How to Calculate String Vibration

Guitar strings each vibrate in several harmonics.

Any string with sufficient tautness will vibrate when plucked. If it vibrates rapidly enough, it produces a sound. In fact, string instruments, such as cello and guitar, are designed for their strings to vibrate at very exact frequencies. The factors that determine strings' frequency of vibration are their tension, length and mass. You can calculate what frequency of vibration (or sound) a string will produce based on these characteristics.

Instructions

Steps

1. Obtain the string's mass, length and tension. If using SI units, you should make sure they are in the units of kilograms, meters and Newtons. If using U.S. units, they should be in pounds-mass, feet and pounds-force.

2. Calculate the string's linear density. This is simply the string's mass divided by its length.

3. Calculate the velocity of the wave on the string. The velocity refers not to how quickly the string moves up and down, but how quickly the string's wave travels forward along its length. Its velocity equals the square root of the ratio of the string's tension to its linear mass. That is, v = sqrt (T / lm), where v = velocity, T = tension and lm = linear mass.

4. Calculate the length of the most prominent wave. An ideal vibrating string actually contains several different waves, called harmonics. The most prominent wave is the fundamental harmonic, which is the whole string moving up and down as a single unit. Because a complete wave consists of a complete up-and-down motion, the length of the string in a fundamental harmonic only contains one-half of a wavelength at any point in time. Therefore, wavelength = L / 2, where L is the string length.

5. Calculate the wavelengths of any additional harmonics of interest. Each harmonic is exemplified by a number of apparent standing arches, or "antinodes" in the string. For example, the second harmonic somewhat looks like two vibrating strings separated by one stationary node. The third harmonic somewhat looks like three vibrating stings separated by two nodes. A wave within a harmonic of degree "n" has a wavelength of n / 2*L.

6. Calculate the vibrational frequency. The frequency equals the velocity divided by wavelength. Note that velocity is consistent, but wavelength varies with the harmonic. Therefore, higher harmonics have a higher frequency, or pitch.

References

• Georgia State University: Vibrating String

Resources

• YouTube Video: Slow-Motion String Vibration