On the course of discussion

[John Delacour]

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