Fundamental frequency
Vibration and standing waves in a string, The fundamental and the first 6 overtones


The fundamental frequency, often referred to simply as the fundamental and abbreviated f0 or F0, 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 non-sinusoidal 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 (T0) and thus the fundamental frequency (F0) is given by the following equation:

 F_0=\frac{1}{T_0}

Where F0 is the fundamental frequency and T0 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:

F_0=\frac{v}{4L}

L can be found using the following equation:

L=\frac{\lambda}{4}

λ (lambda) can be found using the following equation:

\lambda = \frac{v}{F_0}

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:

F_0=\frac{v}{2L}

L can be found using the following equation:

L=\frac{\lambda}{2}

The wavelength, which is the distance in the medium between the beginning and end of a cycle, is found using the following equation:

\lambda=\frac{v}{F_0}

Where:

F0 = 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, ωn, can be found using the following equation:

\omega_n^2 = \frac{k}{m} \,

Where:
k = stiffness of the beam
m = mass of weight
ωn = radian frequency (radians per second)

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

f_n = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \,

Where:
fn = natural frequency in hertz (cycles/second)
k = stiffness of the beam (Newtons/Meter or N/m)
m = mass at the end (kg)
while doing the modal analysis of structures and mechanical equipments, the frequency of 1st mode is called fundamental frequency.

See also


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