This shape class describes long crested waves propagating in infinite water depth.
\[\phi(x, z, t)= \sum_{j=0}^n \mathcal{Re} \Bigl\{c_j(t)\, X_j(x) \Bigr\} Z_j(z)\]
\[X_j(x) = e^{-i k_j x}, \quad Z_j(z) = e^{k_j z}, \quad k_j = j\cdot\Delta k, \quad i=\sqrt{-1}\]
\[\phi(x + \lambda_{\max}, z, t) = \phi(x, z, t), \qquad
\zeta(x + \lambda_{\max}, t) = \zeta(x, t)\]
\[\lambda_{\max} = \frac{2\pi}{\Delta k}, \qquad \lambda_{\min} = \frac{\lambda_{\max}}{n}\]
The actual set of shape functions is uniquely defined by the two input parameters
\(\Delta k\) and \(n\).
4.5.1.1. Kinematics
Given the definitions above we obtain the following explicit kinematics:
\[\phi(\bar{x},\bar{y},\bar{z},\bar{t})= \sum_{j=0}^n \mathcal{Re} \Bigl\{c_j(t)\, X_j(x)\Bigr\} Z_j(z)\]
\[\varphi(\bar{x},\bar{y},\bar{z},\bar{t})= \sum_{j=0}^n \mathcal{Im} \Bigl\{c_j(t)\, X_j(x)\Bigr\} Z_j(z)\]
\[\frac{\partial\phi}{\partial \bar{t}}(\bar{x},\bar{y},\bar{z},\bar{t}) = \sum_{j=0}^n \mathcal{Re}
\Bigl\{\frac{d c_j(t)}{dt} \, X_j(x)\Bigr\} Z_j(z)\]
\[\zeta(\bar{x},\bar{y},\bar{t})= \sum_{j=0}^n \mathcal{Re}
\Bigl\{h_j(t)\, X_j(x)\Bigr\}\]
\[\frac{\partial\zeta}{\partial \bar{t}}(\bar{x},\bar{y},\bar{t}) = \sum_{j=0}^n \mathcal{Re}
\Bigl\{\frac{d h_j(t)}{dt} \, X_j(x)\Bigr\}\]
\[\frac{\partial\zeta}{\partial \bar{x}}(\bar{x},\bar{y},\bar{t}) = \zeta_x\cos\beta, \qquad
\frac{\partial\zeta}{\partial \bar{y}}(\bar{x},\bar{y},\bar{t}) = \zeta_x\sin\beta\]
\[\zeta_x = \sum_{j=0}^n k_j\mathcal{Im} \Bigl\{h_j(t)\, X_j(x)\Bigr\}\]
\[\bar{\nabla}\phi(\bar{x},\bar{y},\bar{z},\bar{t}) = [\phi_x\cos\beta,\phi_x\sin\beta,\phi_z]^T\]
\[\phi_x = \sum_{j=0}^n k_j\mathcal{Im} \Bigl\{c_j(t)\, X_j(x)\Bigr\} \, Z_j(z)\]
\[\phi_z = \sum_{j=0}^n k_j\mathcal{Re} \Bigl\{c_j(t)\, X_j(x)\Bigr\} \, Z_j(z)\]
\[\frac{\partial\bar{\nabla}\phi}{\partial \bar{t}}(\bar{x},\bar{y},\bar{z},\bar{t}) =
[\phi_{xt}\cos\beta,\phi_{xt}\sin\beta,\phi_{zt}]^T\]
\[\phi_{xt} = \sum_{j=0}^n k_j \mathcal{Im} \Bigl\{\frac{d c_j(t)}{dt} \, X_j(x)\Bigr\} Z_j(z)\]
\[\phi_{zt} = \sum_{j=0}^n k_j \mathcal{Re} \Bigl\{\frac{d c_j(t)}{dt} \, X_j(x)\Bigr\} Z_j(z)\]
\[\frac{d\bar{\nabla}\phi}{d\bar{t}}(\bar{x},\bar{y},\bar{z},\bar{t}) =
\frac{\partial\bar{\nabla}\phi}{\partial \bar{t}} +
\bar{\nabla}\phi \cdot \bar{\nabla}\bar{\nabla}\phi\]
\[\begin{split}\bar{\nabla}\bar{\nabla}\phi (\bar{x},\bar{y},\bar{z},\bar{t}) =
\begin{bmatrix}
\phi_{\bar{x},\bar{x}} & \phi_{\bar{x},\bar{y}} & \phi_{\bar{x},\bar{z}} \\
\phi_{\bar{x},\bar{y}} & \phi_{\bar{y},\bar{y}} & \phi_{\bar{y},\bar{z}} \\
\phi_{\bar{x},\bar{z}} & \phi_{\bar{y},\bar{z}} & \phi_{\bar{z},\bar{z}}
\end{bmatrix}\end{split}\]
\[\phi_{\bar{x},\bar{x}} = \phi_{xx}\cos^2\beta, \qquad
\phi_{\bar{x},\bar{y}} = \phi_{xx}\sin\beta\cos\beta, \qquad
\phi_{\bar{x},\bar{z}} = \phi_{xz}\cos\beta\]
\[\phi_{\bar{y},\bar{y}} = \phi_{xx}\sin^2\beta, \qquad
\phi_{\bar{y},\bar{z}} = \phi_{xz}\sin\beta, \qquad
\phi_{\bar{z},\bar{z}} = \phi_{zz} = -\phi_{xx}\]
\[\phi_{xx} = -\sum_{j=0}^n k_j^2 \mathcal{Re} \Bigl\{c_j(t) \, X_j(x)\Bigr\} Z_j(z)\]
\[\phi_{zz} = \sum_{j=0}^n k_j^2 \mathcal{Re} \Bigl\{c_j(t) \, X_j(x)\Bigr\} Z_j(z)
= - \phi_{xx}\]
\[\phi_{xz} = \sum_{j=0}^n k_j^2 \mathcal{Im} \Bigl\{c_j(t) \, X_j(x)\Bigr\} Z_j(z)\]
\[\frac{\partial^2\zeta}{\partial \bar{x}^2}(\bar{x},\bar{y},\bar{t}) = \zeta_{xx}\cos^2\beta
\qquad
\frac{\partial^2\zeta}{\partial \bar{y}^2}(\bar{x},\bar{y},\bar{t}) = \zeta_{xx}\sin^2\beta\]
\[\frac{\partial^2\zeta}{\partial\bar{x}\partial\bar{y}}(\bar{x},\bar{y},\bar{t}) =
\zeta_{xx}\sin\beta\cos\beta\]
\[\zeta_{xx} = -\sum_{j=0}^n k_j^2 \mathcal{Re} \Bigl\{h_j(t) \, X_j(x)\Bigr\}\]
\[p = -\rho\frac{\partial\phi}{\partial \bar{t}}
-\frac{1}{2}\rho\bar{\nabla}\phi\cdot\bar{\nabla}\phi
-\rho g \bar{z}\]
where \(\bar{\nabla}\) denotes gradients with respect to
\(\bar{x}\), \(\bar{y}\) and \(\bar{z}\). The particle acceleration
is labeled \(\frac{d\bar{\nabla}\phi}{d\bar{t}}\).
The stream function \(\varphi\) is related to the velocity potential \(\phi\).
Hence \(\partial \phi/\partial x = \partial \varphi/\partial z\)
and \(\partial \phi/\partial z = -\partial \varphi/\partial x\).