# Fundamental Frequency

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 f0 (or FF), indicating the lowest frequency counting from zero. In other contexts, it is more common to abbreviate it as f1, the first harmonic. (The second harmonic is then f2 = 2⋅f1, etc. In this context, the zeroth harmonic would be 0 Hz.)

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:

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 (f0) is:

Where f0 is the fundamental frequency and T 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:

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 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:

Where:

f0 = fundamental frequency
L = length of the tube
v = wave 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