# Fundamental frequency

(Redirected from Fundamental tone)
Vibration and standing waves in a string, The fundamental and the first 6 overtones

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 f0 (or FF), indicating the lowest frequency counting from zero.[1][2][3] In other contexts, it is more common to abbreviate it as f1, the first harmonic.[4][5][6][7][8] (The second harmonic is then f2 = 2⋅f1, 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:

$x(t) = x(t + T)\text{ for all }t \in \mathbb{R}$

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 (f0) is:

$f_0 = \frac{1}{T}$

Where f0 is the fundamental frequency and T is the fundamental period.

F0leftclosed
F0rightclosed

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

$\lambda_0 = 4L.$

Therefore, using the relation

$\lambda_0 = \frac{v}{f_0}$ ,

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:

$f_0 = \frac{v}{4L}.$
F0bothclosed
F0bothopen

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 2L. By the same method as above, the fundamental frequency is found to be

$f_0 = \frac{v}{2L}.$

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, ωn, can be found using the following equation:

$\omega_\mathrm{n}^2 = \frac{k}{m} \,$

Where:
k = stiffness of the spring
m = mass

From the radian frequency, the natural frequency, fn, can be found by simply dividing ωn by 2π. Without first finding the radian frequency, the natural frequency can be found directly using:

$f_\mathrm{n} = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \,$

Where:
fn = 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.