# Acoustics/How an Acoustic Guitar Works

What are sound vibrations that contribute to sound production? First of all, there are the strings. Any string that is under tension will vibrate at a certain frequency. The weight and length of the string, the tension in the string, and the compliance of the string determine the frequency at which it vibrates. The guitar controls the length and tension of six differently weighted strings to cover a very wide range of frequencies. Second, there is the body of the guitar. The guitar body is very important for the lower frequencies of the guitar. The air mass just inside the sound hole oscillates, compressing and decompressing the compliant air inside the body. In practice this concept is called a Helmholtz resonator. Without this, it would be difficult to produce the wonderful timbre of the guitar.

## The strings

The strings of the guitar vary in linear density, length, and tension. This gives the guitar a wide range of attainable frequencies. The larger the linear density is, the slower the string vibrates. The same goes for the length; the longer the string is the slower it vibrates. This causes a low frequency. Inversely, if the strings are less dense and/or shorter they create a higher frequency. The resonance frequencies of the strings can be calculated by
    $f_{1}={\frac {\sqrt {\frac {T}{\rho _{l}}}}{2L}}\quad {\text{with}}\quad T={\text{string tension}},\ \rho _{l}={\text{linear density of string}},\ L={\text{string length}}.$
The string length, $L$, in the equation is what changes when a player presses on a string at a certain fret. This will shorten the string which in turn increases the frequency it produces when plucked. The spacing of these frets is important. The length from the nut to bridge determines how much space goes between each fret. If the length is 25 inches, then the position of the first fret should be located (25/17.817) inches from the nut. Then the second fret should be located (25−(25/17.817))/17.817 inches from the first fret. This results in the equation
    $d={\frac {L}{17.817}}\quad {\text{with}}\quad d={\text{spacing between frets}},L={\text{length from previous fret to bridge}}.$
When a string is plucked, a disturbance is formed and travels in both directions away from point where the string was plucked. These "waves" travel at a speed that is related to the tension and linear density and can be calculated by
    $c={\sqrt {\frac {T}{\rho _{l}}}}\quad {\text{with}}\quad c={\text{wave speed}},\ T={\text{string tension}},\ \rho _{l}={\text{linear density}}.$
The waves travel until they reach the boundaries on each end where they are reflected back. The link below displays how the waves propagate in a string.
Plucked String @ www.phys.unsw.edu
The strings themselves do not produce very much sound because they are so thin. They can't "push" the air that surrounds them very effectively. This is why they are connected to the top plate of the guitar body. They need to transfer the frequencies they are producing to a large surface area which can create more intense pressure disturbances.

## The body

The body of the guitar transfers the vibrations of the bridge to the air that surrounds it. The top plate contributes to most of the pressure disturbances, because the player dampens the back plate and the sides are relatively stiff. This is why it is important to make the top plate out of a light springy wood, like spruce. The more the top plate can vibrate, the louder the sound it produces will be. It is also important to keep the top plate flat, so a series of braces are located on the inside to strengthen it. Without these braces the top plate would bend and crack under the large stress created by the tension in the strings. This would also affect the magnitude of the sound being transmitted. The warped plate would not be able to "push" air very efficiently. A good experiment to try, in order to see how important this part of the guitar is in the amplification process, is as follows:
* Start with an ordinary rubber band, a large bowl, adhesive tape, and plastic wrap.
* Stretch the rubber band and pluck it a few times to get a good sense for how loud it is.
* Stretch the plastic wrap over the bowl to form a sort of drum.
* Tape down one end of the rubber band to the plastic wrap.
* Stretch the rubber band and pluck it a few times.
* The sound should be much louder than before.

## The air

The final part of the guitar is the air inside the body. This is very important for the lower range of the instrument. The air just inside the soundhole oscillates compressing and expanding the air inside the body. This is just like blowing across the top of a bottle and listening to the tone it produces. This forms what is called a Helmholtz resonator. For more information on Helmholtz resonators go to Helmholtz Resonance. This link also shows the correlation to acoustic guitars in great detail. The acoustic guitar makers often tune these resonators to have a resonance frequency between F#2 and A2 (92.5 to 110.0 Hz). Having such a low resonance frequency is what aids the amplification of the lower frequency strings. To demonstrate the importance of the air in the cavity, simply play an open A on the guitar (the fifth string - second lowest note). Now, as the string is vibrating, place a piece of cardboard over the soundhole. The sound level is reduced dramatically. This is because you've stopped the vibration of the air mass just inside the soundhole, causing only the top plate to vibrate. Although the top plate still vibrates a transmits sound, it isn't as effective at transmitting lower frequency waves, thus the need for the Helmholtz resonator.
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