Heat Equation with Periodic Boundary Conditions

[StretchySurface]

Homework Statement

Find General Solution of Heat Equation on a Ring.

Relevant Equations

See below

I'm solving the heat equation on a ring of radius $R$. The ring is parameterised by $s$, the arc-length from the 3 o'clock position. Using separation of variables I have found the general solution to be:

$$U(s,t) = S(s)T(t) = (A\cos(\lambda s)+B\sin(\lambda s))*\exp(-\lambda^2 kt)$$.

Here is my confusion:
Periodicity demands that $U(s',t) = U(s' + 2\pi R, t).$ This means that
$$\cos(\lambda s') = cos(\lambda s' + \lambda*2\pi R)$.$

This finally gives condition that $\lambda = k/R%$ where $k$ is an integer. This is the correct answer as told by my professor/textbook.

However, periodicity can also demands $U(s',t) = U(s' + 2\pi nR, t)$, i.e revolving around the ring $n$ complete times. But this leads to the condition $2\pi \lambda nR = 2 \pi m$ where $m$ and $n$ are arbitrary integers. This gives us $\lambda = m/(nR)$. This doesn't seem to be the correct answer but I don't understand why and rational number divided by $R$ is not an eigenvalue.

 

[BvU]

Something with a period of $4\pi$ does not necessarily satisfy the condition that $f(\theta)=f(\theta+2\pi)$ !

 

[StretchySurface]

Does that mean the following is untrue: We require $U(s′,t)=U(s′+4πR,t)$?

 

[BvU]

It is a true statement, but:
It's a necessary but insufficient condition.

 

[StretchySurface]

Okay, so I don't really see the point you're trying to convey and how it relates to my question. Below, I'll write out a set steps in reasoning to see if you can pinpoint the error I'm making.

Let's only consider the case $n = 2$ i.e traveling around the ring twice.
Periodicity demands that $U(s',t)=U(s'+4πR,t)$.
This means that $2\lambda R = k$, for any integer $k$.
Thus $\lambda_{k} = k/(2R)$ which includes some values not mentioned in the notes I'm working from.

 

[BvU]

some values not mentioned

Yes. And those do NOT satisfy the condition $U(s′,t)=U(s′+2\pi R,t)$. Do you propose to drop that condition ?

 

[Orodruin]

Put slightly differently, 2pi periodic functions are automatically 4pi periodic but 4pi periodic functions are not necessarily 2pi periodic.

 

[StretchySurface]

Ah okay, now I understand! Thank you very much!

 

[dRic2]

Btw if $\lambda$ is the inverse of a length, and $k$ is an integer (I suppose dimensionless), that exponential doesn't look good to me.

Edit: ok the $k$ in the exponential is the diffusion coefficient (I suppose)... my bad

 

[Orodruin]

Btw if $\lambda$ is the inverse of a length, and $k$ is an integer (I suppose dimensionless), that exponential doesn't look good to me.

Edit: ok the $k$ in the exponential is the diffusion coefficient (I suppose)... my bad

I think you can be excused as the OP used $k$ both for the diffusion coefficient as well as an integer ...