  We study the persistence of the Gevrey class regularity of solutions to
nonlinear wave equations with real analytic nonlinearity. Specifically, it is
proven that the solution remains in a Gevrey class, with respect to some of its
spatial variables, during its whole life-span, provided the initial data is
from the same Gevrey class with respect to these spatial variables. In
addition, for the special Gevrey class of analytic functions, we find a lower
bound for the radius of the spatial analyticity of the solution that might
shrink either algebraically or exponentially, in time, depending on the
structure of the nonlinearity. The standard $L^2$ theory for the Gevrey class
regularity is employed; we also employ energy-like methods for a generalized
version of Gevrey classes based on the $\ell^1$ norm of Fourier transforms
(Wiener algebra). After careful comparisons, we observe an indication that the
$\ell^1$ approach provides a better lower bound for the radius of analyticity
of the solutions than the $L^2$ approach. We present our results in the case of
period boundary conditions, however, by employing exactly the same tools and
proofs one can obtain similar results for the nonlinear wave equations and the
nonlinear Schr\"odinger equation, with real analytic nonlinearity, in certain
domains and manifolds without physical boundaries, such as the whole space
$\mathbb{R}^n$, or on the sphere $\mathbb{S}^{n-1}$.
