The Physics of the Acoustic Guitar

The Physics of the Acoustic Guitar

• The guitar is the most common stringed instrument, and shares many characteristics with other stringed instruments.
• For example, the overtones potentially available on any stringed instrument are the same.
• The guitar sound so much different from a violin because of the overtones that are emphasized in a particular instrument, due to the shape and materials in the resonator (body), strings, how it's played, and other factors.
• The overtones, or harmonics of a string fixed at both ends play a role.

Waves on a String

• A guitar string is a common example of a string fixed at both ends which is elastic and can vibrate. The vibrations of such a string are called standing waves, and they satisfy the relationship between wavelength and frequency that comes from the definition of waves
• The equation for this is v = f,
• V is the speed of the wave, f is the frequency and is the wavelength.
• The speed v of waves on a string depends on the string tension T and linear mass density µ, measured in kg/m.
• Waves travel faster on a tighter string and the frequency is therefore higher for a given wavelength.
• Waves travel slower on a more massive string and the frequency is therefore lower for a given wavelength.
• The relationship between speed, tension and mass density is v = T/µ.
• Since the fundamental wavelength of a standing wave on a guitar string is twice the distance between the bridge and the fret, all six strings use the same range of wavelengths.
• To have different pitches (frequencies) of the strings, then, one must have different wave speeds.
• There are two ways to do this: by having different tension T or by having different mass density µ.
• If one varied pitch only by varying tension, the high strings would be very tight and the low strings would be very loose and it would be very difficult to play.
• It is much easier to play a guitar if the strings all have roughly the same tension and for this reason, the lower strings have higher mass density, by making them thicker and, for the 3 low strings, wrapping them with wire.
• From what you have learned so far, and the fact that the strings are a perfect fourth apart in pitch, you can calculate how much µ increases between strings for T to be constant.

String Harmonics (Overtones) If a guitar string had only a single frequency vibration on it, it would sound a bit boring. What makes a guitar or any stringed instrument interesting is the rich variety of harmonics that are present. Any wave that satisfies the condition that it has nodes at the ends of the string can exist on a string. The fundamental, the main pitch you hear, is the lowest tone, and it comes from the string vibrating with one big arc from bottom to top: fundamental (l = /2) The fundamental satisfies the condition l = /2, where l is the length of the freely vibrating portion of the string. The first harmonic or overtone comes from vibration with a node in the center: 1st overtone (l = 2/2) The 1st overtone satisfies the condition l = . Each higher overtone fits an additional half wavelength on the string: 2nd overtone (l = 3/2) 3rd overtone (l = 4/2) 4th overtone (l = 5/2) Guitar Overtones The thing that makes a guitar note "guitar" is the overtone content and how the note rises and decays in time. This varies with how you play it, such as with a pick or. a finger, or near the bridge vs. in the middle. Summary A guitar string sound consists of standing waves: the fundamental and overtones. The fundamental wavelength is twice the length of the vibrating part of the string. The Western musical scale is based on the overtone series for a string: all the overtones up to the 9th are close to notes of the equal-tempered scale. The timber of a stringed instrument depends on the overtone content of the sound: a "twangy" sound has both odd and even multiples of the fundamental, while a "smooth" sound tends to have only odd multiples. Jared Blatt Period G