How are complex periodic phenomena dealt with?

It appears that periodical phenomena, e.g. waves are of utmost importance in nature. In practice, waves of different frequencies are superimposed so that the resulting periodical phenomenon is not of sinusoidal shape. Then, how can we deal with them mathematically? In 1822, Fourier has discovered that any periodical function can be approximately represented as a linear composition of sines and cosines of ascending frequencies, superimposed with appropriate weights. Each weight is obtained by integrating the function multiplied by the corresponding member of the series. The more that function differ from a sinusoid, the longer series is needed for the approximation to be good enough. Instead both sines an cosines, the complex exponential function can be used in the series making use of Euler’s formula and thereby simplifying calculations. Such series have been named in the honor of Fourier. Furthermore, the Fourier series can be generalized to continues sum, i.e. an integral over frequencies. This fits practical situations since in nature overlapping waves can have arbitrary close frequencies. Such a technique is good for description of a range of phenomena from the heat propagation studied by Fourier to probability waves in quantum mechanics.

Formation of  a periodic function of two overlaid waves.

In acoustics, the overlapping of waves of close frequencies is known as a beat.


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