Directional characteristics of spatially evolving young waves under steady wind

Synchronous single point measurements of the instantaneous surface elevation and of two components of the surface slope in a wind-wave tank are carried out using a high-speed camera and a laser slope gauge. The applied technique allows study of joint statistics between the temporal variation of surface elevation and the slope components at a point, as well as computation of directional wave energy spectra. It is demonstrated that while significant coherence exists between surface elevation and along-wind slope component, the coherence between crosswind slope and surface elevation is almost negligible. Evolution of the directional spectra along the test section is presented for different wind velocities and compared with results obtained in other relevant field and laboratory studies.

Figure 1

Time series of (a) instantaneous surface elevation $\eta$, (b) along-wind slope ${\eta }_x$, and (c) crosswind slope ${\eta }_y$; $U=6.3$ m/s, $x=310$ cm.

Figure 2

Joint probability density function between $\eta$ and (a) ${\eta }_x$ and (b) ${\eta }_y$; $U=6.3$ m/s and $x=310$ cm.

Figure 3

Cross-correlation function between $\eta$ and (a) ${\eta }_x$ and (b) ${\eta }_y$; $U=6.3$ m/s; dashed line: $\tau /T_p=0.25$.

Figure 4

Probability distribution function (pdf) of [(a) and (c)] peak correlation value and [(b) and (d)] nondimensional lag (${\tau }_p/T_p$); between $\eta$ and ${\eta }_x$ (upper row); between $\eta$ and ${\eta }_y$ (lower row); for $U=6.3$ m/s.

Figure 5

Magnitude squared coherency, MSC(f), of $\eta$ and ${\eta }_x$ at (a) $x=130$ cm, (b) $x=230$ cm, and (c) $x=310$ cm. Vertical lines: solid $f_p$ of $\eta$; dashed $f_p$ of ${\eta }_x$.

Figure 6

Phase difference between $\eta$ and ${\eta }_x$, $\mathrm{\Delta }\varphi \left(f\right)$; (black) $x=130$ cm and (blue) $x=310$ cm.

Figure 7

(a) Comparison between measured values of $r_1$ and $r_2$, (b) The normalized spreading parameter $s/s_m$ as a function of the dimensionless frequency $f/f_p$, (c) spreading parameter $s\left(f\right)$, and (d) variation of peak spreading parameter $s_m$ along the channel.

Figure 8

Two-dimensional energy spectrum $\widehat{S}(f,\theta )$ at $x=130$, 230, and 310 cm and $U=4.1$ m/s (first row); $U=6.3$ m/s (second row), and $U=8.5$ m/s (third row).