# Fundamental frequency

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The fundamental frequency, often referred to simply as the fundamental and abbreviated f0 or F0, 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.

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) = x(t + 2T) = x(t + 3T) = ...

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 lowest value of T for which this is true is called the fundamental period (T0) and thus the fundamental frequency (F0) is given by the following equation:

$F_0=\frac{1}{T_0}$

Where F0 is the fundamental frequency and T0 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 both ends 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: WAVELENGTH = Velocity/Frequency or

$\lambda=\frac{v}{F_0}$

Where:

F0 = fundamental Frequency
L = length of the tube
v = 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. The radian frequency, ωn, can be found using the following equation:

Where:
k = stiffness of the beam
m = mass of weight