This shape class describes general short crested spectral waves propagating in constant water depth \(d\).
\[\phi(x,y,z,t) = \sum_{j_x=0}^{n_x}\sum_{j_y=-n_y}^{n_y}
\mathcal{Re} \Bigl\{c_{j_y,j_x}(t)\, X_{j_x}(x)\,Y_{j_y}(y)\Bigr\}\, Z_{j_y,j_x}(z)\]
\[\zeta(x,y,t) = \sum_{j_x=0}^{n_x}\sum_{j_y=-n_y}^{n_y}
\mathcal{Re} \Bigl\{h_{j_y,j_x}(t) \,X_{j_x}(x)\,Y_{j_y}(y)\Bigr\}\]
\[X_{j_x}(x) = e^{- i k_{j_x} x}, \quad
Y_{j_y}(y) = e^{- i k_{j_y} y}, \quad
Z_{j_y,j_x}(z) = \frac{\cosh k_{j_y,j_x}(z+d)}{\cosh k_{j_y,j_x} d}\]
\[k_{j_x} = j_x\cdot\Delta k_x, \quad k_{j_y} = j_y\cdot\Delta k_y, \quad
k_{j_y,j_x} = \sqrt{k_{j_x}^2+k_{j_y}^2}, \quad i = \sqrt{-1}\]
\[\phi(x + L_x, y + L_y, z, t) = \phi(x, y, z, t), \qquad
\zeta(x + L_x, y + L_y, t) = \zeta(x, y, t)\]
\[L_x = \frac{2\pi}{\Delta k_x}, \qquad L_y = \frac{2\pi}{\Delta k_y}, \qquad
\lambda_{\min} = \frac{2\pi}{\sqrt{(n_x \Delta k_x)^2+(n_y \Delta k_y)^2}}\]
4.5.5.1. Kinematics
In order to exploit some symmetric properties in the \(y\)-direction of this class
we apply the following equivalent formulation for more efficient evaluations.
\[\phi(x,y,z,t) = \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
\mathcal{Re} \Bigl\{C_{1,j_y,j_x}(y, t)\, X_{j_x}(x)\Bigr\}\, Z_{j_y,j_x}(z)\]
\[\zeta(x,y,t) = \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
\mathcal{Re} \Bigl\{H_{1, j_y,j_x}(y, t) \,X_{j_x}(x)\Bigr\}\]
\[C_{1,j_y,j_x}(y, t) = c_{1,j_y,j_x}(t)\,Y_{j_y}(y) +
c_{2,j_y,j_x}(t)\, \bar{Y}_{j_y}(y)\]
\[H_{1,j_y,j_x}(y, t) = h_{1,j_y,j_x}(t)\,Y_{j_y}(y) +
h_{2,j_y,j_x}(t)\, \bar{Y}_{j_y}(y)\]
\[\begin{split}c_{1,j_y,j_x}(t) = c_{j_y,j_x}(t), \qquad
c_{2,j_y,j_x}(t) = \begin{cases}
c_{-j_y,j_x}(t), & \text{$j_y>0$}, \\
0, & \text{$j_y=0$}
\end{cases} \\
h_{1,j_y,j_x}(t) = h_{j_y,j_x}(t), \qquad
h_{2,j_y,j_x}(t) = \begin{cases}
h_{-j_y,j_x}(t), & \text{$j_y>0$}, \\
0, & \text{$j_y=0$}
\end{cases}\end{split}\]
where \(\bar{Y}_{j_y}(y)\) denotes the complex conjugate of \(Y_{j_y}(y)\).
Given the definitions above we obtain the following explicit kinematics:
\[\phi(\bar{x},\bar{y},\bar{z},\bar{t})= \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
\mathcal{Re} \Bigl\{C_{1,j_y,j_x}(y, t)\, X_{j_x}(x)\Bigr\} Z_{j_y,j_x}(z)\]
\[\varphi(\bar{x},\bar{y},\bar{z},\bar{t}) \equiv 0\]
\[\frac{\partial\phi}{\partial \bar{t}}(\bar{x},\bar{y},\bar{z},\bar{t}) = \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
\mathcal{Re} \Bigl\{\frac{d C_{1,j_y,j_x}(y, t)}{dt} \, X_{j_x}(x)\Bigr\} Z_{j_y,j_x}(z)\]
\[\zeta(\bar{x},\bar{y},\bar{t})= \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
\mathcal{Re}\Bigl\{H_{1,j_y,j_x}(y, t)\, X_{j_x}(x)\Bigr\}\]
\[\frac{\partial\zeta}{\partial \bar{t}}(\bar{x},\bar{y},\bar{t}) = \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
\mathcal{Re} \Bigl\{\frac{d H_{1,j_y,j_x}(y, t)}{dt} \, X_{j_x}(x)\Bigr\}\]
\[\frac{\partial\zeta}{\partial \bar{x}}(\bar{x},\bar{y},\bar{t}) = \zeta_x\cos\beta - \zeta_y\sin\beta, \qquad
\frac{\partial\zeta}{\partial \bar{y}}(\bar{x},\bar{y},\bar{t}) = \zeta_x\sin\beta + \zeta_y\cos\beta\]
\[\zeta_x =\sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_x} \mathcal{Im} \Bigl\{H_{1,j_y,j_x}(y, t)\, X_{j_x}(x)\Bigr\}\]
\[\zeta_y = -\sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_y} \mathcal{Im} \Bigl\{H_{2,j_y,j_x}(y, t)\, X_{j_x}(x)\Bigr\}\]
\[\bar{\nabla}\phi(\bar{x},\bar{y},\bar{z},\bar{t}) =
[\phi_x\cos\beta - \phi_y\sin\beta, \phi_x\sin\beta + \phi_y\cos\beta,\phi_z]^T\]
\[\phi_x = \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_x}\mathcal{Im} \Bigl\{C_{1,j_y,j_x}(y, t)\, X_{j_x}(x)\Bigr\} \, Z_{j_y,j_x}(z)\]
\[\phi_y = \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_y}\mathcal{Im} \Bigl\{C_{2,j_y,j_x}(y, t)\, X_{j_x}(x)\Bigr\} \, Z_{j_y,j_x}(z)\]
\[\phi_z = \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
\mathcal{Re} \Bigl\{C_{1,j_y,j_x}(y, t)\, X_{j_x}(x)\Bigr\} \, \frac{d Z_{j_y,j_x}(z)}{dz}\]
\[\frac{\partial\bar{\nabla}\phi}{\partial \bar{t}}(\bar{x},\bar{y},\bar{z},\bar{t}) =
[\phi_{xt}\cos\beta - \phi_{yt}\sin\beta, \phi_{xt}\sin\beta + \phi_{yt}\cos\beta,\phi_z]^T\]
\[\phi_{xt} = \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_x}\mathcal{Im} \Bigl\{\frac{d C_{1,j_y,j_x}(y, t)}{dt}\, X_{j_x}(x)\Bigr\} \, Z_{j_y,j_x}(z)\]
\[\phi_{yt} = \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_y}\mathcal{Im} \Bigl\{\frac{d C_{2,j_y,j_x}(y, t)}{dt}\, X_{j_x}(x)\Bigr\} \, Z_{j_y,j_x}(z)\]
\[\phi_{zt} = \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
\mathcal{Re} \Bigl\{\frac{d C_{1,j_y,j_x}(y, t)}{dt}\, X_{j_x}(x)\Bigr\} \, \frac{d Z_{j_y,j_x}(z)}{dz}\]
\[\frac{d\bar{\nabla}\phi}{d\bar{t}}(\bar{x},\bar{y},\bar{z},\bar{t}) =
\frac{\partial\bar{\nabla}\phi}{\partial \bar{t}} +
\bar{\nabla}\phi \cdot \bar{\nabla}\bar{\nabla}\phi\]
\[\begin{split}\bar{\nabla}\bar{\nabla}\phi (\bar{x},\bar{y},\bar{z},\bar{t}) =
\begin{bmatrix}
\phi_{\bar{x},\bar{x}} & \phi_{\bar{x},\bar{y}} & \phi_{\bar{x},\bar{z}} \\
\phi_{\bar{x},\bar{y}} & \phi_{\bar{y},\bar{y}} & \phi_{\bar{y},\bar{z}} \\
\phi_{\bar{x},\bar{z}} & \phi_{\bar{y},\bar{z}} & \phi_{\bar{z},\bar{z}}
\end{bmatrix}\end{split}\]
\[\phi_{\bar{x},\bar{x}} = \phi_{xx}\cos^2\beta - \phi_{xy}\sin(2\beta) + \phi_{yy}\sin^2\beta\]
\[\phi_{\bar{x},\bar{y}} = \phi_{xy}(\cos^2\beta - \sin^2\beta) + (\phi_{xx} - \phi_{yy})\sin\beta\cos\beta\]
\[\phi_{\bar{x},\bar{z}} = \phi_{xz}\cos\beta - \phi_{yz}\sin\beta\]
\[\phi_{\bar{y},\bar{y}} = \phi_{yy}\cos^2\beta + \phi_{xy}\sin(2\beta) + \phi_{xx}\sin^2\beta\]
\[\phi_{\bar{y},\bar{z}} = \phi_{yz}\cos\beta + \phi_{xz}\sin\beta\]
\[\phi_{\bar{z},\bar{z}} = \phi_{zz} = -\phi_{xx} -\phi_{yy}\]
\[\phi_{xx} = - \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_x}^2 \mathcal{Re} \Bigl\{C_{1,j_y,j_x}(y, t) \, X_{j_x}(x)\Bigr\} Z_{j_y,j_x}(z)\]
\[\phi_{xy} = - \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_x} k_{j_y} \mathcal{Re} \Bigl\{C_{2,j_y,j_x}(y, t) \, X_{j_x}(x)\Bigr\} Z_{j_y,j_x}(z)\]
\[\phi_{xz} = \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_x} \mathcal{Im} \Bigl\{C_{1,j_y,j_x}(y, t) \, X_{j_x}(x)\Bigr\} \frac{d Z_{j_y,j_x}(z)}{dz}\]
\[\phi_{yy} = - \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_y}^2 \mathcal{Re} \Bigl\{C_{1,j_y,j_x}(y, t) \, X_{j_x}(x)\Bigr\} Z_{j_y,j_x}(z)\]
\[\phi_{yz} = \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_y} \mathcal{Im} \Bigl\{C_{2,j_y,j_x}(y, t) \, X_{j_x}(x)\Bigr\} \frac{d Z_{j_y,j_x}(z)}{dz}\]
\[\phi_{zz} = \sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_y,j_x}^2 \mathcal{Re} \Bigl\{C_{1,j_y,j_x}(y, t) \, X_{j_x}(x)\Bigr\} Z_{j_y,j_x}(z)
= -\phi_{xx} - \phi_{yy}\]
\[\frac{\partial^2\zeta}{\partial \bar{x}^2}(\bar{x},\bar{y},\bar{t}) =
\zeta_{xx}\cos^2\beta - \zeta_{xy}\sin(2\beta) + \zeta_{yy}\sin^2\beta\]
\[\frac{\partial^2\zeta}{\partial\bar{x}\partial\bar{y}}(\bar{x},\bar{y},\bar{t}) =
\zeta_{xy}(\cos^2\beta - \sin^2\beta) + (\zeta_{xx} - \zeta_{yy})\sin\beta\cos\beta\]
\[\frac{\partial^2\zeta}{\partial\bar{y}^2}(\bar{x},\bar{y},\bar{t}) =
\zeta_{yy}\cos^2\beta + \zeta_{xy}\sin(2\beta) + \zeta_{xx}\sin^2\beta\]
\[\zeta_{xx} = -\sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_x}^2 \mathcal{Re} \Bigl\{H_{1,j_y,j_x}(y, t) \, X_{j_x}(x)\Bigr\}\]
\[\zeta_{xy} = -\sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_x} k_{j_y} \mathcal{Re} \Bigl\{H_{2,j_y,j_x}(y, t) \, X_{j_x}(x)\Bigr\}\]
\[\zeta_{yy} = -\sum_{j_x=0}^{n_x}\sum_{j_y=0}^{n_y}
k_{j_y}^2 \mathcal{Re} \Bigl\{H_{1,j_y,j_x}(y, t) \, X_{j_x}(x)\Bigr\}\]
\[p = -\rho\frac{\partial\phi}{\partial \bar{t}}
-\frac{1}{2}\rho\bar{\nabla}\phi\cdot\bar{\nabla}\phi
-\rho g \bar{z}\]
where \(\bar{\nabla}\) denotes gradients with respect to
\(\bar{x}\), \(\bar{y}\) and \(\bar{z}\).
\(\mathcal{Im}\{\alpha\}\) denotes the imaginary part of a complex number \(\alpha\) and
\[C_{2, j_y, j_x}(y,t) = c_{1, j_y, j_x}(t)\,Y_{j_y}(y) - c_{2, j_y, j_x}(t)\,\bar{Y}_{j_y}(y)\]
\[H_{2, j_y, j_x}(y,t) = h_{1, j_y, j_x}(t)\,Y_{j_y}(y) - h_{2, j_y, j_x}(t)\,\bar{Y}_{j_y}(y)\]
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.