Abstract: The exact theoretical expression describing the
probability distribution of nonlinear sea-surface elevations derived
from the second-order narrowband model has a cumbersome form
that requires numerical computations, not well-disposed to theoretical
or practical applications. Here, the same narrowband model is reexamined
to develop a simpler closed-form approximation suitable
for theoretical and practical applications. The salient features of the
approximate form are explored, and its relative validity is verified
with comparisons to other readily available approximations, and
oceanic data.
Abstract: Occurrences of spurious crests on the troughs of large,
relatively steep second-order Stokes waves are anomalous and not an
inherent characteristic of real waves. Here, the effects of such
occurrences on the statistics described by the standard second-order
stochastic model are examined theoretically and by way of
simulations. Theoretical results and simulations indicate that when
spurious occurrences are sufficiently large, the standard model leads
to physically unrealistic surface features and inaccuracies in the
statistics of various surface features, in particular, the troughs and
thus zero-crossing heights of large waves. Whereas inaccuracies can
be fairly noticeable for long-crested waves in both deep and
shallower depths, they tend to become relatively insignificant in
directional waves.