Re: Harmonics

["Bob Sciamanda" <trebor@velocity.net>]

Hi Ed,

The textbook theory says yes to your question:  the time development of
the Fourier series = the time development of the string.  But the
assumptions in that derivation, besides the "small displacement"
restriction built into the linear wave equation itself, assume that the
vibrating string is isolated from the rest of the universe (except for
the end constraints).  The resonances of the bodies of these instruments
are designed to treat different frequencies differently.  The interaction
between the string and the instrument body will necessarily affect the
string (presumably both through the body and through the air) - the
string is purposely, and designedly, not isolated.

I guess it comes down to saying that the normal modes of the instrument
are not the normal modes of the isolated string.  (A good student
research project, using real instruments and comparing their outputs to
that of an "isolated" string.)

Bob Sciamanda
Physics, Edinboro Univ of PA (ret)
trebor@velocity.net
http://www.velocity.net/~trebor
-----Original Message-----
From: Ed Schweber <edschweb@ix.netcom.com>
To: phys-l@atlantis.uwf.edu <phys-l@atlantis.uwf.edu>
Date: Monday, May 18, 1998 11:24 PM
Subject: Harmonics


> Ed Schweber (edschweb@ix.netcom.com)
> Physics Teacher at The Solomon Schechter Day School, West Orange, NJ
> To obtain free resources for creative physics teachers visit:
> http://www.physicsweb.com
>
> Hi All:
>
>    If a guitar string is deformed and then released, my very unreliable
> intuition tells my that the relative amplitudes of the Fourrier
components
> of the deformed shape of the string should also be the relative
amplitudes
> of the harmonics in which the string subsequently vibrates and that the
> different timbres of a guitar and banjo should then be due to which of
these
> harmonics create resonances in the rest of the instrument.
>
>   But I also recall reading somewhere that the initial waveform of the
> sound wave produced after a string is plucked is different than the
waveform
> that evolves after a short time interval and that what we hear as the
> difference between a banjo and guitar playing the same note is really
due to
> the differences in this initial waveform. According to what I read, if
the
> initial waveforms in each note of a recording of a guitar are
electronically
> replaced by the corresponding waveforms from a banjo (but the subsequent
> "steady state" waveforms are left alone) the recording will sound more
> banjo-like than guitar-like. But if this is true, it would seem to be at
> odds with my intuition since that idea would seem to be unable to
account
> for the any evolution in the waveforms.
>
>   Is there any connection between the harmonics and the Fourrier
components
> of the original shape? any insight would be appreciated.
>
>   Thanks
>
> Ed Schweber