  A Rayleigh wave is a type of surface wave that propagates in the boundary of
an elastic solid with traction (or Neumann) boundary conditions. Since the
1980s much work has been done on the problem of constructing a leading term in
an \emph{approximate} solution to the rather complicated second-order
quasilinear hyperbolic boundary value problem with fully nonlinear Neumann
boundary conditions that governs the propagation of Rayleigh waves. The
question has remained open whether or not this leading term approximate
solution is really close in a precise sense to the exact solution of the
governing equations. We prove a positive answer to this question for the case
of Rayleigh wavetrains in any space dimension $d\geq 2$. The case of Rayleigh
pulses in dimension $d=2$ has already been treated by Coulombel and Williams.
For highly oscillatory Rayleigh wavetrains we are able to construct high-order
approximate solutions consisting of the leading term plus an arbitrary number
of correctors. Using those high-order solutions we then perform an error
analysis which shows (among other things) that for small wavelengths the
leading term is close to the exact solution in $L^\infty$ on a fixed time
interval independent of the wavelength. The error analysis is carried out in a
somewhat general setting and is applicable to other types of waves for which
high order approximate solutions can be constructed.
