A radial three-dimensional wave reduces to one dimension

Forward direction. For a radial function $u(x,t)=v(r,t)$, $r=\mathrm{\mid }x\mathrm{\mid }$, the chain rule gives ${\mathrm{\partial }}_{j}u=v_r{x}_j\mathrm{/}r$ and ${\mathrm{\partial }}_j^{2}u=v_{rr}{x}_j^2\mathrm{/}{r}^2+v_r(1\mathrm{/}r-x_j^2\mathrm{/}{r}^3)$ for each $j$, so $\mathrm{\Delta }u={\sum }_{j=1}^3{\mathrm{\partial }}_j^{2}u=v_{rr}+\frac{2}{r}{v}_r=\frac{1}{r}{\mathrm{\partial }}_r^2(rv)=\frac{1}{r}{w}_{rr}$ by the product rule. Therefore $0=u_{tt}-c^2\mathrm{\Delta }u=\frac{1}{r}(w_{tt}-c^2{w}_{rr})$ and, since $r>0$, the one-dimensional equation $w_{tt}=c^2{w}_{rr}$ follows.

Converse direction, regularity at the origin. Suppose $w\in C^3([0,\mathrm{\infty })\times \mathbb{R})$ solves $w_{tt}=c^2{w}_{rr}$ on $r>0$ and $w(0,t)=0$ for all $t$. By Every continuous function on an interval has a primitive; two primitives differ by a constant; and ${\int }_a^{b}f=G(b)-G(a)$ for any primitive $G$ and $w(0,t)=0$, $w(r,t)=r{\int }_0^1{w}_r(sr,t)ds$, hence $v(r,t):=w(r,t)\mathrm{/}r={\int }_0^1{w}_r(sr,t)ds$; as $w\in C^3$, Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral applied locally to each derivative permits differentiation under the integral and shows that $v\in C^2([0,\mathrm{\infty })\times \mathbb{R})$ with $v(0,t)=w_r(0,t)$ and ${\mathrm{\partial }}_{r}v(0,t)=\frac{1}{2}{w}_{rr}(0,t)=0$, the last equality because $w_{rr}(0,t)=c^{-2}{w}_{tt}(0,t)$ by continuity of the equation up to $r=0$ and $w(0,\cdot )\equiv 0$.

The equation extends to the origin. For $r>0$ the product rule gives $v_r=\frac{1}{r}{w}_r-\frac{1}{r^2}w$, $v_{rr}=\frac{1}{r}{w}_{rr}-\frac{2}{r^2}{w}_r+\frac{2}{r^3}w$ and $v_{tt}=\frac{1}{r}{w}_{tt}=\frac{c^2}{r}{w}_{rr}$, so the radial three-dimensional Laplacian of $v$ is $v_{rr}+\frac{2}{r}{v}_r=\frac{1}{r}{w}_{rr}=c^{-2}{v}_{tt}$. For the limits at $r\downarrow 0$, the integral formulas and $w_{rr}(sr,t)={\int }_0^{sr}{w}_{rrr}(a,t)da$, since $w_{rr}(0,t)=0$, give $v_{rr}(r,t)={\int }_0^1{s}^2{w}_{rrr}(sr,t)ds\to \frac{1}{3}{w}_{rrr}(0,t)$ and $v_r(r,t)={\int }_0^{1}s{w}_{rr}(sr,t)ds=\frac{r}{3}{w}_{rrr}(0,t)+o(r)$, so that $\frac{2}{r}{v}_r(r,t)\to \frac{2}{3}{w}_{rrr}(0,t)$; hence $(v_{rr}+\frac{2}{r}{v}_r)(r,t)\to w_{rrr}(0,t)$. On the other hand, differentiating $w_{tt}=c^2{w}_{rr}$ in $r$ and letting $r\downarrow 0$ gives $v_{tt}(0,t)=w_{rtt}(0,t)=c^2{w}_{rrr}(0,t)$. These limits are uniform for $t$ in compact intervals by continuity of the derivatives of $w$. For $r>0$, $u_{ij}=(v_{rr}-v_r\mathrm{/}r)x_i{x}_j\mathrm{/}{r}^2+(v_r\mathrm{/}r){\delta }_{ij}$, and both $v_{rr}$ and $v_r\mathrm{/}r$ tend to $w_{rrr}(0,t)\mathrm{/}3$, so $u_{ij}$ extends continuously with this value times ${\delta }_{ij}$. The gradient is $v_{r}x\mathrm{/}r$ and is differentiable at $x=0$ by the same limits. Also $v_r(0,t)=0$ implies $v_{rt}(0,t)=0$; hence $u_{it}=v_{rt}{x}_i\mathrm{/}r\to 0$, agreeing with the derivative in $t$ of $u_i(0,t)=0$. Together with the integral formula for $v_{tt}$, this proves $u(x,t):=v(\mathrm{\mid }x\mathrm{\mid },t)$ is $C^2$ on ${\mathbb{R}}^3\times \mathbb{R}$ and satisfies $u_{tt}=c^2\mathrm{\Delta }u$ at every point, including the origin, where both sides equal $c^2{w}_{rrr}(0,t)$.

Both directions are proved: a radial three-dimensional solution corresponds to a one-dimensional solution $w=rv$ on the half-line, and a one-dimensional solution vanishing at $r=0$ gives back a $C^2$ radial three-dimensional solution, so radial three-dimensional data may be propagated by the one-dimensional formula.