9-012-S-PDEGuitarTuning

By Brian Winkel

SIMIODE

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Abstract

We present a derivation of  a partial differential equation which models the motion of a string held at both ends, a case of the one-dimensional wave equation. We immediately offer  numerical solutions in a computer algebra system (we use Mathematica, but any computer algebra system should do) to get feedback on the reasonableness of our model.  Applying data on a guitar string we then give  students a chance to tune the guitar string to a given frequency and compare the sound of their tuned guitar string to that of a frequency generator using both auditory and visual feedback. We use Mathematica's Play command but the tuning can be graphically done, matching the frequency plots over small intervals of time [0,3/440] seconds, say. We do not analytically solve this wave equation here, rather we build meaning and support to what the equation is and can do. At another time we solve the wave equation analytically. There too we do sound and an immediate application. Finally, we acknowledge the ideal nature of this first wave equation model and suggest some damping, again with numerical validation using  sound and graphic output.

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Researchers should cite this work as follows:

  • Brian Winkel (2015), "9-012-S-PDEGuitarTuning," https://www.simiode.org/resources/671.

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