Frequency of a String

Calculates the string frequency from diameter, length, density and tension of a string (or chord). A string oscillates, when being drawn (e.g. violin), plucked (e.g. guitar) or struck (e.g. piano), with a certain fundamental frequency and, in theory, infinite many harmonic overtones, which are integer multiples of the fundamental frequency. Those frequencies result from the physical properties of the string. The fundamental frequency determines the note, the ratios of the strengths of the overtones determine the timbre, which can't be calculated here. The wavelength refers to the fundamental frequency.

The formula for the frequency is: f = √ ψ / ( π * ρ ) / ( d * l )

The wavelength of the fundamental frequency λ is twice the string length.

The unit for the tension is newton, for the frequencies the unit is hertz.

An example with fictitious values: a string with a diameter of one millimeter, a length of 60 centimeters, and a density of 3 grams per cubic centimeter is tensioned to 20 Newtons. The sound produced on it then has a frequency of approximately 77 Hertz and a wavelength of 1.20 meters. The overtones are multiples of the fundamental frequency; the first overtone has twice the frequency, the second has three times the frequency, and so on.

Larger diameter and greater length decrease the fundamental frequency linearly. Increasing density decreases the fundamental frequency less, expressed by the square root function. Increasing tension increases the fundamental frequency, also according to the square root function. A higher string density can therefore be compensated by a correspondingly increased tension. Likewise, a higher diameter can be compensated by a shorter length, or vice versa.

Diameter d:
Length l:
Density ρ:		g/cm³
Tension ψ:		N
Fundamental frequency f:		Hz
Wavelength λ:		m
1. overtone f₁:		Hz
2. overtone f₂:		Hz
3. overtone f₃:		Hz
4. overtone f₄:		Hz
5. overtone f₅:		Hz