INTRODUCTION: The acoustic guitar is a stringed musical instrument frequently used in popular music. Because it is common in dorm rooms across the country and can be analyzed using a simple linear model, we thought it a fitting instrument for a theoretical and experimental analysis.In this project, we will attempt to quantify certain aspects that affect the way sound is produced by the acoustic guitar. In the remainder of this section, we will summarize the theoretical model and the signal analysis techniques we will use. In Section 2, we will cover the affects that picking location has on the harmonic distribution of a guitar note. In addition, we show that by comparing the results of the signal analysis with the theoretical model for picking location, we can approximate the picking location used to create a recorded note. In Section 3, we look into the subtle implications of tuning systems used in western music and how they influence the guitar. In Section 4, we examine natural harmonics, which, in addition to fretting, are a way to create new pitches on a guitar string.
THEORATICAL ANALYSIS: Since the acoustic guitar is a stringed instrument, we plan to carry out theoretical analysis using the one dimensional wave equation. This is a completely linear model of a vibrating string, so the harmonic implications derived from it ignores non-linearities such as string stiffness, dullness introduced by skin oil and aging, and other material properties of the strings beyond linear density and tension.
The 1-D wave equation is shown below, where T represents the string tension, ρL represents its linear density, and c the wave speed along the string: The general complex-form solution to the above differential equation is shown below, where the leading constants are complex-valued: For a guitar string that is fixed at both ends (x=0 and x=L), we have boundary conditions that restrict the solution space further. When we plug in these boundary conditions and take the real part of the possible solutions, we are left with a solution that can be written as a linear combination of harmonically-related sinusoids. In this case, the fundamental frequency of the string is and all other possible frequencies are multiples of this frequency. We will use these implications of the 1D wave equation as a starting point for our theoretical analysis.