The fundamental frequency, often referred to simply as the fundamental, is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. 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.)
Explanation
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)\text{ for all }t \in \mathbb{R}
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 largest possible value of T for which this is true is called the fundamental period and the fundamental frequency (f_{0}) is:

f_0 = \frac{1}{T}
Where f_{0} is the fundamental frequency and T is the fundamental period.
F0leftclosed
F0rightclosed
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,

\lambda_0 = 4L.
Therefore, using the relation

\lambda_0 = \frac{v}{f_0} ,
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:

f_0 = \frac{v}{4L}.
F0bothclosed
F0bothopen
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 2
L. By the same method as above, the fundamental frequency is found to be

f_0 = \frac{v}{2L}.
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
In music
In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. The fundamental may be created by vibration over the full length of a string or air column, or a higher harmonic chosen by the player. The fundamental is one of the harmonics. A harmonic is any member of the harmonic series, an ideal set of frequencies that are positive integer multiples of a common fundamental frequency. The reason a fundamental is also considered a harmonic is because it is 1 times itself. ^{[9]}
The fundamental is the frequency at which the entire wave vibrates. Overtones are other sinusoidal components present at frequencies above the fundamental. All of the frequency components that make up the total waveform, including the fundamental and the overtones, are called partials. Together they form the harmonic series. Overtones which are perfect integer multiples of the fundamental are called harmonics. When an overtone is near to being harmonic, but not exact, it is sometimes called a harmonic partial, although they are often referred to simply as harmonics. Sometimes overtones are created that are not anywhere near a harmonic, and are just called partials or inharmonic overtones.
The fundamental frequency is considered the first harmonic and the first partial. The numbering of the partials and harmonics is then usually the same; the second partial is the second harmonic, etc. But if there are inharmonic partials, the numbering no longer coincides. Overtones are numbered as they appear above the fundamental. So strictly speaking, the first overtone is the second partial (and usually the second harmonic). As this can result in confusion, only harmonics are usually referred to by their numbers, and overtones and partials are described by their relationships to those harmonics.
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:

\omega_\mathrm{n}^2 = \frac{k}{m} \,
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:

f_\mathrm{n} = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \,
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
References
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