Editing Fundamental frequency (section)

==Explanation==
All sinusoidal and many non-sinusoidal waveforms repeat exactly over time – they are periodic. The period of a waveform is the smallest positive value <math>T</math> for which the following is true:

{{block indent|1=<math> x(t) = x(t + T)\text{ for all }t \in \mathbb{R} </math>}}

Where <math>x(t)</math> is the value of the waveform <math>t</math>. This means that the waveform's values over any interval of length <math>T</math> is all that is required to describe the waveform completely (for example, by the associated [[Fourier series]]). Since any multiple of period <math>T</math> also satisfies this definition, the fundamental period is defined as the smallest period over which the function may be described completely. The fundamental frequency is defined as its reciprocal:

{{block indent|1=<math> f_0 = \frac{1}{T}</math>}}

When the units of time are seconds, the frequency is in <math>s^{-1}</math>, also known as [[Hertz]].

===Fundamental frequency of a pipe===

For a pipe of length <math>L</math> with one end closed and the other end open the wavelength of the fundamental harmonic is <math>4L</math>, as indicated by the first two animations. Hence,
{{block indent|1=<math>\lambda_0 = 4L</math>}}

Therefore, using the relation

{{block indent|1=<math> \lambda_0 = \frac{v}{f_0}</math>}}

where <math>v</math> is the speed of the wave, the fundamental frequency can be found in terms of the speed of the wave and the length of the pipe:

{{block indent|1=<math> f_0 = \frac{v}{4L}</math>}}

If the ends of the same pipe are now both closed or both opened, the wavelength of the fundamental harmonic becomes <math>2L</math> . By the same method as above, the fundamental frequency is found to be

{{block indent|1=<math> f_0 = \frac{v}{2L}</math>}}