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Physics Tutorial: Guitar String Harmonics

The Universal Wave Equation is a simple formula, which not only applies to all waves, but can be used to solve a range of problems concerning wave motion. The equation is as follows:

v = fL                     

where:

v = the velocity or speed of the wave

f = the frequency of the wave (in Hertz or Hz)

L = the wavelength

In real life, this equation may be used in conjunction with guitar string harmonics. With the knowledge of

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harmonics, the ability to comprehend wavelengths is greatly increased. A guitar string stretched to a particular length and tautness vibrates with characteristic frequency or pitch called its fundamental frequency. This is the note we hear as we listen to the vibrating string. However, if we listen carefully we also hear overtones, which are caused by higher frequency (and quieter) vibrations of the string. By plucking at different places on the string, different combinations of overtones are produced and we can hear the same note being played with either mellow or sharp tones. These higher frequency vibrations are called harmonics.

 

The pitch of a plucked string is directly dependant on three variables: the length of the string, the tension of the string and the mass density of the string. When length is decreased or when tension is increased (ex. pulling on a string), the pitch increases. When mass density (mass divided by string length) is decreased then the frequency goes up. This is why the thinner strings produce higher sounds.

 

Harmonics result from the capacity of a guitar string to vibrate at several frequencies at the same time.

The relationship between frequency and wavelength is also one of importance. As frequencies increase, then wavelengths become shorter. This can be better explained through the use of natural harmonics.In the diagram, nodes are simple areas where the string is unable to vibrate. These are found where the ends of the string are attached to the guitar and wherever a finger applies pressure on the guitar sting when in use. Harmonics occur at the same time as the fundamental frequency.The 1st harmonic (or fundamental) shows a string where the wave is allowed to move from one end of the guitar to the other. The second harmonic occurs when a node resides on the center of the string. This doubles the frequency, but halves the wave length. For example,a guitar sting with a length of 50 cm and a fundamental frequency of a 100 Hz may also vibrate with a frequency of 200 Hz and have a node 25 cm from one end. Combinations and relative amplitudes of harmonics predestine the tone of the notes we hear.

 

In the 1st harmonic, there is only half a wave in the string. In the 3rd harmonic there are one and a half waves per string.Harmonics occur when waves having multiples of one-half wavelength form standing waves on a guitar string.

So when figuring out wave problems for guitar strings, that must be accounted for. The following equation can be used.

L   = (harmonic # / 2) L

(L = string length)

 

Question: The velocity waves on a guitar string is 345m/s. What is the 2nd harmonic frequency of a 1.2 m guitar string?

 

Answer:

Step 1: Figure out the wave length using L = (harmonic # / 2) L

L  = (2/2) 1.2m

L  = 1.2m

Step 2: Use the Universal Wave Equation (v = f L)

v = f L  

f = (v / L)

f = (345 /1.2)

f = 287.5 Hz

 

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The template for the diagrams was obtained at http://www.harmony-central.com/Guitar/harmonics.html. Alterations were done by myself.

-Amanda Taphorn