19. The transverse displacement $u(x, t)$ of a string obeys the wave equation
$\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$,
where $t$ and $x$ denote time and position respectively and $c$ is a constant. Show that
a possible solution is given by
$u(x, t) = f(x - ct) + g(x + ct)$,
where $f$ and $g$ are any functions that can be differentiated twice.
Suppose that the string stretches to infinity in both directions, and satisfies the
intial conditions,
$u(x, 0) = 0$,
$\frac{\partial u}{\partial t}(x, 0) = \frac{x}{(1 + x^2)^2}$.
Show that the initial conditions can be satisfied by taking $f = -g$ for suitable $f$.
Find the solution $u(x, t)$ and sketch its behaviour.
