Wave Equation
Summary
In this lecture we begin the treatment of the wave equation $$ u_{tt} - \Delta u =0 $$ on $\mathbb{R}^n \times (0,\infty)$. We first consider the case in one spatial dimensions and obtain D’Alembert’s solution formula for the solution of the initial-value problem
$$ \left\{ \begin{alignedat}{2} u_{tt} -\Delta u &= 0\quad &&\text{in}~\mathbb{R} \times(0,\infty)\\ u=g\quad u_t&=h&&\text{on}~\mathbb{R} \times \{t=0\} \end{alignedat} \right. $$
We then move on to the initial-boundary value problem posed on the half-line, that is,
$$ \left\{ \begin{alignedat}{2} u_{tt} -\Delta u &= 0\quad &&\text{in}~\mathbb{R} \times(0,\infty)\\ u=g,\quad u_t&=h&&\text{on}~\mathbb{R} \times \{t=0\}\\ u&=0, &&\text{on}~\{x=0\}\times (0,\infty) \end{alignedat} \right. $$
where $\mathbb{R}_+ :=(0,\infty)$, and obtain a representation for its solution.
Spherical Means
Finally, we introduce the spherical means $$ \begin{aligned} U(x;r,t)&:=\frac{1}{\mathrm{surf}(\partial B(x;r))}\int\limits_{\partial B(x;r)} u(y,t)\, \mathrm{d}S_y\\ G(x;r)&:=\frac{1}{\mathrm{surf}(\partial B(x;r))}\int\limits_{\partial B(x;r)} g(y)\, \mathrm{d}S_y\\ H(x;r)&:=\frac{1}{\mathrm{surf}(\partial B(x;r))}\int\limits_{\partial B(x;r)} h(y)\, \mathrm{d}S_y \end{aligned} $$
for the problem (IVP) posed on $\mathbb{R}^n \times (0,\infty)$ $$ \left\{ \begin{alignedat}{2} u_{tt} -\Delta u &= 0\quad &&\text{in}~\mathbb{R}^n \times(0,\infty)\\ u=g\quad u_t&=h&&\text{on}~\mathbb{R}^n \times \{t=0\} \end{alignedat} \right. $$ and show that they satisfy Euler-Poisson-Darboux equation:
Fix $x\in\mathbb{R}^n$, let $u\in C^m(\mathbb{R}^n\times[0,\infty])$ satisfy (IVP). Then $U\in C^m\left(\overline{\mathbb{R}}_+\times[0,\infty]\right)$ and
$$ \left\{ \begin{alignedat}{2} U_{tt} -U_{rr} -\left(\frac{n-1}{r}\right) U_r &= 0\quad &&\text{in}~\mathbb{R}_+\times(0,\infty)\\ u=G\quad u_t&=H&&\text{on}~{\mathbb{R}_+} \times \{t=0\} \end{alignedat} \right. $$