Chapter 19

Visualizing PDE's, an introduction.

Traveling wave solutions to the wave equation.

Consider a horizontal guitar string vibrating after being plucked. If $U(t,x)$ is the vertical displacement (in centimeters) of the string over the horizontal position at distance $x$ cm from one end of the string at $t$ seconds past the initial time then the (1-dimensional) wave equation is \begin{equation} \frac{\partial^2 U}{\partial t^2} = k \frac{\partial^2 U}{\partial x^2}, \end{equation} where $k>0$ (with units $\textrm{cm}^2/\textrm{s}^2$) is related to the string's mass and tension and $k=c^2$ where $c$ is the wave speed.

Recall that for a PDE with independent variables $x$ and $t$, a solution is a traveling wave if it has the form $u(t,x) = w(x-c t)$ for a constant $c$ (called the speed of the wave) and a function $w(x)$ depending only one one variable. As in the textbook, we can find two traveling wave solutions to the wave equation \[ u_r(t,x) = \sin (x-\sqrt{k}t) \qquad \textrm{and} \qquad u_l(t,x) = \sin (x+\sqrt{k} t). \] Note that $u_r$ is moving to the right with speed $\sqrt{k}$ while $u_l$ is moving to the left with speed $\sqrt{k}$. In particular, we can identify the constant $k$ with the square root of the speed of waves.

Below is a visualization of the two traveling waves $u_r$ and $u_l$. Choose the parameter $k$ and hit the button "Run the traveling waves!" to see how the two waves travel. As you increase $k$, the speed of the traveling waves increases.

Heat and wave equations with Dirichlet boundary conditions.

Like the initial value problem for ODE’s, when solving PDE’s it is necessary to prescribe the behavior of the solution at some initial time. This is known as an initial condition. Because a PDE may take place on a region in space with data fixed at the boundary (such as the edge of the batter head of a drum), it is common to also prescribe the behavior at certain points. This is known as the boundary values (or boundary conditions). Let’s illustrate this for the heat and wave equations.

The heat equation. Consider the heat equation on a finite interval $0 \le x \le L$. This models the evolution of the temperature of a long skinny metal rod of length $L$. We impose the Dirichlet boundary condtions $u(t,0) = 0 = u(t,L)$ and intial condtion $u(0,x) = u_0(x)$:

\begin{equation} \label{eq:heat-equation-dir-bd-cond} \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2};\,\,\, u(t,0) = 0 = u(t,L);\,\,\, u(0,x) = u_0(x), \end{equation}

Define $\omega = \pi/L$. As checked in the textbook, $$u_{\text{heat}}(t,x) = e^{-\omega^2 t} \sin\left(\omega x\right)$$ is a solution to this "initial and boundary value" problem. The corresponding initial condition is $u_0(x) = \sin(\omega x)$.

A couple of salient features of solutions to the heat equation are present in this example. First of all, for $t$ very large the factor $e^{-\omega^2 t}$ becomes very small. In particular, $u(t,x) \approx 0$ for large $t$. Physically, this corresponds to the (reasonable) fact that if the ends of a rod are kept at a common fixed temperature then the rod will eventually settle down to that uniform temperature. For larger $L$ the coefficient $-\omega^2$ in the exponential is "less negative", so a long rod will settle down more slowly than a short rod.

The wave equation. An important difference between the wave equation and the heat equation is that the wave equation has a second-order time derivative (as opposed to first-order for the heat equation). Inspired by our experience with second-order ODE's, for which an IVP required specifying the value of the function and its derivative at the initial time, we are led to expect the need to specify the initial displacement $u(0,x) = u_0(x)$ and the initial velocity $(\partial u/\partial t)(0,x) = u_1(x)$ at all points in order to uniquely pin down a solution. In the case of plucking a guitar string, such combined initial conditions to describe subsequent motion of the string is quite reasonable. And as with the heat equation, we will need to impose boundary conditions, such as pinning the string down at its ends. In particular, Dirichlet boundary conditions are very natural for the wave equation. Her we consider

\[ \frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2};\,\,\, u(t,0) = 0= u(t,L);\,\,\, \,\,u(0,x) = \sin(\omega x),\,\, \frac{\partial u}{\partial t}(0,x) = 0. \]

Define $\omega = \pi/L$. As checked in the textbook, $$u_{\text{wave}}(t,x) = \cos(\omega t) \sin(\omega x)$$ satisfies the above wave equation with the given boundary and initial conditions.

Comparing with the heat equation, there is a significant difference: the exponential damping factor $e^{-\omega^2 t}$ in the heat equation solution has been replaced with the oscillatory function $\cos(\omega t)$, so the solution oscillates forever with an unchanging amplitude of $\sin(\omega x_0)$ over each point $x_0$.

Below is a visualization of $u_{\text{heat}}$ and $u_{\text{wave}}$. See how their long time behaviors differ. Observe that $u_{\rm{heat}}$ decays whereas $u_{\rm{wave}}$ oscillates as $t$ grows.