# Fundamental frequency

File:Overtone.jpg
Vibration and standing waves in a string, The fundamental and the first 6 overtones

The fundamental tone, often referred to simply as the fundamental and abbreviated fo, is the lowest frequency in a harmonic series.

The fundamental frequency (also called a natural frequency) of a periodic signal is the inverse of the pitch period length. The pitch period is, in turn, the smallest repeating unit of a signal. One pitch period thus describes the periodic signal completely. The significance of defining the pitch period as the smallest repeating unit can be appreciated by noting that two or more concatenated pitch periods form a repeating pattern in the signal. However, the concatenated signal unit obviously contains redundant information.

In terms of a superposition of sinusoids (for example, fourier series), the fundamental frequency is the lowest frequency sinusoidal in the sum.

To find the fundamental frequency of a sound wave in a tube that has a closed end you will use the equation:

$F={\frac {V}{4L}}$ To find L you will use:

$L={\frac {\lambda }{4}}$ To find λ (lambda) you will use:

$\lambda ={\frac {V}{F}}$ To find the fundamental frequency of a sound wave in a tube that has open ends you will use the equation:

$F={\frac {V}{2L}}$ To find L you will use:

$L={\frac {\lambda }{2}}$ To find Wavelength which is the distance in the medium between the beginning and end of a cycle and is found using the following equation: WAVELENGTH = Velocity/Frequency or

$\lambda ={\frac {V}{F}}$ At 70 °F the speed of sound in air is approximately 1130 ft/s or 340 m/s. This speed is temperature dependent and does increase at a rate of 1.1 ft/s for each degree Fahrenheit increase in temperature, or 0.6 m/s for every increase of 1 °C.

The velocity of a sound wave at different temperatures:

• V = 343.7 m/s at 20 °C
• V = 331.5 m/s at 0 °C

WHERE:

F = fundamental Frequency
L = length of the tube
V = velocity of the sound wave
λ = wavelength

## 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 circular natural frequency, ωn, can be found using the following equation:

$\omega _{n}^{2}=k/m\,$ where:
k = stiffness of the beam
m = mass of weight
ωn = circular natural frequency (radians per second)
fn = natural frequency in hertz (1/seconds)

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

$f_{n}=(1/2\pi )(k/m)^{1/2}\,$  