Wavy PDE Intro

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

A string?


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)


There's a lovely solution:

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

Configurable strings


Now the conditions look like

  • $u_{tt}+D u_t=c^2u_{xx}$
  • $u(0,t)=u(1,t)=0$
  • $u(x,0)=f(x)$
  • $u_t(x,0)=g(x)$,

where $f$ and $g$ are arbitrary specifications of initial position and velocity.


And the lovely solution is..

$$\begin{aligned} u(x,t) &= 2 \sum_{n=1}^{\infty} e^{-dt/2} \,\Biggl(\cos\!\left(\frac{t}{2}\sqrt{4c^2n^2\pi^2-d^2}\right) \int_0^1 \sin(n\pi \chi)u_0(\chi)d\chi \cdots \\[25mu] &\qquad + \frac{\sin\!\left(\frac{t}{2}\sqrt{4c^2n^2\pi^2-d^2}\right)}{\sqrt{4c^2n^2\pi^2-d^2}} \int_0^1 \sin(n\pi \chi) (d u_0(\chi) + 2v_0(\chi)) d\chi \Biggr) \sin(n\pi x). \end{aligned}$$

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.

Solution on $I$

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

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

Animated solution on $I$

Some drums