Science:Math Exam Resources/Courses/MATH307/April 2012/Question 06 (b)

MATH307 April 2012

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Question 06 (b)
Explain the physical significance of the values $|c_{n}|$ and how these relate to $\omega _{n}$. If t is measure in days, and the time period T encompasses several years, which values of cn might you expect to be the largest absolute value? (Think about what time scales you expect temperature to fluctuate over.)

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Hint

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Since $c_{n}$ is a complex number with real and imaginary part, we write $c_{n}$ in its complex exponential form $c_{n}$ = $\left|c_{n}{}\right|e^{2\pi i\phi _{n}}$ Therefore $c_{n}$ reflects the original temperature function’s frequency spectrum. If we put the above expression back into the Fourier series representation of y(t), we have $y(t)=\sum _{n=-\propto }^{\propto }c_{n}e^{2\pi i{\frac {n}{T}}t}=\sum _{n=-\propto }^{\propto }\left|c_{n}\right|e^{2\pi i\phi _{n}}e^{2\pi i{\frac {n}{T}}t}=\sum _{n=-\propto }^{\propto }\left|c_{n}\right|e^{2\pi i(\omega _{n}t+\phi _{n})}$ t=0 or t=T would make $c_{n}$ the largest. Because both t=0 and t=T would make $e^{2\pi i{\frac {n}{T}}t}$ = 1. So $c_{0}$ and $c_{T}$ have the largest absolute value. The values of $c_{n}$ might you expect to have the largest absolute value is the frequency for the maximum temperature swings over the course of years. This would probably be seasonal fluctuations, so it would be $c_{n}$ for about 12 months.