At 9:33 AM +0100 1/21/02, Richard Brekne wrote: >I found this article that has a several basic definitions and some >good illustrations. Amoung them right at the start of the article a >good illustration of a transverse wave in solids. This illustration >along with some descriptions I have recieved recently from a few >kind souls satisfy one of the problems I was having with this >transverse wave explanation. > >http://www.geology.uiuc.edu/~hsui/classes/geo351/LectureNotes/waves.pdf I downloaded this paper and found no mention of bending waves. I was already doubtful before I downloaded it, having seen geology in the URI. In seismic phenomena there are types of wave that do not concern us. I have sent you off-list an illustration of a Bending Wave and emphasized that the use of the expression "transverse bending wave" is a contradiction in terms. The term Flexural Wave is used interchangably with Bending Wave, but it is quite distinct from a Transverse Wave, as I will explain below. LONGITUDINAL WAVE (Compression Wave, Sound Wave) In a Longitudinal Wave or Compression Wave or Sound Wave the molecules (never mind fibres!) oscillate, in an ideal visualization, parallel to the direction of the wave appearing as vertical columns moving horizontally as the wave passes horizontally. Any differential element of a compression wave will have this configuration. I stress that this is an idealized and simplified picture of such waves in an isotropic medium that considers only one direction of wave propagation. A picture of how a longitudinal wave radiates from a point of excitation through an orthotropic or anisotropic medium would look very different. The simplified picture is shown as an animated gif at the Dan Russell, which those interested have seen. TRANSVERSE WAVE In a Transverse Wave, the particles or molecules oscillate orthogonally to the direction of the wave. Such a phenomenon is seen in the oscillations of a flicked rope or in a vibrating piano string. Both are essentially "floppy" in that _ideally_ they have no stiffness. The difference between the rope and the string is that the rope is of "infinite" length whereas the piano string is terminated at both ends; thus the visible wave in the rope is seen to move in one direction along the rope (unless you're a very clever flicker) wheras in the case of the piano string the wave is reflected at the terminations and becomes a standing wave rather than a travelling wave. In principle, however the wave in the rope is the same as the wave in the string, and of course the rope, qua clothes-line will behave like a piano string. The point is that the particles are moving up and down while the wave moves hosizontally in one or both directions. The fact that a piano string is not ideal, means that it is not perfectly "floppy", and wire stiffness is responsible for inharmonicity etc. as we all know. However this is not the issue and the transverse excursions of the string are so slight that essentially the particles or molecules are moving up and down vertically in columns of negligeable height. >It is desecribed to me as a transverse bending wave. I will refer to these as Bending Waves for the reasons given above. > As the wave propgates through the wood this bending stretches the >fibres of the soundboard on the convex side and compresses the >fibres on the opposite side. This compression and stretching of >fibres alternates with the frequency of the fundamental (of the >string). The wave itself however, is not a compression wave. Before I continue, I'd like to pick up the points here. The Fibres are an unnecessary ingredient in the discussion. We can equally well consider an isotropic elastic soundboard made of metal or plastic and the principles will be the same. Whether spruce or titanium or plexiglas, it is the molecules of the medium that are affected. And yes, a Bending Wave is not a Compression Wave because there is a clear distinction between them just as there is between a Transverse Wave and a Bending Wave. The fact that there may be transversally moving particles in a real compression wave and locaslized parallel disturbances in a grand piano string is irrelevant. We have here three distinct phenomena with three distinct recognized names to identify them. BENDING WAVE Acoustic Radiation from the surfaces of the soundboard compresses and rarefies the air and causes the sound we hear. What for the pianomaker is tone production, for the architect is a noise problem and a great deal more serious scientific work has been done on the reduction of acoustic radiation in structures and panels than has been done in perfecting it in the piano soundboard. It is the bending wave that is PRIMARILY reponsible for acoustic radiation and the _sources_ and _behaviour_ of the Bending Wave are extremely complicated. In fact all wave phenomena are extremely complicated requiring very difficult mathematics for their quantification and analysis, but that is no impediment to understanding the principles and I (with a very basic grasp of mathematics) find I have been able to advance a long way in the understanding of waves notwithstanding. I also find that a lot of mediocre scientists take a lot for granted and often gloss over basic principles. Consider a differential element of the structure shown below _____ | | | | |_____| In the longitudinal wave this is deformed in the following way: ___ _________ | | | | | | ---> | | |___| |_________| And for the transverse wave: |\ /| | \ / | | \ / | \ | ---> | / \ | | / \| |/ And for the bending wave: ___ _______ / \ \ / / \ ---> \ / /_______\ \___/ So you take the horizontal lines of the ascii art to be the top surface and the bottom surface of a small cross-section the soundboard, and know that this wave pattern is to produce a compression/rarefaction of the air at these surfaces, then you can interpolate a convex bottom to the left hand figure and a convex top to the right hand figure as though the enclosed areas were of silicone rubber and the sides of steel and that this superficial deformation travels to the boundary where it is either reflected or absorbed or both. Another way to think of it would be consider a whole stack of linear compression waves one molecule thick, with the "squeezed" line at the top and the "stretched" line of section at the bottom and all lines in between differentially stretched or squeezed. The molecular disturbances resulting from this will, it easy to conceive cause the elastic solid to bulge out into the air at the bottom and become concave at the top and vice versa. This alternating bulging and contraction is what compresses and rarefied the air and gives rise to the sound waves in the air. The Bending Wave is a travelling wave and it can be seen as travelling through the soundboard (longitudinally or radially) and being reflected or absorbed according to the structure of the system. There is also the complication of "dispersion" which I think is in some way analogous to the refraction of light through a prism, where the different frequencies move at different speeds. I personally cannot at the moment form a mental picture of this and need to work on the concept. It will also be clear (pity we haven't got a movie of it) that there will be a longitudinal and a transverse component to the oscillation of any particle affected by a bending wave, but that it is neither a longitudinal wave (except in its direction of travel) nor a transverse wave. >This is not at all far removed from the clothsline illustration, >though I get a different picture then one of surface ripples and >little or nothing happening in the middle of the wood. It is a long way from the clothes-line illustration, as I've described above. Any similarity ends with the visual image. > This description satisfies nicely the difficulty I was having >concerning the 3 dimensionality of the wood. I couldnt see how any >wave could fail to travel through the full thickness of the wood >horizontally. And of course they do. Yes. There are such things as "surface waves" in seismic phenomena (Love Waves and Rayleigh Waves) but these are not significant for us. >This leaves me still wondering about how all the strings partials >end up driving the sound of the board, but I suppose I might get a >further description to that as well at some point. I don't like the word "driving" -- it's pretty meaningless and vague in the way it's been used from time to time in this discussion. What we have, so far as I at present understand it, is (mainly vertical but in fact complex) propagation of sound as a compression wave through the bridge from the string termination. This vibration (after a few milliseconds) hits the soundboard at a right angle, which is, and must be to a degree flexible and mobile, for reasons we have discussed. Exactly what then happens, I think you will search a long while to discover but I believe we can build up quite a good picture given time and patience. One thing that happens is well described in the paper below and this is certainly an important part of the story, which you will find most interesting: "The Calculation and Measurement of Flexural and Longitudinal Structural Power Flow on a Tee Shaped Beam" by Richard P. Szwerc, Stephen A. Hambric <http://jmc.oe.fau.edu/docs/szwercREP96.pdf> However, I can conceive of other simultaneous causes that give rise to the bending, or flexural, waves and these I hope to investigate and think through in more detail. In all the above, I have not considered the _Modal_ resonances of the soundboard which are a separate issue and where Transverse vibration of the soundboard at its fundamental frequency (say 110 Hertz) and many different modes comes into the whole picture. None of these things can be considered in isolation, because there is constant interaction and exchange of energy between the different wave forms. The compression wave down through the bridge (to put it at its very simplest) will, on reaching the soundboard, BOTH excite the natural resonances of the board AND give rise to the Bending Waves that are primarily responsible for Acoustic Radiation, which radiates the sound we hear into the air. That just about sums up my current understanding of the thing. JD
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