Heat equation, periodic heating of a surface

[bobred]

Homework Statement

The temperature variation at the surface is described by a Fourier
series
$$\theta(t)=\sum^\infty_{n=-\infty}\theta_n e^{2\pi i n t /T}$$
find an expression for the complex Fourier
series of the temperature at depth $d$ below the surface

Homework Equations

Solution of the diffusion equation
$$\theta(x,t)=\cos\left(\phi+\omega t-\sqrt{\dfrac{\omega}{2D}}x\right)\exp\left(-\sqrt{\dfrac{\omega}{2D}}x\right)$$

The Attempt at a Solution

A the surface $x=0$ so
$$\theta(0,t)=\cos\left(\omega t + \phi\right)$$

To find the coefficients $\theta_n$ I'm guessing I use the Fourier formula

$$\theta_n=\frac{1}{T}\int_0^T dt \, \theta(0,t) \exp\left( -2\pi i n t/T \right)$$

 

[mfb]

I don't think you can calculate the $\theta_n$. They are some given (but unknown) constants, and you have to modify your solution to fit this constraint.

You can start with an easy case: imagine $\theta_1=1$ and $\theta_n=0$ for all other n. That makes the temperature at x=0 a sine. Can you find the temperature at depth d?
What happens with $\theta_2=3$ and $\theta_n=0$ for all other n? What happens in the general case?

 

[bobred]

Typo, $\theta(x,t)$ should be
$\theta(x,t)=A\cos\left(\phi+\omega t-\sqrt{\dfrac{\omega}{2D}}x\right)\exp\left(-\sqrt{\dfrac{\omega}{2D}}x\right)$ (c)
Advice from my tutor was
'The temperature is a superposition of solutions found in (c).
On the surface x=0...use this to determine the coefficients in the above and complete.'
Assuming I have got (c) correct!
James

 

[bobred]

Appologies, $A=7.5^{\circ}$C, $\omega=\frac{\pi}{43200} s^{-1}$, $\phi=\frac{4}{3}$ and $D=5\times10^{-7} m^2/s$

 

[mfb]

Where do those numbers come from? You cannot fix them like that for the problem.

A superposition of those solutions is the right approach.

 

[bobred]

These were given earlier in the question and you are right have no baring on this.

So to get the coefficients I should set x=0 and use the Fourier formula to find them?

 

[mfb]

Yes.

 

[bobred]

I have done some calculations but getting trivial answers.
I am taking
$$\theta(0,t)=\cos\left(2\pi n \ t/T + \phi\right)$$
and converting it to its complex exponential form and inserting into
$$\theta_n=\frac{1}{T}\int_0^T dt \, \theta(0,t) \exp\left( -2\pi i n t/T \right)$$
Should phi be included?

 

[pasmith]

You probably want to use the solution of the diffusion equation in the form $$Ce^{i\omega t}\exp\left(-\sqrt{\frac{\omega}{2D}}x - i\sqrt{\frac{\omega}{2D}}x \right)$$ with $x$ measuring distance below the surface.

 

[mfb]

You'll need phi if θ_n can be complex.

I still don't see what "inserting into [long equation]" is supposed to mean, but I agree that you don't need long calculations.

 

[bobred]

By inserting into, I mean to work out the coefficients.
I'm a bit lost at the moment.

 

[pasmith]

By inserting into, I mean to work out the coefficients.
I'm a bit lost at the moment.

You should treat the coefficients $\theta_n$ as known. You are asked for "an expression for the complex Fourier
series of the temperature at depth d below the surface" so you are looking for an expression of the form $$\theta(d,t) = \sum_{n=-\infty}^\infty e^{2n\pi it/T}f_n(d)$$ where $e^{2n\pi it/T}f_n(x)$ is a solution of the diffusion equation with $f_n(0) = \theta_n$.

 

[bobred]

So would
$$f_n(0)=\cos\left( 2\pi n t /T + \phi \right)$$