How to Find Resonant Frequencies

By Marcia Spear; Updated April 25, 2017
Resonance frequencies exist for many objects and systems, including waves, springs and pendulums.

A resonant frequency is the natural vibrating frequency of an object and is usually denoted as a f with a subscript zero (f0). This type of resonance is found when an object is in equilibrium with acting forces and could keep vibrating for a long time under perfect conditions. One example of a resonance frequency is seen when pushing a child on a swing. If you pull back and let it go it will swing out and return at its resonant frequency. A system of many objects can have more than one resonance frequency.

A spring is a great object to use to calculate a resonance frequency.

Use the formula f0 = [(1/2π) x (√(k/m)] to find a resonance frequency of a spring. "π" is a long number, but for calculation purposes it can be rounded down to 3.14. The letter "m" stands for the mass of the spring, whereas "k" represents the spring constant, which can be given in a problem. This formula states the resonance frequency is equal to one half "π" multiplied by the square root of the spring constant divided by the mass of the spring.

Use the formula v = λf to find the resonance frequency of a single continuous wave. The letter "v" stands for the wave velocity, whereas "λ" represents the distance of the wavelength. This formula states that the wave velocity equals the distance of the wavelength multiplied by the resonance frequency. In manipulating this equation, resonance frequency equals wave velocity divided by the distance of the wavelength.

There can be multiple resonant frequencies to calculate for in a given situation.

Use another set of formulas to find multiple resonance frequencies for different waves moving at the same time. The resonance frequency of each vibration can be found using the formula fn = (v/ λn) = (nv/2L). The term λn stands for (2L/n) and the term L represents (n(λn)/2). In these equations, n denotes the frequency number currently being calculated; if there are five different resonant frequencies, n would equal one, two, three, four and five, respectively. The term "L" corresponds to the length of the wave.

Basically, this formula states the resonance frequency is equal to the wave velocity divided by the distance of the wavelength multiplied by the resonance frequency number the user is calculating for. This formula also equals the resonance frequency number the user is calculating for multiplied by the velocity then divided by two multiplied by the length of the wave.

About the Author

Marcia Spear holds a Master of Science in biomedical sciences with an emphasis in Immunology and microbiology. Spear has taught science classes for Weatherford College, Tarrant County College, Dallas County College and HIll College since 2010. Her classes include Biology Majors, Biology Non-Majors, Microbiology, and Anatomy and Physiology.