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Homework Statement
One face of a thick uniform layer is subject to a sinusoidal temperature variation of angular frequency ω. SHow that the damped sinusoidal temperature oscillation propagate into eh layer and give an expression for the decay length of the oscillation amplitude.
A cellar is built underground covered by a ceiling which 3m thick made of limestone. The Outside temperature is subject to daily fluctuations of amplitude 10 C and annual fluctuations of 20 C. Estimate the magnitude of the daily and annual temperature variation within the cellar.
Homework Equations
The Attempt at a Solution
I am unable to solve the first part of this question.
Take the diffusion equation
[tex] \frac{\partial T}{\partial t} = -D\frac{\partial^2T}{\partial x^{2}} [/tex]
Using separation of variable method:
Let [tex] T=X(x)F(t) [/tex]
[tex]X \frac{dF}{dt} = -DF\frac{d^2X}{DX^2} [/tex]
[tex]-D\frac{dF}{dt}=\frac{1}{X}\frac{d^2X}{DX^2}= k [/tex]
where k is the separation constant.
These separate into two equation which I solve to give
[tex] X=Ae^{\sqrt{k}x}+Be^{-\sqrt{k}x} [/tex]
[tex]F=Ce^{-Dkt} [/tex]
By superpositon principle
[tex] T=\sum (A_ke^{\sqrt{k}x}+B_ke^{-\sqrt{k}x})e^{-Dkt} + A_0 + B_0x [/tex]
where C has been absorbed into A and B.
Then taking the boundary conditions:
At x →∞ T→0, which shows A_k → 0
[tex] T=\sum B_ke^{-\sqrt{k}x}e^{-Dkt} + B_0x [/tex]
Then apply conditon that at x=0 T [itex]\propto[/itex] sinωt.
[tex] e^{-Dkt} = sinwt [/tex]
However here is where I am stuck, I do not see how proceed further. Is my solution so far correct, it does not seem so as I seem to have the wrong form. Or have I chosen the wrong form for my separation coefficient?
Any help would be greatly appreciated.