Wavy PDE Intro

Partial differential equations are the to use if you want to study quantities varying over time and space. This intro presents a rather high-level introduction in the context of vibration.

Pluck

Conditions

If the vertical displacement of the string at position $x$ and time $t$ is $u(x,t)$, then $u$ should satisfy

  • $u(0,t)=u(1,t)=0$ (fixed endpoints)
  • $u_t(x,0)=g(x)=0$ (zero initial velocity)
  • $\displaystyle u(x,0)= f(x) = A\begin{cases} \frac{x}{x_0} & \text{if } 0 \leq x \leq x_0 \\ \frac{x-1}{x_0-1} & \text{if } x_0 \leq x \leq 1. \end{cases}$
    (sawtooth initial condition)
  • $u_{tt}+D u_t=c^2u_{xx}$ (damped wave equation)

Solution

There's a lovely solution:

$$u_{A,x_0}(x,t) = -2 A e^{-d\,t/2} \sum_{n=1}^{\infty} \left(\frac{\sin (\pi n x) \sin (\pi n x_0)\cos\left(\frac{t}{2} \sqrt{4 \pi ^2 c^2 n^2-d^2}\right)}{\pi ^2 n^2 \, x_0 (x_0-1)} + \cdots \right.$$
$$\left. \cdots \frac{\sin (\pi n x) \sin (\pi n x_0)\sin \left(\frac{t}{2} \sqrt{4 \pi ^2 c^2 n^2-d^2}\right)}{\pi ^2 n^2 \, x_0 (x_0-1) \sqrt{4 \pi ^2 c^2 n^2-d^2}}\right).$$

The basic wave equation

$$u_{tt} = c^2\,u_{xx}.$$

There are lots of solutions. The important ones depend largely on the physical problem under consideration.

Some solutions on $I=[0,1]$

Some fundamental solutions of the wave equation on the unit interval look like

$$u(x,t) = \cos(c n \pi t)\sin(n \pi x).$$

It's easy to check!

You can also sorta see these in the "lovely solution" to the "pluck" problem.

Animated solution on $I$

Two dimensional vibration

Many of these ideas extend into two dimensions. The damped wave equation, for example, looks like

$$u_{tt} + D u_t = c^2(u_{xx}+u_{yy}).$$

Rotationally symmetric fundamental modes

Drums

Advection waves

The first PDE models that we'll play with in detail will be first order. The simplest example is the so-called advection model: $$\begin{aligned} u_t + cu_x &= 0 \\ u(x,0) &= F(x). \end{aligned}$$ It's not hard to show that $u(x,t) = F(x-ct)$ satisfies both the PDE and the inital condition.

Example

If $c=1$ and $F(x) = e^{-x^2}$, the our solution is $$u(x,t) = e^{-(x-t)^2}.$$ If we animate this as a function of time, we generate a right travelling wave: