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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