Wave equation, Cauchy data and wave speed

Let $n\ge 1$, $T>0$ and $c>0$, let ${\mathrm{\partial }}_t,{\mathrm{\partial }}_j$ be the partial derivatives of Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ and let $\mathrm{\Delta }={\sum }_{j=1}^n{\mathrm{\partial }}_j^2$ be the Laplacian of The Laplacian of a $C^2$ function and of a $C^2$ vector field. The wave operator of speed $c$ is ${\mathrm{\square }}_c:={\mathrm{\partial }}_t^2-c^2\mathrm{\Delta },$ and the wave equation with source $f$ is the equation ${\mathrm{\square }}_{c}u=f$ on a slab ${\mathbb{R}}^n\times (0,T)$; the equation is homogeneous when $f=0$. A classical solution on the time slab ${\mathbb{R}}^n\times [0,T)$ is a function $u$ of class $C^2$ on ${\mathbb{R}}^n\times (0,T)$ in the sense of $C^k$ maps and multi-index derivative notation in Euclidean space which satisfies the equation at every point of ${\mathbb{R}}^n\times (0,T)$. This is the classical-solution and Cauchy-data vocabulary of Scalar partial differential equations, order, and classical solutions, and ${\mathrm{\square }}_c$ is a linear second-order operator in the sense of Linear, semilinear, quasilinear, and fully nonlinear partial differential equations.

The Cauchy problem for ${\mathrm{\square }}_{c}u=f$ prescribes a displacement $u_0:{\mathbb{R}}^n\to \mathbb{R}$ and a velocity $u_1:{\mathbb{R}}^n\to \mathbb{R}$ and asks for a classical solution attaining them as $t\downarrow 0$ in the pointwise sense: $u(x,t)\to u_0(x)$ and ${\mathrm{\partial }}_{t}u(x,t)\to u_1(x)$ for every $x\in {\mathbb{R}}^n$ as $t\downarrow 0$. Only the limits are part of the data, so $u$ need not be defined at $t=0$ a priori.

Unit-speed rescaling convention. A function $u$ solves ${\mathrm{\square }}_{c}u=0$ on ${\mathbb{R}}^n\times [0,T)$ with data $(u_0,u_1)$ if and only if $v(x,s):=u(x,s\mathrm{/}c)$ solves ${\mathrm{\partial }}_s^{2}v=\mathrm{\Delta }v$ on ${\mathbb{R}}^n\times [0,cT)$ with data $(u_0,u_1\mathrm{/}c)$; equivalently $u(x,t)=v(x,ct)$. Indeed continuous coordinate partial derivatives imply total differentiability (If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative), so the chain rule (The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$) gives ${\mathrm{\partial }}_{t}u(x,t)=c{\mathrm{\partial }}_{s}v(x,ct)$ and ${\mathrm{\partial }}_t^{2}u(x,t)=c^2{\mathrm{\partial }}_s^{2}v(x,ct)$ while spatial derivatives are unchanged, so ${\mathrm{\square }}_{c}u=c^2({\mathrm{\partial }}_s^{2}v-\mathrm{\Delta }v)$ evaluated at $s=ct$, and the initial displacement limits agree, while the initial velocity limit of $u$ is $c$ times that of $v$. The cited treatments state their formulas at unit speed, and this convention is the only place where the general speed enters those formulas.

Sphere normalisation. Let $\sigma$ be the polar surface measure on the unit sphere (The polar surface set function on the unit sphere). Write $V_m:=\mathrm{\mid }{B}_1^m\mathrm{\mid }$ for the unit ball and ${\omega }_m:=\sigma (S^m)=(m+1)V_{m+1}>0$ for the total polar measure of the unit sphere $S^m\subseteq {\mathbb{R}}^{m+1}$ (Sphere and ball measures scale in Rn); thus the boundary sphere of a ball in ${\mathbb{R}}^n$ has total measure ${\omega }_{n-1}=n{V}_n$. For an integer $m\ge 0$ put $(2m+1)!!:=(2m+1)(2m-1)\cdots 3\cdot 1$ and $(2m)!!:=(2m)(2m-2)\cdots 2$, so that $0!!=1!!=1$ and $(n-2)!!$ is defined for every odd $n\ge 3$. All sphere and weighted-ball integrals on this page are read under the Axiom of Countable Choice of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), under which the polar measure and its integrals are supplied.