4.5.4.1. Shape class 4, impl=1¶
This implementation of Shape 4 evaluates the more general non-symmetric spatial resolution where \(\Delta k_x\) may differ from \(\Delta k_y\) or \(n_x\) from \(n_y\).
However, since \(Y_{-j_y}(y) = \bar{Y}_{j_y}(y)\), where \(\bar{Y}_{j_y}(y)\) denotes the complex conjugate of \(Y_{j_y}(y)\), and \(Z_{-j_y,j_x}(z) = Z_{j_y,j_x}(z)\) we may always apply the following equivalent wave field formulation.
\[\phi(x,y,z,t) = \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
\mathcal{Re} \Bigl\{C_{1,j_y,j_x}(y, t)\, X_{j_x}(x)\Bigr\}\, Z_{j_y,j_x}(z)\]
\[\zeta(x,y,t) = \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
\mathcal{Re} \Bigl\{H_{1, j_y,j_x}(y, t) \,X_{j_x}(x)\Bigr\}\]
\[C_{1,j_y,j_x}(y, t) = c_{1,j_y,j_x}(t)\,Y_{j_y}(y) +
c_{2,j_y,j_x}(t)\, \bar{Y}_{j_y}(y)\]
\[H_{1,j_y,j_x}(y, t) = h_{1,j_y,j_x}(t)\,Y_{j_y}(y) +
h_{2,j_y,j_x}(t)\, \bar{Y}_{j_y}(y)\]
\[\begin{split}c_{1,j_y,j_x}(t) = c_{j_y,j_x}(t), \qquad
c_{2,j_y,j_x}(t) = \begin{cases}
c_{-j_y,j_x}(t), & \text{$j_y>0$}, \\
0, & \text{$j_y=0$}
\end{cases} \\
h_{1,j_y,j_x}(t) = h_{j_y,j_x}(t), \qquad
h_{2,j_y,j_x}(t) = \begin{cases}
h_{-j_y,j_x}(t), & \text{$j_y>0$}, \\
0, & \text{$j_y=0$}
\end{cases}\end{split}\]
4.5.4.1.1. Kinematics¶
Given the definitions above we obtain the following explicit kinematics:
\[\phi(\bar{x},\bar{y},\bar{z},\bar{t})= \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
\mathcal{Re} \Bigl\{C_{1,j_y,j_x}(y, t)\, X_{j_x}(x)\Bigr\} Z_{j_y,j_x}(z)\]
\[\varphi(\bar{x},\bar{y},\bar{z},\bar{t}) \equiv 0\]
\[\frac{\partial\phi}{\partial \bar{t}}(\bar{x},\bar{y},\bar{z},\bar{t}) = \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
\mathcal{Re} \Bigl\{\frac{d C_{1,j_y,j_x}(y, t)}{dt} \, X_{j_x}(x)\Bigr\} Z_{j_y,j_x}(z)\]
\[\zeta(\bar{x},\bar{y},\bar{t})= \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
\mathcal{Re}\Bigl\{H_{1,j_y,j_x}(y, t)\, X_{j_x}(x)\Bigr\}\]
\[\frac{\partial\zeta}{\partial \bar{t}}(\bar{x},\bar{y},\bar{t}) = \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
\mathcal{Re} \Bigl\{\frac{d H_{1,j_y,j_x}(y, t)}{dt} \, X_{j_x}(x)\Bigr\}\]
\[\frac{\partial\zeta}{\partial \bar{x}}(\bar{x},\bar{y},\bar{t}) = \zeta_x\cos\beta - \zeta_y\sin\beta, \qquad
\frac{\partial\zeta}{\partial \bar{y}}(\bar{x},\bar{y},\bar{t}) = \zeta_x\sin\beta + \zeta_y\cos\beta\]
\[\zeta_x =\sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_x} \mathcal{Im} \Bigl\{H_{1,j_y,j_x}(y, t)\, X_{j_x}(x)\Bigr\}\]
\[\zeta_y = -\sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_y} \mathcal{Im} \Bigl\{H_{2,j_y,j_x}(y, t)\, X_{j_x}(x)\Bigr\}\]
\[\bar{\nabla}\phi(\bar{x},\bar{y},\bar{z},\bar{t}) =
[\phi_x\cos\beta - \phi_y\sin\beta, \phi_x\sin\beta + \phi_y\cos\beta,\phi_z]^T\]
\[\phi_x = \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_x}\mathcal{Im} \Bigl\{C_{1,j_y,j_x}(y, t)\, X_{j_x}(x)\Bigr\} \, Z_{j_y,j_x}(z)\]
\[\phi_y = \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_y}\mathcal{Im} \Bigl\{C_{2,j_y,j_x}(y, t)\, X_{j_x}(x)\Bigr\} \, Z_{j_y,j_x}(z)\]
\[\phi_z = \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_y,j_x} \mathcal{Re} \Bigl\{C_{1,j_y,j_x}(y, t)\, X_{j_x}(x)\Bigr\} \, Z_{j_y,j_x}(z)\]
\[\frac{\partial\bar{\nabla}\phi}{\partial \bar{t}}(\bar{x},\bar{y},\bar{z},\bar{t}) =
[\phi_{xt}\cos\beta - \phi_{yt}\sin\beta, \phi_{xt}\sin\beta + \phi_{yt}\cos\beta,\phi_z]^T\]
\[\phi_{xt} = \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_x}\mathcal{Im} \Bigl\{\frac{d C_{1,j_y,j_x}(y, t)}{dt}\, X_{j_x}(x)\Bigr\} \, Z_{j_y,j_x}(z)\]
\[\phi_{yt} = \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_y}\mathcal{Im} \Bigl\{\frac{d C_{2,j_y,j_x}(y, t)}{dt}\, X_{j_x}(x)\Bigr\} \, Z_{j_y,j_x}(z)\]
\[\phi_{zt} = \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_y,j_x} \mathcal{Re} \Bigl\{\frac{d C_{1,j_y,j_x}(y, t)}{dt}\, X_{j_x}(x)\Bigr\} \, Z_{j_y,j_x}(z)\]
\[\frac{d\bar{\nabla}\phi}{d\bar{t}}(\bar{x},\bar{y},\bar{z},\bar{t}) =
\frac{\partial\bar{\nabla}\phi}{\partial \bar{t}} +
\bar{\nabla}\phi \cdot \bar{\nabla}\bar{\nabla}\phi\]
\[\begin{split}\bar{\nabla}\bar{\nabla}\phi (\bar{x},\bar{y},\bar{z},\bar{t}) =
\begin{bmatrix}
\phi_{\bar{x},\bar{x}} & \phi_{\bar{x},\bar{y}} & \phi_{\bar{x},\bar{z}} \\
\phi_{\bar{x},\bar{y}} & \phi_{\bar{y},\bar{y}} & \phi_{\bar{y},\bar{z}} \\
\phi_{\bar{x},\bar{z}} & \phi_{\bar{y},\bar{z}} & \phi_{\bar{z},\bar{z}}
\end{bmatrix}\end{split}\]
\[\phi_{\bar{x},\bar{x}} = \phi_{xx}\cos^2\beta - \phi_{xy}\sin(2\beta) + \phi_{yy}\sin^2\beta\]
\[\phi_{\bar{x},\bar{y}} = \phi_{xy}(\cos^2\beta - \sin^2\beta) + (\phi_{xx} - \phi_{yy})\sin\beta\cos\beta\]
\[\phi_{\bar{x},\bar{z}} = \phi_{xz}\cos\beta - \phi_{yz}\sin\beta\]
\[\phi_{\bar{y},\bar{y}} = \phi_{yy}\cos^2\beta + \phi_{xy}\sin(2\beta) + \phi_{xx}\sin^2\beta\]
\[\phi_{\bar{y},\bar{z}} = \phi_{yz}\cos\beta + \phi_{xz}\sin\beta\]
\[\phi_{\bar{z},\bar{z}} = \phi_{zz} = -\phi_{xx} -\phi_{yy}\]
\[\phi_{xx} = - \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_x}^2 \mathcal{Re} \Bigl\{C_{1,j_y,j_x}(y, t) \, X_{j_x}(x)\Bigr\} Z_{j_y,j_x}(z)\]
\[\phi_{xy} = - \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_x} k_{j_y} \mathcal{Re} \Bigl\{C_{2,j_y,j_x}(y, t) \, X_{j_x}(x)\Bigr\} Z_{j_y,j_x}(z)\]
\[\phi_{xz} = \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_x} k_{j_y,j_x} \mathcal{Im} \Bigl\{C_{1,j_y,j_x}(y, t) \, X_{j_x}(x)\Bigr\} Z_{j_y,j_x}(z)\]
\[\phi_{yy} = - \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_y}^2 \mathcal{Re} \Bigl\{C_{1,j_y,j_x}(y, t) \, X_{j_x}(x)\Bigr\} Z_{j_y,j_x}(z)\]
\[\phi_{yz} = \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_y} k_{j_y,j_x} \mathcal{Im} \Bigl\{C_{2,j_y,j_x}(y, t) \, X_{j_x}(x)\Bigr\} Z_{j_y,j_x}(z)\]
\[\phi_{zz} = \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_y,j_x}^2 \mathcal{Re} \Bigl\{C_{1,j_y,j_x}(y, t) \, X_{j_x}(x)\Bigr\} Z_{j_y,j_x}(z)
= -\phi_{xx} - \phi_{yy}\]
\[\frac{\partial^2\zeta}{\partial \bar{x}^2}(\bar{x},\bar{y},\bar{t}) =
\zeta_{xx}\cos^2\beta - \zeta_{xy}\sin(2\beta) + \zeta_{yy}\sin^2\beta\]
\[\frac{\partial^2\zeta}{\partial\bar{x}\partial\bar{y}}(\bar{x},\bar{y},\bar{t}) =
\zeta_{xy}(\cos^2\beta - \sin^2\beta) + (\zeta_{xx} - \zeta_{yy})\sin\beta\cos\beta\]
\[\frac{\partial^2\zeta}{\partial\bar{y}^2}(\bar{x},\bar{y},\bar{t}) =
\zeta_{yy}\cos^2\beta + \zeta_{xy}\sin(2\beta) + \zeta_{xx}\sin^2\beta\]
\[\zeta_{xx} = -\sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_x}^2 \mathcal{Re} \Bigl\{H_{1,j_y,j_x}(y, t) \, X_{j_x}(x)\Bigr\}\]
\[\zeta_{xy} = -\sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_x} k_{j_y} \mathcal{Re} \Bigl\{H_{2,j_y,j_x}(y, t) \, X_{j_x}(x)\Bigr\}\]
\[\zeta_{yy} = -\sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_y}^2 \mathcal{Re} \Bigl\{H_{1,j_y,j_x}(y, t) \, X_{j_x}(x)\Bigr\}\]
\[p = -\rho\frac{\partial\phi}{\partial \bar{t}}
-\frac{1}{2}\rho\bar{\nabla}\phi\cdot\bar{\nabla}\phi
-\rho g \bar{z}\]
where \(\bar{\nabla}\) denotes gradients with respect to \(\bar{x}\), \(\bar{y}\) and \(\bar{z}\). We also apply
\[C_{2, j_y, j_x}(y,t) = c_{1, j_y, j_x}(t)\,Y_{j_y}(y) - c_{2, j_y, j_x}(t)\,\bar{Y}_{j_y}(y)\]
\[H_{2, j_y, j_x}(y,t) = h_{1, j_y, j_x}(t)\,Y_{j_y}(y) - h_{2, j_y, j_x}(t)\,\bar{Y}_{j_y}(y)\]
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.
4.5.4.1.2. 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.
\[Z_{j_y, j_x}(z) = 1 + \sum_{p=1}^{q-1}\frac{(k_{j_y, j_x} z)^p}{p!}, \qquad z > 0\]