Fundamental frequency
This article's introduction may be too long for the overall article length. (December 2013) 
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 f_{0} (or FF), indicating the lowest frequency counting from zero.^{1}^{2}^{3} In other contexts, it is more common to abbreviate it as f_{1}, the first harmonic.^{4}^{5}^{6}^{7}^{8} (The second harmonic is then f_{2} = 2⋅f_{1}, etc. In this context, the zeroth harmonic would be 0 Hz.)
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:
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 (f_{0}) is:
Where f_{0} is the fundamental frequency and T is the fundamental period.
For a tube of length L with one end closed and the other end open the wavelength of the fundamental harmonic is 4L, as indicated by the top two animations on the right. Hence,
Therefore, using the relation
 ,
where v is the speed of the wave, we can find the fundamental frequency in terms of the speed of the wave and the length of the tube:
If the ends of the same tube are now both closed or both opened as in the bottom two animations on the right, the wavelength of the fundamental harmonic becomes 2L. By the same method as above, the fundamental frequency is found to be
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 spring, 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, ω_{n}, can be found using the following equation:
Where:
k = stiffness of the spring
m = mass
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:
Where:
f_{n} = natural frequency in hertz (cycles/second)
k = stiffness of the spring (Newtons/meter or N/m)
m = mass(kg)
while doing the modal analysis of structures and mechanical equipment, the frequency of 1st mode is called fundamental frequency.
See also
 Missing fundamental
 Natural frequency
 Oscillation
 Hertz
 Electronic tuner
 Scale of harmonics
 Pitch detection algorithm
References
 ^ "sidfn". Phon.ucl.ac.uk. Retrieved 20121127.
 ^ "Phonetics and Theory of Speech Production". Acoustics.hut.fi. Retrieved 20121127.
 ^ "Fundamental Frequency of Continuous Signals". Fourier.eng.hmc.edu. Retrieved 20121127.
 ^ "Standing Wave in a Tube II  Finding the Fundamental Frequency". Nchsdduncanapphysics.wikispaces.com. Retrieved 20121127.
 ^ "Physics: Standing Waves". Physics.kennesaw.edu. Retrieved 20121127.
 ^ "Phys 1240: Sound and Music". Colorado.edu. Retrieved 20121127.
 ^ "Standing Waves on a String". Hyperphysics.phyastr.gsu.edu. Retrieved 20121127.
 ^ [1]^{dead link}


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