Fundamental frequency

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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 f₀ or F₀, is the lowest frequency in a harmonic series.

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

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

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 = 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 circular natural frequency, ω_n, can be found using the following equation:

Where:
k = stiffness of the beam
m = mass of weight
ω_n = circular natural frequency (radians per second)

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

Where:
f_n = natural frequency in hertz (1/seconds)
k = stiffness of the beam (Newton/Meters or N/m)
m = mass of weight (kg)

See also

• Missing fundamental
• Oscillation
• Hertz
• Electronic tuner
• Scale of harmonics