## Wave propagation

Determine a formula for the position $u(x,t)$ each point $x$ of the string at time $t$ in the form of a trigonometric series. Also, determine a formula for the coefficients of the series for $f$ and $g$.

$a^2 \frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2}$ and $0; $t>0$

$u(0,t) = 0$; $u(L,t) = 0$; $t>0$

$u(x,0) = f(x)$ and $\frac{\partial u}{\partial t}(x,0) = 0$

$f(x)$ is continuous split and regular period 2L which describes the initial position of the string