Arbitrary Waveforms Propagating to the Left or Right

The first solution we can discern by noting that the wave equation equates a second derivative in time to a second derivative in space. Suppose we write the solution as $f(u)$ where $u$ is an unknown function of $x$ and $t$ and substitute it into the differential equation and use the chain rule:

$$\frac{d^2f}{du^2}(\frac{du}{dt})^2 - v^2 \frac{d^2f}{du^2}(\frac{du}{dx})^2 = 0$$ (108)

or

$$\frac{d^2f}{du^2}\left\{(\frac{du}{dt})^2 - v^2 (\frac{du}{dx})^2 \right\}= 0$$ (109)

$$\frac{du}{dt} = \pm v \frac{du}{dx}$$ (110)

with a simple solution:

$$u = x \pm vt$$ (111)

What this tells us is that any function

$$y(x,t) = f(x \pm vt)$$ (112)

satisfies the wave equation. Any shape of wave created on the string and propagating to the right or left is a solution to the wave equation, although not all of these waves will vanish at the ends of a string.