# How to Find Resonant Frequencies ••• Furtseff/iStock/GettyImages

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.

Use the formula

f_0=\frac{1}{2\pi}\sqrt{\frac{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.

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.

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

f_n=\frac{v}{\lambda_n}=\frac{nv}{2L}

The term λn stands for the wavelength of the nth frequency, and L is the length of the string.

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.

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