Consider a semi-infinite string, initially level and at rest. We jiggle it up and down at the end sinusoidally. The string vibration problem can be formulated as follows:
$u_{tt} = c^2 u_{xx}$, $0 < x < \infty$, $t > 0$
$u(x, 0) = u_t(x, 0) = 0$, $0 < x < \infty$
$u(0, t) = \sin(2\pi t)$, $t > 0$
Solve the above problem by using the general solution
$u(x, t) = F(x - ct) + G(x + ct)$
Find the analytical expression and sketch the solution in the $x - t$ plane for $c = 1$ and at time instants
$t = 0.5, 1.0, 1.5, \text{and } 2.$
