4.5.4.2. Shape class 4, impl=2¶
This implementation of Shape 4 is designed for exploiting symmetric spatial resolutions.
\[\Delta k_x = \Delta k_y, \qquad n_x = n_y\]
For short, we will apply the notation \(\Delta k=\Delta k_x=\Delta k_y\) and \(n = n_x = n_y\). It follows that
\[Y_{-j_y}(y) = \bar{Y}_{j_y}(y), \qquad
Z_{-j_y,j_x}(z) = Z_{j_y,j_x}(z), \qquad Z_{j_x,j_y}(z) = Z_{j_y,j_x}(z)\]
where \(\bar{Y}_{j_y}(y)\) denotes the complex conjugate of \(Y_{j_y}(y)\). Consequently, we apply the following equivalent wave field formulation.
\[\phi(x,y,z,t) = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Re} \Bigl\{C_{j_y,j_x}(x, y, t)\Bigr\}\, Z_{j_y,j_x}(z)\]
\[\zeta(x,y,t) = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Re} \Bigl\{H_{ j_y,j_x}(x, y, t)\Bigr\}\]
\[\begin{split}C_{j_y,j_x}(x, y, t) = &\bigl( c_{1,j_y,j_x}(t)\,Y_{j_y}(y) +
c_{2,j_y,j_x}(t)\, \bar{Y}_{j_y}(y)\bigr) X_{j_x}(x) +\\
&\bigl( c_{3,j_y,j_x}(t)\,Y_{j_x}(y) +
c_{4,j_y,j_x}(t)\, \bar{Y}_{j_x}(y)\bigr) X_{j_y}(x)\end{split}\]
\[\begin{split}H_{j_y,j_x}(x, y, t) = &\bigl( h_{1,j_y,j_x}(t)\,Y_{j_y}(y) +
h_{2,j_y,j_x}(t)\, \bar{Y}_{j_y}(y)\bigr) X_{j_x}(x) +\\
&\bigl( h_{3,j_y,j_x}(t)\,Y_{j_x}(y) +
h_{4,j_y,j_x}(t)\, \bar{Y}_{j_x}(y)\bigr) X_{j_y}(x)\end{split}\]
\[\begin{split}c_{1,j_y,j_x}(t) = c_{j_y,j_x}(t), \qquad
c_{2,j_y,j_x}(t) = \begin{cases}
c_{-j_y,j_x}(t), & \text{$j_y>0$}, \\
0, & \text{$j_y=0$}
\end{cases} \\
c_{3,j_y,j_x}(t) = \begin{cases}
c_{j_x,j_y}(t), & \text{$j_y<j_x$}, \\
0, & \text{$j_y=j_x$}
\end{cases}, \qquad
c_{4,j_y,j_x}(t) = \begin{cases}
c_{-j_x,j_y}(t), & \text{$j_y<j_x$}, \\
0, & \text{$j_y=j_x$}
\end{cases} \\
h_{1,j_y,j_x}(t) = h_{j_y,j_x}(t), \qquad
h_{2,j_y,j_x}(t) = \begin{cases}
h_{-j_y,j_x}(t), & \text{$j_y>0$}, \\
0, & \text{$j_y=0$}
\end{cases} \\
h_{3,j_y,j_x}(t) = \begin{cases}
h_{j_x,j_y}(t), & \text{$j_y<j_x$}, \\
0, & \text{$j_y=j_x$}
\end{cases}, \qquad
h_{4,j_y,j_x}(t) = \begin{cases}
h_{-j_x,j_y}(t), & \text{$j_y<j_x$}, \\
0, & \text{$j_y=j_x$}
\end{cases}\end{split}\]
\[k_{j_x} = j_x\cdot\Delta k, \quad k_{j_y} = j_y\cdot\Delta k, \quad
k_{j_y,j_x} = \sqrt{j_x^2+ j_y^2}\, \Delta k, \quad i = \sqrt{-1}\]
4.5.4.2.1. Kinematics¶
Given the definitions above we obtain the following explicit kinematics:
\[\phi(\bar{x},\bar{y},\bar{z},\bar{t})= \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Re} \Bigl\{C_{j_y,j_x}(x, y, t)\Bigr\} Z_{j_y,j_x}(z)\]
\[\varphi(\bar{x},\bar{y},\bar{z},\bar{t}) \equiv 0\]
\[\frac{\partial\phi}{\partial \bar{t}}(\bar{x},\bar{y},\bar{z},\bar{t}) = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Re} \Bigl\{\frac{d C_{j_y,j_x}(x, y, t)}{dt}\Bigr\} Z_{j_y,j_x}(z)\]
\[\zeta(\bar{x},\bar{y},\bar{t})= \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Re}\Bigl\{H_{j_y,j_x}(x, y, t)\Bigr\}\]
\[\frac{\partial\zeta}{\partial \bar{t}}(\bar{x},\bar{y},\bar{t}) = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Re} \Bigl\{\frac{d H_{j_y,j_x}(x, y, t)}{dt}\Bigr\}\]
\[\frac{\partial\zeta}{\partial \bar{x}}(\bar{x},\bar{y},\bar{t}) = \zeta_x\cos\beta - \zeta_y\sin\beta, \qquad
\frac{\partial\zeta}{\partial \bar{y}}(\bar{x},\bar{y},\bar{t}) = \zeta_x\sin\beta + \zeta_y\cos\beta\]
\[\zeta_x =\sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Im} \Bigl\{H_{j_y,j_x}^{1,0}(x, y, t)\Bigr\}\]
\[\zeta_y = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Im} \Bigl\{H_{j_y,j_x}^{0,1}(x, y, t)\Bigr\}\]
\[\bar{\nabla}\phi(\bar{x},\bar{y},\bar{z},\bar{t}) =
[\phi_x\cos\beta - \phi_y\sin\beta, \phi_x\sin\beta + \phi_y\cos\beta,\phi_z]^T\]
\[\phi_x = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Im} \Bigl\{C_{j_y,j_x}^{1,0}(x, y, t)\Bigr\} \, Z_{j_y,j_x}(z)\]
\[\phi_y = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Im} \Bigl\{C_{j_y,j_x}^{0,1}(x, y, t)\Bigr\} \, Z_{j_y,j_x}(z)\]
\[\phi_z = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
k_{j_y,j_x} \mathcal{Re} \Bigl\{C_{j_y,j_x}(x, y, t)\Bigr\} \, Z_{j_y,j_x}(z)\]
\[\frac{\partial\bar{\nabla}\phi}{\partial \bar{t}}(\bar{x},\bar{y},\bar{z},\bar{t}) =
[\phi_{xt}\cos\beta - \phi_{yt}\sin\beta, \phi_{xt}\sin\beta + \phi_{yt}\cos\beta,\phi_z]^T\]
\[\phi_{xt} = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Im} \Bigl\{\frac{d C_{j_y,j_x}^{1,0}(x, y, t)}{dt}\Bigr\} \, Z_{j_y,j_x}(z)\]
\[\phi_{yt} = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Im} \Bigl\{\frac{d C_{j_y,j_x}^{0,1}(x, y, t)}{dt}\Bigr\} \, Z_{j_y,j_x}(z)\]
\[\phi_{zt} = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
k_{j_y,j_x} \mathcal{Re} \Bigl\{\frac{d C_{j_y,j_x}(x, y, t)}{dt}\Bigr\} \, Z_{j_y,j_x}(z)\]
\[\frac{d\bar{\nabla}\phi}{d\bar{t}}(\bar{x},\bar{y},\bar{z},\bar{t}) =
\frac{\partial\bar{\nabla}\phi}{\partial \bar{t}} +
\bar{\nabla}\phi \cdot \bar{\nabla}\bar{\nabla}\phi\]
\[\begin{split}\bar{\nabla}\bar{\nabla}\phi (\bar{x},\bar{y},\bar{z},\bar{t}) =
\begin{bmatrix}
\phi_{\bar{x},\bar{x}} & \phi_{\bar{x},\bar{y}} & \phi_{\bar{x},\bar{z}} \\
\phi_{\bar{x},\bar{y}} & \phi_{\bar{y},\bar{y}} & \phi_{\bar{y},\bar{z}} \\
\phi_{\bar{x},\bar{z}} & \phi_{\bar{y},\bar{z}} & \phi_{\bar{z},\bar{z}}
\end{bmatrix}\end{split}\]
\[\phi_{\bar{x},\bar{x}} = \phi_{xx}\cos^2\beta - \phi_{xy}\sin(2\beta) + \phi_{yy}\sin^2\beta\]
\[\phi_{\bar{x},\bar{y}} = \phi_{xy}(\cos^2\beta - \sin^2\beta) + (\phi_{xx} - \phi_{yy})\sin\beta\cos\beta\]
\[\phi_{\bar{x},\bar{z}} = \phi_{xz}\cos\beta - \phi_{yz}\sin\beta\]
\[\phi_{\bar{y},\bar{y}} = \phi_{yy}\cos^2\beta + \phi_{xy}\sin(2\beta) + \phi_{xx}\sin^2\beta\]
\[\phi_{\bar{y},\bar{z}} = \phi_{yz}\cos\beta + \phi_{xz}\sin\beta\]
\[\phi_{\bar{z},\bar{z}} = \phi_{zz} = -\phi_{xx} -\phi_{yy}\]
\[\phi_{xx} = - \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Re} \Bigl\{C_{j_y,j_x}^{2,0}(x, y, t)\Bigr\} Z_{j_y,j_x}(z)\]
\[\phi_{xy} = - \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Re} \Bigl\{C_{j_y,j_x}^{1,1}(x, y, t)\Bigr\} Z_{j_y,j_x}(z)\]
\[\phi_{xz} = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
k_{j_y,j_x} \mathcal{Im} \Bigl\{C_{j_y,j_x}^{1,0}(x, y, t)\Bigr\} Z_{j_y,j_x}(z)\]
\[\phi_{yy} = - \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Re} \Bigl\{C_{j_y,j_x}^{0,2}(x, y, t)\Bigr\} Z_{j_y,j_x}(z)\]
\[\phi_{yz} = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
k_{j_y,j_x} \mathcal{Im} \Bigl\{C_{j_y,j_x}^{0,1}(x, y, t)\Bigr\} Z_{j_y,j_x}(z)\]
\[\phi_{zz} = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
k_{j_y,j_x}^2 \mathcal{Re} \Bigl\{C_{j_y,j_x}(x, y, t)\Bigr\} Z_{j_y,j_x}(z)
= -\phi_{xx} - \phi_{yy}\]
\[\frac{\partial^2\zeta}{\partial \bar{x}^2}(\bar{x},\bar{y},\bar{t}) =
\zeta_{xx}\cos^2\beta - \zeta_{xy}\sin(2\beta) + \zeta_{yy}\sin^2\beta\]
\[\frac{\partial^2\zeta}{\partial\bar{x}\partial\bar{y}}(\bar{x},\bar{y},\bar{t}) =
\zeta_{xy}(\cos^2\beta - \sin^2\beta) + (\zeta_{xx} - \zeta_{yy})\sin\beta\cos\beta\]
\[\frac{\partial^2\zeta}{\partial\bar{y}^2}(\bar{x},\bar{y},\bar{t}) =
\zeta_{yy}\cos^2\beta + \zeta_{xy}\sin(2\beta) + \zeta_{xx}\sin^2\beta\]
\[\zeta_{xx} = -\sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Re} \Bigl\{H_{j_y,j_x}^{2,0}(x, y, t)\Bigr\}\]
\[\zeta_{xy} = -\sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Re} \Bigl\{H_{j_y,j_x}^{1,1}(x, y, t)\Bigr\}\]
\[\zeta_{yy} = -\sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Re} \Bigl\{H_{j_y,j_x}^{0,2}(x, y, t)\Bigr\}\]
\[p = -\rho\frac{\partial\phi}{\partial \bar{t}}
-\frac{1}{2}\rho\bar{\nabla}\phi\cdot\bar{\nabla}\phi
-\rho g \bar{z}\]
where \(\bar{\nabla}\) denotes gradients with respect to \(\bar{x}\), \(\bar{y}\) and \(\bar{z}\). We also apply the notation
\[\begin{split}\frac{\partial^{i+j}C_{j_y,j_x}(x, y, t)}{\partial x^i \partial y^j} =
(-i)^{i+j} C_{j_y,j_x}^{i,j}(x, y, t) \\
\frac{\partial^{i+j}H_{j_y,j_x}(x, y, t)}{\partial x^i \partial y^j} =
(-i)^{i+j} H_{j_y,j_x}^{i,j}(x, y, t)\end{split}\]
\[\begin{split}C_{j_y,j_x}^{i,j}(x, y, t) = &k_{j_x}^ik_{j_y}^j\bigl( c_{1,j_y,j_x}(t)\,Y_{j_y}(y) +(-1)^j
c_{2,j_y,j_x}(t)\, \bar{Y}_{j_y}(y)\bigr) X_{j_x}(x) +\\
&k_{j_y}^i k_{j_x}^j\bigl( c_{3,j_y,j_x}(t)\,Y_{j_x}(y) + (-1)^j
c_{4,j_y,j_x}(t)\, \bar{Y}_{j_x}(y)\bigr) X_{j_y}(x)\end{split}\]
\[\begin{split}H_{j_y,j_x}^{i,j}(x, y, t) = &k_{j_x}^ik_{j_y}^j\bigl( h_{1,j_y,j_x}(t)\,Y_{j_y}(y) +(-1)^j
h_{2,j_y,j_x}(t)\, \bar{Y}_{j_y}(y)\bigr) X_{j_x}(x) +\\
&k_{j_y}^i k_{j_x}^j\bigl( h_{3,j_y,j_x}(t)\,Y_{j_x}(y) + (-1)^j
h_{4,j_y,j_x}(t)\, \bar{Y}_{j_x}(y)\bigr) X_{j_y}(x)\end{split}\]
The particle acceleration is labeled \(\frac{d\bar{\nabla}\phi}{d\bar{t}}\).
The stream function \(\varphi\) is not relevant for short crested seas. Hence, we apply the dummy definition \(\varphi=0\) for all locations.
4.5.4.2.2. Implementation notes¶
Evaluation of costly transcendental functions (\(\cos\), \(\sin\), \(\exp\), …) is significantly reduced by exploiting the following recursive relations
\[X_{j_x}(x) = X_1(x)\, X_{j_x-1}(x), \qquad
Y_{j_y}(y) = Y_1(y)\, Y_{j_y-1}(y)\]
It should be noted that contrary to long crested seas, there are no trivial recursive relations for the \(z\)-dependent term \(Z_{j_y,j_x}(z)\). This makes calculations of surface elevations significantly faster than calculations of other kinematics for short crested seas.
In case the wave generator applies a perturbation theory of order \(q\) we apply the following Taylor expansion above the calm free surface.
\[Z_{j_y, j_x}(z) = 1 + \sum_{p=1}^{q-1}\frac{(k_{j_y, j_x} z)^p}{p!}, \qquad z > 0\]