  This work is devoted to the analysis of high frequency solutions to the
equations of nonlinear elasticity in a half-space. We consider surface waves
(or more precisely, Rayleigh waves) arising in the general class of isotropic
hyperelastic models, which includes in particular the Saint Venant-Kirchhoff
system. Work has been done by a number of authors since the 1980s on the
formulation and well-posedness of a nonlinear evolution equation whose (exact)
solution gives the leading term of an \emph{approximate} Rayleigh wave solution
to the underlying elasticity equations. This evolution equation, which we refer
to as "the amplitude equation", is an integrodifferential equation of nonlocal
Burgers type. We begin by reviewing and providing some extensions of the theory
of the amplitude equation. The remainder of the paper is devoted to a rigorous
proof in 2D that exact, highly oscillatory, Rayleigh wave solutions $u^\eps$ to
the nonlinear elasticity equations exist on a fixed time interval independent
of the wavelength $\eps$, and that the approximate Rayleigh wave solution
provided by the analysis of the amplitude equation is indeed close in a precise
sense to $u^\eps$ on a time interval independent of $\eps$. The paper focuses
mainly on the case of Rayleigh waves that are \emph{pulses}, which have
profiles with continuous Fourier spectrum, but our method applies equally well
to the case of wavetrains, whose Fourier spectrum is discrete.
