[Fourier and PDEs] Heat and Wave Equations

1. Nondimensionalizing the Heat Equation

An infinite slab of material of constant thermoconductivity $\kappa$ occupies the region of $\mathbb{R}^3$ defined by $0 \leq x \leq a$. Initially, the temperature $T(x,t)$ of the slab is given by \[ T(x,0) = 0. \]

Subsequently the face at $x = 0$ is kept at zero temperature at the face at $x = a$ is kept at unit temperature.

a) Removing the steady-state profile

Write down the boundary conditions satisfied by $T$. Show that \[ U(x,t) = T(x,t) - \frac{x}{a}, \]

satisfies the heat equation and deduce the boundary conditions on $U$.

Spoiler

b) Fourier series solution

Show that for $t > 0$, \[ T(x,t) = \frac{x}{a} + \sum_{n=1}^\infty B_n \exp \left( -\frac{n^2\pi^2 \kappa t}{a^2}\right) \sin \left( \frac{n\pi x}{a}\right), \]

and given an explicit expression for $B_n$. Compute (in series form) the heat flux on the face $x = a$.

Spoiler

2. Heat Equation: uniqueness and discontinuous initial conditions

A slab of conducting material occupies the region $0 \leq x \leq L$, bounded by the planes $x = 0$ and $x = L$, and the temperature $T(x,t)$ is a solution of the heat equation $T_t = \kappa T_{xx}$. The initial temperature is \[ T(x,0) = \begin{cases} 0 & \text{for } 0 \leq x < a \text{ and } b < x \leq L, \\ 1 & \text{for } a \leq x \leq b \end{cases} \]

where $0 < a < b < L$, and the faces $x = 0$ and $x = L$ are thermally insulated.

What conditions does $T$ satisfy on the boundaries? Express the solution $T$ as a sum of an infinite series and prove that $T$ is uniquely determined. What is the behaviour of the temperature in the limit as $t \to \infty$?

Spoiler

3. Wave equation: derivation and energy

Derive the wave equation $T y_{xx} = \rho y_{tt}$ for the transverse displacement $y(x,t)$ of a stretched string. Let \[ E \equiv \frac{1}{2} \left[ T y_x^2 + \rho y_t^2\right] \quad \text{and} \quad P \equiv T y_x y_t. \]

Show that \[ \pd{E}{t} = \pd{P}{x} \quad \text{and} \quad T\pd{E}{x} = \rho \pd{P}{t}, \]

and deduce that $P$ and $E$ are solutions of the wave equation. What are the physical interpretations of $E$ and $P$?

Spoiler
1) The potential energy of a string or spring is $T (l - \delta x)^2$ where $l$ is the stretched length of string in $\delta x$