Martinez's Fourier Assignment: Difference between revisions
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(Created page with 'The purpose of this assignment is to solve any problem using Fourier series or Laplace transforms. I will compare the hit between a baseball bat and the baseball and the sound wa…') |
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Prove that a<sub>3</sub> = 0 for the waveform below: |
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The purpose of this assignment is to solve any problem using Fourier series or Laplace transforms. I will compare the hit between a baseball bat and the baseball and the sound wave it creates. It's exciting to see your favorite baseball team and hear that popping noise when the baseball is hit. When the ball hits the bat a vibration is form causing waves to form. |
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[[Image:Circuit8.png|thumb|400px|center]] |
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<math>\begin{align} |
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T &= 6 seconds\\ |
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a_n &= \frac{2}{T}\int_0^T f(t)\cos(n\omega_0t)\, dt\\ |
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b_n &= \frac{2}{T}\int_0^T f(t)\sin(n\omega_0t)\, dt\\ |
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\frac{2\pi}{\omega_0}\ &= 6\\ |
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\because \!\, T &= 6\\ |
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\therefore \!\,\omega_o &= \frac{pi}{3}\\ |
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a_3 &= \frac{2}{6}\int_0^6 f(t)\cos(3\omega_0t)\, dt\\ |
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a_3 &= \frac{1}{3}[\int_2^3 \cos(\pi*t)\, dt + \int_3^4 \cos(3\omega_ot)\, dt\\ |
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\because \!\, \omega_o &= \frac{pi}{3}\\ |
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a_3 &= \frac{1}{3}[\frac{10}{pi}(sin(3\pi)-sin(2\pi)+\frac{5}{pi}(sin(4\pi)-sin(3\pi)]\\ |
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a_3 &= \frac{1}{3}[\frac{10}{pi}(0)+\frac{5}{pi}(0)]\\ |
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a_3 &= 0\\ |
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\end{align} |
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Latest revision as of 01:03, 13 December 2010
Prove that a3 = 0 for the waveform below:
<math>\begin{align} T &= 6 seconds\\ a_n &= \frac{2}{T}\int_0^T f(t)\cos(n\omega_0t)\, dt\\ b_n &= \frac{2}{T}\int_0^T f(t)\sin(n\omega_0t)\, dt\\ \frac{2\pi}{\omega_0}\ &= 6\\ \because \!\, T &= 6\\ \therefore \!\,\omega_o &= \frac{pi}{3}\\ a_3 &= \frac{2}{6}\int_0^6 f(t)\cos(3\omega_0t)\, dt\\ a_3 &= \frac{1}{3}[\int_2^3 \cos(\pi*t)\, dt + \int_3^4 \cos(3\omega_ot)\, dt\\ \because \!\, \omega_o &= \frac{pi}{3}\\ a_3 &= \frac{1}{3}[\frac{10}{pi}(sin(3\pi)-sin(2\pi)+\frac{5}{pi}(sin(4\pi)-sin(3\pi)]\\ a_3 &= \frac{1}{3}[\frac{10}{pi}(0)+\frac{5}{pi}(0)]\\ a_3 &= 0\\ \end{align}