  Strong nonlinear effects are known to contribute to the wave run-up caused
when a progressive wave impinges on a vertical surface piercing cylinder. The
magnitude of the wave run-up is largely dependent on the coupling of the
cylinder slenderness, $ka$, and wave steepness, $kA$, parameters. This present
work proposes an analytical solution to the free-surface elevation around a
circular cylinder in plane progressive waves. It is assumed throughout that the
horizontal extent of the cylinder is much smaller than the incident wavelength
and of the same order of magnitude as the incident wave amplitude. A
perturbation expansion of the velocity potential and free-surface boundary
condition is invoked and solved to third-order in terms of $ka$ and $kA$. The
validity of this approach is investigated through a comparison with canonical
second-order diffraction theory and existing experimental results. We find that
for small $ka$, the long wavelength theory is valid up to $kA \approx 0.16-0.2$
on the up wave side of the cylinder. However, this domain is significantly
reduced to $kA<0.06$ when an arbitrary position around the cylinder is
considered. An important feature of this work is an improved account of the
first-harmonic of the free-surface elevation over linear diffraction theory.
