(a) By introducing the variables $\xi=x+c t$ and $\eta=x-c t$ (where $c$ is a constant), derive d'Alembert's solution of the initial value problem for the wave equation:

$$u_{t t}-c^{2} u_{x x}=0, \quad u(x, 0)=\phi(x), \quad u_{t}(x, 0)=\psi(x)$$

where $-\infty<x<\infty, t \geqslant 0$ and $\phi$ and $\psi$ are given functions (and subscripts denote partial derivatives).

(b) Consider the forced wave equation with homogeneous initial conditions:

$$u_{t t}-c^{2} u_{x x}=f(x, t), \quad u(x, 0)=0, \quad u_{t}(x, 0)=0$$

where $-\infty<x<\infty, t \geqslant 0$ and $f$ is a given function. You may assume that the solution is given by

$$u(x, t)=\frac{1}{2 c} \int_{0}^{t} \int_{x-c(t-s)}^{x+c(t-s)} f(y, s) d y d s$$

For the forced wave equation $u_{t t}-c^{2} u_{x x}=f(x, t)$, now in the half space $x \geqslant 0$ (and with $t \geqslant 0$ as before), find (in terms of $f$ ) the solution for $u(x, t)$ that satisfies the (inhomogeneous) initial conditions

$$u(x, 0)=\sin x, \quad u_{t}(x, 0)=0, \quad \text { for } x \geqslant 0$$

and the boundary condition $u(0, t)=0$ for $t \geqslant 0$.