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.

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:

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

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

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.

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

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

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: