for point and angle for angle moment by moment. The frequencies of the string are, it goes without saying, transmitted to the soundboard and radiated into the air without any alteration in frequency, but the mechanisms by which this happens are what we have been trying to discuss. My rubber bridge experiment was designed to present a practically visual demonstration of the principles without straying too far away from the piano into such simple (but relevant and useful) pendulum demonstrations as Benade begins with. There has been no response to this demonstration and all we hear is the same tired mantra "The string moves the bridge and the bridge moves the soundboard", as though that empty statement were supposed to provide all the enlightenment nedded for an understanding of the acoustics of the piano. JD
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