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Homework Help: Application of Fourier series to pressure waves

  1. May 20, 2010 #1
    1. The problem statement, all variables and given/known data

    Assume that a pressure wave produces a change in pressure at a point in space [tex]\Delta P(t)[/tex] which is proportional to a sawtooth function of frequency f = 1/2 Hz.

    (i) If the amplitude of the pressure wave is [tex]\Delta P_{0}[/tex], write down an expression for [tex]\Delta P(t)[/tex].

    (ii) Two oscillators, designed to respond to changes in pressure, resonate at a frequency f = 3/2 Hz and f = 5/4 Hz respectively. When the pressure wave encounters the oscillators, which of these will resonate and why?

    2. Relevant equations

    The Fourier expansion of a sawtooth function [tex] f(x) = x, -1 < x < 1 [/tex] is given by

    [tex] f(x) = \sum_{r=1}^{\infty} \frac{-2(-1)^{r}}{\pi r} \sin{\pi r x} [/tex]


    3. The attempt at a solution

    Is the previous equation simply the answer to part (i) with the x's replaced with t's? Or is there a bit more to it? I think I'm missing something pretty obvious.

    For part (ii), we know that [tex] r \pi t = 2 \pi ft [/tex] and hence [tex]r = 2f[/tex]. For f = 3/2 Hz, r = 3 and for f = 5/4 Hz, r = 5/2. Would only the first resonate as it is the only one that produces r that is an integer and hence satisfies the expansion above?
     
  2. jcsd
  3. May 22, 2010 #2

    vela

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    You have to get the amplitude right as well.
    Yes. The sawtooth pressure wave contains harmonics with frequencies equal to integral multiples of 1/2 Hz, so it has a 3/2-Hz component but lacks a 5/4-Hz component and will excite only the one resonator.
     
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