Vibrating strings and bridges

[Robin Hufford]

Phil
     I am still interested for one.  This site, however, is, I think a draft of a book about digitally modelling waveguide behavior and as such there are certain underlying
assumptions, one of this is linearity of response, as evidenced by the assumption apparent in the phrase "since the bridge and string move together".
    Also, I quote from the section  of the reflection transfer function as relates to the string and bridge.  I am not able to copy the symbolic notation and have instead placed
multiple parentheses around the verbal rendition of such notation.  The single parentheses are original material.

((( S subscript b,(z))) is the reflection transfer function at the bridge
(((R subscript b,(z))) is the bridge reflectance

     A complicated expression to complicated to reproduce verbally here, is given, in which he has "thus related the impedance of the termination to quanties whole within the
string wave state." Which is a very interesting expression worthy of study, and then continues:

"Note that this same result can be obtained from the general formula for scattering at a loaded waveguide junction for the case of (N=1) a single waveguide  terminated by a lumped
load.

Because (((R subscript b (z)))  is positive real, (((S subscript b(z)))   is a Schur function, i.e., (the absolute value of (((S subscript b(z)))  is less than or equal to1 for
absolute values of z which are less than are equal to 1.   Schur functions become allpass filters as damping goes to zero, and they cannot provide gain at any frequency, i.e., the
gain is less than or equal to 1, as needed for string loop stability. Note that reflection filters always have an equal number of poles and zeros, as can be seen from  the above
expression."     There are caveats that the entire analysis is made in a "simulation context".  This qualification should be taken seriously I am sure but to what degree escapes
me.

     The entire section models a complicated question, but, I think, it is a  reasonably good expression of the nature of the transfer function at the bridge in spite of its linear
approach.  Were the gain he refers to here greater than one then the bridge would indeed "displace the node and confuse the frequencies produced by the string" as string loop
stability would be impaired, a point I have repeatedly tried to make.


"The reflection transfer function is defined for force waves. Note that as the bridge impedance goes to infinity (becomes rigid)   (((S subscript b,( z)))      approaches 1 ,
a result which agrees with an analysis of rigid terminations.  SINCE TYPICAL BRIDGES ARE QUITE RIGID, (caps mine, rh)    (((S subscript b,(z)))    { the symbol for approximately
equal to is given(rh)} 1   IN ALL PRACTICAL CASES.   Similarly, if the bridge impedance goes to zero,   (((s subscript b,(x)))   goes to   -1   which also agrees with  the physics
of a string with a free end."

     In this model the Schur function he uses is approximately equal to 1, which indicates perfect ridigity, of course something that cannot exist, but nevertheless, he takes pains
with the phrase which I repeat "Since typical bridges are quite rigid in all practical cases."  to  indicate  in reality that this is  their functional aspect and that  again, this
is required for "string loop stability".

     Additionally, this short section gives a proper treatment of the transfer function in which the concept of  impedance, the use of which is very casually bandied about on this
list and elsewhere,  can be seen to be but part of a larger expression.  I have criticized the metaphoric treatment  seen here at other times for this reason, among others.

     There are other comments I have on this but time at the moment is short and I will have to come back to this later.
Regards, Robin Hufford

> For those that are still interested in this topic I found what I think is an
> interesting site.  Apparently it's the draft of a book about the behavior
> of musical instruments.
>
> http://www-ccrma.stanford.edu/~jos/waveguide/waveguide.html
>
> This is from a Stanford site.  Also here's another link to a page giving
> many tutorials, etc.  A wealth of information if you have the time:
>
> http://www-ccrma.stanford.edu/~jos/pubs.html
>
> I haven't spent much time looking at this as yet.  However, here's one little
> quote which I thought was interesting because it mentions bridge rigidity
> and also the relative rigidity of electric guitar bodies (for which I took some
> flak for mentioning some time back).  The quote:
>
> No vibrating string in musical acoustics is rigidly terminated, since such a string would produce no sound through the body of the instrument. (Electric guitars with magnetic
> pickups have nearly rigid terminations, but even then, coupling phenomena are clearly observed, especially above the sixth harmonic.) Furthermore, it is typically the case that
> vertical transverse waves are transduced differently at the bridge. For example, the bridge on a piano is much easier to ``push'' into the soundboard than it is to ``shear''
> sidewise along the soundboard.
>
> Happy Reading,
>
> Phil F
> ---
> Phillip Ford
> Piano Service & Restoration
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> San Francisco, CA  94124
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