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For $f:(0,\infty )\to \mathbb{R}$, its odd extension to the real line isThe zero value is forced by oddness, whether or not the extension is continuous. If $f$ has a finite right limit at zero, the extension is continuous there exactly when that limit is zero. Thus extending $f(x)=1$ gives a jump, while extending $f(x)=x$ gives the smooth function $\tilde{f}(x)=x$.

More generally, suppose $f$ has continuous right derivatives through order $k$ at zero. Its odd extension is $C^k$ exactly when every even-order endpoint derivative through order $k$, including $f(0)$, is zero. Differentiating the reflected formula shows that the left order-$j$ limit is $(-1{)}^{j+1}{f}^{(j)}(0+)$. In the radial reduction of the three-dimensional wave equation, the factor $r\varphi (t,r)$ has a smooth odd extension when $\varphi$ is a smooth radial field at the origin.

For a radial function, the Laplacian is $\Delta \varphi ={\varphi }_{rr}+2{\varphi }_r/r$. Introduce the retarded and advanced null coordinates and the radial reduction of the three-dimensional wave equation:The chain rule gives ${\partial }_t={\partial }_u+{\partial }_v$ and ${\partial }_r=-{\partial }_u+{\partial }_v$. Consequently the inhomogeneous wave equation isMultiplying the radial wave equation by $r$ removes its first-order radial derivative:Thus the particularly useful retarded and advanced null coordinates form isThese formulas hold for $r>0$. At the axis, smooth spherical symmetry imposes ${\varphi }_r(t,0)=0$ and $w(t,0)=0$; equivalently $w$ has a smooth odd extension across $r=0$.

Use the radial reduction of the three-dimensional wave equation $w=r\varphi$ and extend $w$ oddly in $r$. Write $w=w_h+w_F$, where $w_h$ is the homogeneous wave equation solution with the prescribed Cauchy data, and $w_F$ has zero Cauchy data. With $w_0(r)=r{\varphi }_0(r)$ and $w_1(r)=r{\varphi }_1(r)$, the D'Alembert formula givesBoth odd initial profiles have compact support, soThe Duhamel principle, applied to $w_{tt}-w_{rr}=-rF$, givesNow write $r=t+a$ with $1/2\le a\le 1$. The lower integration limit is $s+a>0$, so this domain of dependence never crosses the axis. Intersecting its integration interval with the support of the source leaves a subset of $[s+a,s+1]$, of length at most one. On this interval,The outgoing shell source estimate for a radial wave therefore yieldsFinally $r\ge t+1/2\ge (1+t)/2$, and thusThis constant depends only on the Cauchy data and the fixed unit bound for the source. The logarithm comes from integrating the shell's accumulated forcing, whereas the factor $(1+t{)}^{-1}$ comes from division by $r$.

For a radial function in three dimensions,Set $w=ru$. Multiplying the wave equation by $r$ gives the radial reduction of the three-dimensional wave equationwith the regularity boundary condition $w(t,0)=0$.