(Solved): Consider the one-dimensional wave equation and initial conditions \[ \begin{aligned} u_{t t} &=4 u ...

Consider the one-dimensional wave equation and initial conditions \[ \begin{aligned} u_{t t} &=4 u_{x x} \\ u(x, 0) &=u_{0}(x)=\sin (x) \\ u_{t}(x, 0) &=u_{1}(x)=0 \\ u(0, t) &=0 \end{aligned} \] with \( x \in[0, \infty) \) and \( t \in[0, \infty) \). Find the solution \( u(x, t) \). Clearly state any modifications you need to make to the given functions and evaluate any integrals.