ncts

By definition, a wave is a $C^\infty$ solution $u(x,t)$ of the wave equation on $\mathbb R^n$ , and a snapshot of the wave $u$ at time $t$ is the function $u_t$ on $\mathbb R^n$ given by $u_t(x)=u(x,t)$ . We show that there are infinitely many waves with given $C^\infty$ snapshots $f_0$ and $f_1$ at times $t=0$ and $t=1$ respectively, but that all such waves have the same snapshots at integer times. We present necessary and sufficient conditions for the existence and uniqueness of a wave $u$ to have three given snapshots at three different times, and we show how this leads to problems in Diophantine approximations and "small denominators", which dates back to the early study of the $n$ -body problem in $\mathbb R^3$ . We consider generalizations to the Euler-Poisson-Darboux equation and to modified wave equations on spheres and symmetric spaces, as well as some open questions.