Although the concept of a simple resonator applies to a number of different simple systems (e.g. a pendulum, a tuning fork, a mass on a spring), there is no single formula which allows to calculate what the natural frequency of a resonator will be given some measurement of its size or composition. For example: the natural frequency of a pendulum is controlled exclusively by the length of the pendulum; however the natural frequency of a mass on a spring will depend on the weight of the mass and the stiffness of the spring, and the natural frequency of tuning fork will depend on both the length and the stiffness of the tines.

For a pendulum, the formula for its period is actually quite simple:

where T=period(s), l=length(m) and g=acceleration due to gravity (9.81ms^-2).

But on the whole, we are better off measuring the resonant frequency rather than calculating it, using the method of forced oscillation.

What we mean by a resonator is simply a system that exhibits a preference for the frequencies at which likes to vibrate. This is the key to measuring its resonant frequency: shake it at different frequencies and find out which one causes the resonator to vibrate most. This is called measurement by 'forced oscillation'. In the laboratory we measure the resonant frequency of the acoustic resonator by feeding a sinusoidal pressure wave into the cavity and measuring the size of the resulting pressure variations in the cavity. Since we can assume that our sinewave generator produces the same amplitude vibrations for every frequency, we can simply say that the frequency at which we get most output amplitude is the resonant frequency.

Our definition of the period of something is simply how long it takes, and is measured in seconds. For a periodic waveform, we define its fundamental period (known simply as its period) as the time it takes to comple one cycle of vibration.

Our definition of the frequency of something is simply how many times it occurs within some space of time (for example the frequency of a bus service is expressed in 'number of buses per hour'). For sound waves, the vibrations are often very rapid and we get a large number of vibrations occurring within one second. So we measure the fundamental frequency of a periodic sound in terms of how many cycles of vibration occur within one second, or in other words, units of 'per second' or s^-1. However, this unit also has a special name in the S.I. system, called Hertz (Hz).

Given these two definitions, then, we can say the the fundamental frequency in Hertz of a periodic waveform is simply the number of fundamental periods it completes in one second, i.e.

Fundamental frequency is the correct name given to the repetition frequency of a complex periodic waveform, i.e. how many cycles of the waveform occur in one second.

Resonant frequency, or natural frequency, is the name given to the frequency which is 'most preferred' by a simple resonator, i.e. the frequency which it most likes to vibrate at, or equivalently, the stimulating frequency which gives the biggest response.

Otherwise, we should only use the term 'frequency' to describe simple periodic waveforms: i.e. sinewaves. This is why we can say that the fundamental frequency of a vowel is X Hz, but not that the frequency of a vowel is X Hz, because vowels are not simple periodic waveforms.

If we think of a periodic sound being generated by a loudspeaker, and the sound pressure waves travelling out from the speaker into space, then it is easy to see that in one second the sound will have travelled a distance numerically equal to the speed of sound, i.e. if the speed of sound is 330ms^-1, then in one second it will have travelled 330m.

If the sound has a fundamental frequency of f Hz, then in that 330m of sound spead out from the speaker there will be exactly f cycles, or in other words, each cycle will be spread along 330/f metres. This distance is called the wavelength:

or, in symbols:

Another way to think about this is to say that the wavelength must be equal to the distance the sound travels in one period, or:

i.e.