(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. 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.

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# Homework Help: Problem solving Heat Diffusion Equation

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