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| The conceptual model proposed for the case of waves only (Equation 90) can be extended to the interaction between waves and current. Assuming that ''U''<sub>''s''</sub> is proportional to the shear velocity at the bottom, a dependence on <math>{ {\left| {{\theta }_{cw,onshore}}+{{\theta}_{cw,offshore}}\right| }^{1/2} }</math> may be assumed for ''U''<sub>''s''</sub>, where the interaction between waves and current is taken into account. The representative shear stresses θ<sub>''cw'',''onshore''</sub> and θ<sub>''cw'',''offshore''</sub> are defined based on the instantaneous Shields parameter in the direction of the wave for positive and negative values of θ<sub>''cw''</sub>(''t''), respectively (Figure 29). For an arbitrary angle ( between the waves and the current, this yields the same equations as Equation 88, where θ<sub>''w''</sub> is replaced by θ<sub>''cw''</sub>, and ''T''<sub>''wc''</sub> and ''T''<sub>''wc''</sub> are the half-periods where the instantaneous velocity <math>u\left( t \right)={{U}_{c}}\cos \varphi \ +{{u}_{w}}\left( t \right)</math> (or instantaneous Shields parameter) is onshore (''u''(''t'') > 0) or offshore (''u''(''t'') < 0), respectively (Figure 29). The representative shear stresses θ<sub>''cw,onshore''</sub> and θ<sub>''cw'',''offshore''</sub> are defined as quadratic values of the instantaneous Shields parameter in the direction of the wave for positive and negative values of θ<sub>''cw''</sub>(''t''), respectively (Figure 29). For an arbitrary angle ( between the waves and the current, this yields:
| | \begin{equation}\tag{176} k_b = 0.062 \left[ 1-0.9 \tanh \left( 0.25\frac{u_{*w}}{W_s} \right) \right] |
| | \end{equation} |
| | |
| | \begin{equation} |
| | {{\sigma }_{E}}=\left\{ \begin{align} |
| | & 0.7+3.6{{\sin }^{2}}\left( \frac{\pi }{2}\frac{{{W}_{s}}}{{{u}_{*c}}} \right) |
| | \text{if}\frac{{{W}_{s}}}{{{u}_{*c}}}\le 1 \tag{136} \\ |
| | & 1.0+3.3{{\sin }^{2}}\left( \frac{\pi }{2}\frac{{{u}_{*c}}}{{{W}_{s}}} \right) |
| | \text{if}\frac{{{W}_{s}}}{{{u}_{*c}}}>1 \\ |
| | \end{align} \right. |
| | \end{equation} |
Latest revision as of 15:00, 6 October 2011
\begin{equation}\tag{176} k_b = 0.062 \left[ 1-0.9 \tanh \left( 0.25\frac{u_{*w}}{W_s} \right) \right]
\end{equation}
\begin{equation}
{{\sigma }_{E}}=\left\{ \begin{align}
& 0.7+3.6{{\sin }^{2}}\left( \frac{\pi }{2}\frac{{{W}_{s}}}{{{u}_{*c}}} \right)
\text{if}\frac{{{W}_{s}}}{{{u}_{*c}}}\le 1 \tag{136} \\
& 1.0+3.3{{\sin }^{2}}\left( \frac{\pi }{2}\frac{{{u}_{*c}}}{{{W}_{s}}} \right)
\text{if}\frac{{{W}_{s}}}{{{u}_{*c}}}>1 \\
\end{align} \right.
\end{equation}