Editing Harmonic analysis (section)

==Applied harmonic analysis==

[[File:Bass Guitar Time Signal of open string A note (55 Hz).png|thumb|400 px| Bass-guitar time signal of open-string A note (55&nbsp;Hz)]]
[[File:Fourier Transform of bass guitar time signal.png|thumb|400 px| Fourier transform of bass-guitar time signal of open-string A note (55&nbsp;Hz)<ref>{{Cite web |date=2015-07-07 |title=A More Accurate Fourier Transform |url=https://sourceforge.net/projects/amoreaccuratefouriertransform/ |access-date=2024-08-26 |website=SourceForge |language=en}}</ref>]]

Many applications of harmonic analysis in science and engineering begin with the idea or hypothesis that a phenomenon or signal is composed of a sum of individual oscillatory components.  Ocean [[tide]]s and vibrating [[String (music)|strings]] are common and simple examples.  The theoretical approach often tries to describe the system by a [[differential equation]] or [[system of equations]] to predict the essential features, including the amplitude, frequency, and phases of the oscillatory components.  The specific equations depend on the field, but theories generally try to select equations that represent significant principles that are applicable.

The experimental approach is usually to [[Data collection|acquire data]] that accurately quantifies the phenomenon.  For example, in a study of tides, the experimentalist would acquire samples of water depth as a function of time at closely enough spaced intervals to see each oscillation and over a long enough duration that multiple oscillatory periods are likely included.  In a study on vibrating strings, it is common for the experimentalist to acquire a sound waveform sampled at a rate at least twice that of the highest frequency expected and for a duration many times the period of the lowest frequency expected.

For example, the top signal at the right is a sound waveform of a bass guitar playing an open string corresponding to an A note with a fundamental frequency of 55&nbsp;Hz.  The waveform appears oscillatory, but it is more complex than a simple sine wave, indicating the presence of additional waves.  The different wave components contributing to the sound can be revealed by applying a mathematical analysis technique known as the [[Fourier transform]], shown in the lower figure.  There is a prominent peak at 55&nbsp;Hz, but other peaks at 110&nbsp;Hz, 165&nbsp;Hz, and at other frequencies corresponding to integer multiples of 55&nbsp;Hz.  In this case, 55&nbsp;Hz is identified as the fundamental frequency of the string vibration, and the integer multiples are known as [[harmonic]]s.