Fourier series: Difference between revisions
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==Propagating waveforms== | ==Propagating waveforms== | ||
Fourier's theorem states that any | Fourier's theorem states that any real-valued periodic function can be expressed as a sum of sinusoidal functions with periods related to ''P'':<ref name=Schaum/> | ||
:<math>f(\xi)=a_0 +\sum_1^\infty a_n\cos\left(\frac{2\pi}{P/n}\xi+\varphi_n\right) \ , </math> | :<math>f(\xi)=a_0 +\sum_1^\infty a_n\cos\left(\frac{2\pi}{P/n}\xi+\varphi_n\right) \ , </math> | ||
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:<math>f(t)=a_0 +\sum_1^\infty a_n \cos \left( n\omega_0t +\varphi_n \right) \ , </math> | :<math>f(t)=a_0 +\sum_1^\infty a_n \cos \left( n\omega_0t +\varphi_n \right) \ , </math> | ||
where ω<sub>0</sub> = 2π/''T'' is called the fundamental frequency and its multiples 2ω<sub>0</sub>, 3ω<sub>0</sub>,... are called ''harmonic frequencies'' and the terms | where ω<sub>0</sub> = 2π/''T'' is called the fundamental frequency and its multiples 2ω<sub>0</sub>, 3ω<sub>0</sub>,... are called ''harmonic frequencies'' and the cosine terms are called ''harmonics'' of ƒ. A function ''ƒ(x)'' of spatial period λ, can be synthesized as a sum of harmonic functions whose wavelengths are integral sub-multiples of λ (''i.e.'' λ, λ/2, λ/3, ''etc.''):<ref name=Schaum/> | ||
:<math>f(x)=a_0 +\sum_1^\infty a_n\cos\left(\frac{2\pi}{\lambda/n}x+\varphi_n\right) \ . </math> | :<math>f(x)=a_0 +\sum_1^\infty a_n\cos\left(\frac{2\pi}{\lambda/n}x+\varphi_n\right) \ . </math> |
Revision as of 12:19, 4 June 2012
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In mathematics, the Fourier series, named after Joseph Fourier (1768—1830), refers to an infinite series representation of a periodic function ƒ of a real variable ξ, of period P:
In the case of a complex-valued function ƒ(ξ), Fourier's theorem states that an infinite series, known as a Fourier series, is equivalent (in some sense) to such a function:
where the coefficients {cn} are defined by
In what sense it may be said that this series converges to ƒ(ξ) is a complicated question.[1][2] However, physicists being less delicate than mathematicians in these matters, simply write
and usually do not worry too much about the conditions to be imposed on the arbitrary function ƒ(ξ) of period P in order that this expansion converge to the function.
Propagating waveforms
Fourier's theorem states that any real-valued periodic function can be expressed as a sum of sinusoidal functions with periods related to P:[3]
a series of cosines with various phases {φn}.
If the function is a fixed waveform propagating in time, we may take ξ as:
where x is a position in space, v is the speed of propagation and t is the time. The period in space at a fixed instant in time is called the wavelength λ=P, and the period in time at a fixed position in space is called the period T=λ/v. Thus, a function periodic in time with period T can be expressed as a Fourier series:[4]
where ω0 = 2π/T is called the fundamental frequency and its multiples 2ω0, 3ω0,... are called harmonic frequencies and the cosine terms are called harmonics of ƒ. A function ƒ(x) of spatial period λ, can be synthesized as a sum of harmonic functions whose wavelengths are integral sub-multiples of λ (i.e. λ, λ/2, λ/3, etc.):[3]
References
- ↑ G. H. Hardy, Werner Rogosinski (1999). “Chapter IV: Convergence of Fourier series”, Fourier Series, Reprint of Cambridge University Press 1956 ed. Courier Dover Publications, pp. 37 ff. ISBN 0486406814.
- ↑ For an historical account, see Hans Niels Jahnke (2003). “§6.5 Convergence of Fourier series”, A History of Analysis. American Mathematical Society, pp. 178 ff. ISBN 0821826239.
- ↑ 3.0 3.1 Eugene Hecht (1975). Schaum's Outline of Theory and Problems of Optics. McGraw-Hill Professional. ISBN 0070277303.
- ↑ A.V.Bakshi U.A.Bakshi (2008). Circuit Analysis. Technical Publications. ISBN 8184310579.