How To Calculate Fundamental Frequency

The fundamental frequency is the lowest frequency in a resonating system. It is a vital concept in musical instruments and many aspects of engineering. The harmonics of a given wave, for example, are all based on the fundamental frequency. In order to calculate a fundamental frequency, you need the length of the system or wave as well as a handful of other measurements.

Measure the length of the system. This is one half of the wavelength of the wave it carries. For a tube, measure the length of the tube. For a string, measure the length of the string, etc. Record length in meters. If you must measure with millimeters, centimeters, or another unit, make sure that your velocity uses the same length units. For example, use meters if your velocity is in meters per second.

Divide the velocity of your wave by twice the length of the system. If your tube is closed at one end, divide the velocity by four times the length. The result is the fundamental frequency, in cycles per second, or hertz (Hz). The velocity of a sound wave in air at 20 degrees Celsius is 343 meters per second. For example:

For an open tube of length = 0.5 m

The fundamental frequency of a sound wave in the tube is:

\(\frac{343}{2\times 0.5}=\frac{343}{1}=343\text{ Hz}\)

Calculate the velocity for a wave on a string by dividing the tension by its mass per unit length. Make sure that the units of mass in your measurement of tension are the same as the units in which you notate the mass itself. For example, if you use newtons as your unit of tension, express your mass in kilograms. Take the square root of this quotient. Divide that result by twice the length. The result is the fundamental frequency. For example:

For a piano string of mass 0.02 kg and length 0.5m,
mass per unit length = 0.02 kg÷0.5m = 0.04 kg/m

With a tension of 1500 N,

\(v^2=\frac{1500}{0.04}=37500\text{ }\v=193.65\text{ }\frac{193.65}{2\times 0.5}=193.65\text{ Hz}\)