The fundamental frequency, often referred to simply as the fundamental and abbreviated f_{0} or F_{0}, 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 nonsinusoidal 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 (T_{0}) and thus the fundamental frequency (F_{0}) is given by the following equation:
Where F_{0} is the fundamental frequency and T_{0} 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:
L can be found using the following equation:
λ (lambda) can be found using the following equation:
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:
L can be found using the following equation:
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
Where:
F_{0} = 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, ω_{n}, can be found using the following equation:
Where:
k = stiffness of the beam
m = mass of weight
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 2π. Without first finding the radian frequency, the natural frequency can be found directly using:
Full article ▸
