  We consider the wave equation on a closed Riemannian manifold. We observe the
restriction of the solutions to a measurable subset $\omega$ along a time
interval $[0, T]$ with $T>0$. It is well known that, if $\omega$ is open and if
the pair $(\omega,T)$ satisfies the Geometric Control Condition then an
observability inequality is satisfied, comparing the total energy of solutions
to their energy localized in $\omega \times (0, T)$. The observability constant
$C\_T({\omega})$ is then defined as the infimum over the set of all nontrivial
solutions of the wave equation of the ratio of localized energy of solutions
over their total energy. In this paper, we provide estimates of the
observability constant based on a low/high frequency splitting procedure
allowing us to derive general geometric conditions guaranteeing that the wave
equation is observable on a measurable subset $\omega$. We also establish that,
as $T\rightarrow+\infty$, the ratio $C\_T({\omega})/T$ converges to the minimum
of two quantities: the first one is of a spectral nature and involves the
Laplacian eigenfunctions, the second one is of a geometric nature and involves
the average time spent in $\omega$ by Riemannian geodesics.
