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Lecture 11: Fourier Analysis

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Lecture Topics

On a piece of graph paper, a flattened disc has been drawn. Two curved red lines come out from the bottom. A small dome is on top.
  • Fourier Analysis
  • Time Evolution of Pulses
  • Spectrum

Learning Objectives

By the end of this lecture, you should:

  • write the general expression for a Fourier series.
  • determine the terms of the Fourier series of a simple function.
  • understand how even and odd functions contribute to Fourier integrals.
  • be able to sketch the build-up of a function from the parts of its Fourier series.
  • understand how the normal modes, each at its own frequency, contribute to development of a traveling wave.
  • describe why the place where a musical instrument string is plucked affects the sound.
  • construct a power density spectrum from the Fourier components, making a spectrum.

Lecture Activities

Check Yourself

  • The term "Fourier analysis" that is the title of this lecture is generally used to include all of the techniques demonstrated here. In fact, there are two distinct and complementary operations done from a practical (physics) point of view. What are they?

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The first step actually is analysis, which means breaking something down into parts. Operations such as \[{A_m} = \tfrac{1}{L}\int_0^{2L} {f(x)\cos \tfrac{{mx\pi }}{L}dx}\] provide "parts" from which the function is made when viewed as a Fourier series. Formation of the Fourier series is actually synthesis, or building up, of a function. The general form\[f(x) = \tfrac{{{A_0}}}{2} + \sum\limits_{m = 1}^\infty {{A_m}\cos mx} + \sum\limits_{m = 1}^\infty {{B_m}\sin mx} \]puts the pieces back together to rebuild the function from the coefficients.

 

  • Spectra of stars were discussed earlier in the course. In these, the light intensity over a range of wavelengths of light is displayed. At wavelengths characteristic of resonance in the various atoms in the star, there may be, for example, "absorption lines". Discuss the aspects of the Fourier series that are like a spectrum, and some aspects that are different.

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A display of light spread out by wavelength is another way of spreading it out by frequency (it just happens that optical tools usually more naturally work with wavelengths). Similarly, a Fourier synthesis incorporates strengths (the A and B coefficients’ values) at different frequencies since each m corresponds to a harmonic term m times the fundamental. A major difference is that the light display is continuous while only discrete frequencies are possible in the Fourier analysis. However, it can be helpful to think, carefully, of the Fourier coefficients giving a "spectrum". If a body can move in many normal modes, and for a given motion one of the coefficients is larger than the others, the "spectrum" would let you know that that is the dominant mode for that given motion.

 

  • We saw previously that a single traveling Gaussian pulse can satisfy the wave equation. Can we use Fourier series to analyze such a pulse?

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Strictly speaking, no. The general form \[f(x) = \tfrac{{{A_0}}}{2} + \sum\limits_{m = 1}^\infty {{A_m}\cos \tfrac{{mx\pi }}{L}} + \sum\limits_{m = 1}^\infty {{B_m}\sin \tfrac{{mx\pi }}{L}}\]is for functions with period 2L and is itself a periodic function with that period. Since the Gaussian pulse is not periodic, it cannot be analyzed. There may be circumstances where the tools could be applied to a periodic extension of the signal, and give some information, but this would need to be done with care.

 

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