# Fundamental frequency

The **fundamental frequency**, often referred to simply as the **fundamental**, is defined as the lowest frequency of a periodic waveform. In terms of a superposition of sinusoids (e.g. Fourier series), the fundamental frequency is the lowest frequency sinusoidal in the sum. In some contexts, the fundamental is usually abbreviated as ** f_{0}** (or

**FF**), indicating the lowest frequency counting from zero.

^{[1]}

^{[2]}

^{[3]}In other contexts, it is more common to abbreviate it as

**, the first harmonic.**

*f*_{1}^{[4]}

^{[5]}

^{[6]}

^{[7]}

^{[8]}(The second harmonic is then f

_{2}= 2⋅f

_{1}, etc. In this context, the zeroth harmonic would be 0 Hz.)

## Explanation

All sinusoidal and many non-sinusoidal waveforms are periodic, which is to say they repeat exactly over time. A single period is thus the smallest repeating unit of a signal, and one period describes the signal completely. We can show a waveform is periodic by finding some period *T* for which the following equation is true:

Where *x*(*t*) is the function of the waveform.

This means that for multiples of some period T the value of the signal is always the same. The least possible value of T for which this is true is called the fundamental period and the fundamental frequency (*f*_{0}) is:

Where *f*_{0} is the fundamental frequency and *T* is the fundamental period.

For a tube of length *L* with one end closed and the other end open the wavelength of the fundamental harmonic is 4*L*, as indicated by the top two animations on the right. Hence,

Therefore, using the relation

where *v* is the speed of the wave, we can find the fundamental frequency in terms of the speed of the wave and the length of the tube:

If the ends of the same tube are now both closed or both opened as in the bottom two animations on the right, the wavelength of the fundamental harmonic becomes 2*L*. By the same method as above, the fundamental frequency is found to be

At 20 °C (68 °F) the speed of sound in air is 343 m/s (1129 ft/s). This speed is temperature dependent and does increase at a rate of 0.6 m/s for each degree Celsius increase in temperature (1.1 ft/s for every increase of 1 °F).

The velocity of a sound wave at different temperatures:-

- v = 343.2 m/s at 20 °C
- v = 331.3 m/s at 0 °C

## Mechanical systems

Consider a spring, fixed at one end and having a mass attached to the other; this would be a single degree of freedom (SDoF) oscillator. Once set into motion it will oscillate at its natural frequency. For a single degree of freedom oscillator, a system in which the motion can be described by a single coordinate, the natural frequency depends on two system properties: mass and stiffness; (providing the system is undamped). The radian frequency, *ω*_{n}, can be found using the following equation:

Where:

*k* = stiffness of the spring

*m* = mass

*ω*_{n} = radian frequency (radians per second)

From the radian frequency, the natural frequency, *f*_{n}, can be found by simply dividing *ω*_{n} by 2*π*. Without first finding the radian frequency, the natural frequency can be found directly using:

Where:

*f*_{n} = natural frequency in hertz (cycles/second)

*k* = stiffness of the spring (Newtons/meter or N/m)

*m* = mass(kg)

while doing the modal analysis of structures and mechanical equipment, the frequency of 1st mode is called fundamental frequency.

## See also

- Electronic tuner
- Hertz
- Missing fundamental
- Natural frequency
- Oscillation
- Harmonic series (music)#Terminology
- Pitch detection algorithm
- Scale of harmonics