Engineering Acoustics/Acoustic Guitars

I plan to discuss the workings of an acoustic guitar, and how the topics that we have studied apply. This will largely be vibrations of strings and vibrations of cavities.

Introduction
The acoustic guitar is one of the most well known musical instruments. Although precise dates are not known, the acoustic guitar is generally thought to have originated sometime during the Renaissance in the form of a lute, a smaller fretless form of what is known today. After evolving over the course of about 500 years, the guitar today consists of a few major components: the strings and neck, the bridge, soundboard, head, and internal cavity.

Strings, Neck, and Head
The strings are what actually create vibration on the guitar. On a standard acoustic, there are six strings, each with a different constant linear density. Strings run along the length of the neck, and are wound around adjustable tuning pegs located on the head. These tuning pegs can be turned to adjust the tension in the string. This allows a modification of the wave speed, governed by the equation

$$c^2=T/\rho$$

where c is the wave speed [m/s] as a function of tension [N], T, and rho is the linear density [kg/m^3]. The string is assumed to fixed at the head (x=0) and mass loaded at the bridge (x=L).

To determine the vibrating frequency of an open string, a general harmonic solution (GHS) is assumed, $$y(x,t)=Aexp(j(wt-kx))+Bexp(j(wt-kx))$$

To solve for coefficients A and B, boundary conditions at x=0 and x=L are evaluated. At x=0, string velocity (dy/dx) must be zero at all times because that end is assumed to be fixed. Applying this knowledge to the GHS produces

$$y(x,t)=-2jAsin(kx)*exp(jwt)$$

Alternatively, at the bridge (a.k.a the mass load at x=L), the bridge and soundboard (along with any other piece that may vibrate) is assumed to be a lumped element of mass m. The overall goal with this boundary condition is to determine the velocity of the mass. From Newton's second law (F=ma), the only force involved is the tension force in the string. The y-component of this force divided by mass m equals the acceleration. Knowing that acceleration equals velocity times jw (a=jwu),

$$u(L,t)=-T/(j*w*m)*(dy/dx)$$

evaluated at x=L. Combining the two boundary equations and simplifying, a final equation can be obtained

$$cot(kL)=(m/ms)kL$$

where k is the wavenumber (w/c), L is the string length, m is the lumped mass of the guitar body, ms is the total mass of the string (linear density times length), w is the frequency, and c is the wave speed. If the ratio of m/ms is large (which in a guitar's case, it is), these frequencies are designated by kL=n*pi. Simplified, the fundamental frequency can be given by

$$f=sqrt(T/rho) / 2L$$

Therefore to adjust the resonance frequency of the string, either change the tension (turn the tuning knob), change the linear density (play a different string), or adjust the length (use the fretboard).

To determine the location of the frets, musical notes must be considered. In the musical world, it is common practice to use a tempered scale. In this scale, an A note is set at 440 Hz. To get the next note in the scale, multiply that frequency by the 12th root of 2 (approximately 1.059), and an A-sharp will be produced. Multiply by the same factor for the next note, and so on. With this in mind, to increase f by a factor of 1.059, a corresponding factor should be applied to L. That factor is 1/17.817, with L in inches. For example, consider an open A string, vibrating at 440 Hz. For a 26 inch string, the position of the first fret is (26/17.817=1.459) inches from the head. The second fret will be ((26-1.459)/17.817) inches from the first, and so on.

Bridge
The bridge is the connection point between the strings and the soundboard. The vibration of the string moves the assumed mass load of the bridge, which vibrates the soundboard, described next.

Soundboard
The soundboard increases the surface area of vibration, increasing the initial intensity of the note, and is assisted by the internal cavity.

Internal Cavity
The internal cavity acts as a Helmholtz resonator, and helps to amplify the sound. As the sound board vibrates, the sound wave is able to resonate inside.