  Solutions to the wave equation with constant coefficients in $\mathbb{R}^d$
can be represented explicitly in Fourier space. We investigate a reconstruction
formula, which provides an approximation of solutions $u(.,t)$ to initial data
$u_0(.)$ for large times. The reconstruction consists of three steps: 1) Given
$u_0$, initial data for a profile equation are extracted. 2) A profile
evolution equation determines the shape of the profile at time $\tau =
\varepsilon^2 t$. 3) A shell reconstruction operator transforms the profile to
a function on $\mathbb{R}^d$. The sketched construction simplifies the wave
equation, since only a one-dimensional problem in an $O(1)$ time span has to be
solved. We prove that the construction provides a good approximation to the
wave evolution operator for times $t$ of order $\varepsilon^{-2}$.
