- #1
bobred
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Homework Statement
The temperature variation at the surface is described by a Fourier
series
[tex]\theta(t)=\sum^\infty_{n=-\infty}\theta_n e^{2\pi i n t /T}[/tex]
find an expression for the complex Fourier
series of the temperature at depth [itex]d[/itex] below the surface
Homework Equations
Solution of the diffusion equation
[tex]\theta(x,t)=\cos\left(\phi+\omega t-\sqrt{\dfrac{\omega}{2D}}x\right)\exp\left(-\sqrt{\dfrac{\omega}{2D}}x\right)[/tex]
The Attempt at a Solution
A the surface [itex]x=0[/itex] so
[tex]\theta(0,t)=\cos\left(\omega t + \phi\right)[/tex]
To find the coefficients [itex]\theta_n[/itex] I'm guessing I use the Fourier formula
[tex]\theta_n=\frac{1}{T}\int_0^T dt \, \theta(0,t) \exp\left( -2\pi i n t/T \right) [/tex]