# Fundamental frequency

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.)

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:

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

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:

$f_0 = \frac{1}{T}$[citation needed]

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

The fundamental frequency of a sound wave in a tube with a single CLOSED end can be found using the following equation:

$f_0 = \frac{v}{4L}$

L can be found using the following equation:

$L = \frac{\lambda}{4}$

λ (lambda) can be found using the following equation:

$\lambda = \frac{v}{f_0}$

The fundamental frequency of a sound wave in a tube with either BOTH ends OPEN or CLOSED can be found using the following equation:

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

L can be found using the following equation:

$L = \frac{\lambda}{2}$

The wavelength, which is the distance in the medium between the beginning and end of a cycle, is found using the following equation:

$\lambda = \frac{v}{f_0}$

Where:

f0 = fundamental frequency
L = length of the tube
v = wave velocity of the sound wave
λ = wavelength

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 beam, 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 beam
m = mass of weight

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 beam (Newtons/meter or N/m)
m = mass at the end (kg)
while doing the modal analysis of structures and mechanical equipment, the frequency of 1st mode is called fundamental frequency.