(i) Show that an arbitrary $C^{2}$ solution of the one-dimensional wave equation $u_{t t}-u_{x x}=0$ can be written in the form $u=F(x-t)+G(x+t)$.

Hence, deduce the formula for the solution at arbitrary $t>0$ of the Cauchy problem

$$u_{t t}-u_{x x}=0, \quad u(0, x)=u_{0}(x), \quad u_{t}(0, x)=u_{1}(x)$$

where $u_{0}, u_{1}$ are arbitrary Schwartz functions.

Deduce from this formula a theorem on finite propagation speed for the onedimensional wave equation.

(ii) Define the Fourier transform of a tempered distribution. Compute the Fourier transform of the tempered distribution $T_{t} \in \mathcal{S}^{\prime}(\mathbb{R})$ defined for all $t>0$ by the function

$$T_{t}(y)= \begin{cases}\frac{1}{2} & \text { if }|y| \leqslant t \\ 0 & \text { if }|y|>t\end{cases}$$

that is, $\left\langle T_{t}, f\right\rangle=\frac{1}{2} \int_{-t}^{+t} f(y) d y$ for all $f \in \mathcal{S}(\mathbb{R})$. By considering the Fourier transform in $x$, deduce from this the formula for the solution of $(*)$ that you obtained in part (i) in the case $u_{0}=0$.