This shape class describes long crested waves propagating in infinite water depth.
$$\phi(x, z, t)= \sum_{j=0}^n \mathcal{Re} \Bigl\{c_j(t)\, X_j(x) \Bigr\} Z_j(z)$$
$$\zeta(x, t)= \sum_{j=0}^n \mathcal{Re} \Bigl\{h_j(t)\, X_j(x) \Bigr\}$$
$$X_j(x) = e^{-i k_j x}, \quad Z_j(z) = e^{k_j z}, \quad k_j = j\cdot\Delta k, \quad i=\sqrt{-1}$$
The set of real constants kj resemble wave numbers. It follows that the kinematics is periodic in space
ϕ(x + λmax, z, t) = ϕ(x, z, t), ζ(x + λmax, t) = ζ(x, t)
$$\lambda_{\max} = \frac{2\pi}{\Delta k}, \qquad \lambda_{\min} = \frac{\lambda_{\max}}{n}$$
where λmin and λmax are the shortest and longest wave lengths resolved respectively.
The actual set of shape functions is uniquely defined by the two input parameters Δk and n.
The fields related to j = 0 are uniform in space (DC bias). Non-zero values of h0(t) violates mass conservation. The amplitude c0(t) adds a uniform time varying ambient pressure field not influencing the flow field. Consequently, these components will by default be suppressed in the kinematic calculations. However, there is an option in the API for including all DC values provided by the wave generator<wave-generator>.
The fields related to j = n are expected to correspond to the Nyquist frequency of the physical resolution applied in the wave generator<wave-generator>. Hence, typical n = ⌊nfft/2⌋ where nfft is the physical spatial resolution applied in the wave generator<wave-generator>.
Given the definitions above we obtain the following explicit kinematics:
$$\phi(\bar{x},\bar{y},\bar{z},\bar{t})= \sum_{j=0}^n \mathcal{Re} \Bigl\{c_j(t)\, X_j(x)\Bigr\} Z_j(z)$$
$$\varphi(\bar{x},\bar{y},\bar{z},\bar{t})= \sum_{j=0}^n \mathcal{Im} \Bigl\{c_j(t)\, X_j(x)\Bigr\} Z_j(z)$$
$$\frac{\partial\phi}{\partial \bar{t}}(\bar{x},\bar{y},\bar{z},\bar{t}) = \sum_{j=0}^n \mathcal{Re}
\Bigl\{\frac{d c_j(t)}{dt} \, X_j(x)\Bigr\} Z_j(z)$$
$$\zeta(\bar{x},\bar{y},\bar{t})= \sum_{j=0}^n \mathcal{Re}
\Bigl\{h_j(t)\, X_j(x)\Bigr\}$$
$$\frac{\partial\zeta}{\partial \bar{t}}(\bar{x},\bar{y},\bar{t}) = \sum_{j=0}^n \mathcal{Re}
\Bigl\{\frac{d h_j(t)}{dt} \, X_j(x)\Bigr\}$$
$$\frac{\partial\zeta}{\partial \bar{x}}(\bar{x},\bar{y},\bar{t}) = \zeta_x\cos\beta, \qquad
\frac{\partial\zeta}{\partial \bar{y}}(\bar{x},\bar{y},\bar{t}) = \zeta_x\sin\beta$$
$$\zeta_x = \sum_{j=0}^n k_j\mathcal{Im} \Bigl\{h_j(t)\, X_j(x)\Bigr\}$$
$$\bar{\nabla}\phi(\bar{x},\bar{y},\bar{z},\bar{t}) = [\phi_x\cos\beta,\phi_x\sin\beta,\phi_z]^T$$
$$\phi_x = \sum_{j=0}^n k_j\mathcal{Im} \Bigl\{c_j(t)\, X_j(x)\Bigr\} \, Z_j(z)$$
$$\phi_z = \sum_{j=0}^n k_j\mathcal{Re} \Bigl\{c_j(t)\, X_j(x)\Bigr\} \, Z_j(z)$$
$$\frac{\partial\bar{\nabla}\phi}{\partial \bar{t}}(\bar{x},\bar{y},\bar{z},\bar{t}) =
[\phi_{xt}\cos\beta,\phi_{xt}\sin\beta,\phi_{zt}]^T$$
$$\phi_{xt} = \sum_{j=0}^n k_j \mathcal{Im} \Bigl\{\frac{d c_j(t)}{dt} \, X_j(x)\Bigr\} Z_j(z)$$
$$\phi_{zt} = \sum_{j=0}^n k_j \mathcal{Re} \Bigl\{\frac{d c_j(t)}{dt} \, X_j(x)\Bigr\} Z_j(z)$$
$$\frac{d\bar{\nabla}\phi}{d\bar{t}}(\bar{x},\bar{y},\bar{z},\bar{t}) =
\frac{\partial\bar{\nabla}\phi}{\partial \bar{t}} +
\bar{\nabla}\phi \cdot \bar{\nabla}\bar{\nabla}\phi$$
$$\begin{aligned}
\bar{\nabla}\bar{\nabla}\phi (\bar{x},\bar{y},\bar{z},\bar{t}) =
\begin{bmatrix}
\phi_{\bar{x},\bar{x}} & \phi_{\bar{x},\bar{y}} & \phi_{\bar{x},\bar{z}} \\\
\phi_{\bar{x},\bar{y}} & \phi_{\bar{y},\bar{y}} & \phi_{\bar{y},\bar{z}} \\\
\phi_{\bar{x},\bar{z}} & \phi_{\bar{y},\bar{z}} & \phi_{\bar{z},\bar{z}}
\end{bmatrix}
\end{aligned}$$
ϕx̄, x̄ = ϕxxcos2β, ϕx̄, ȳ = ϕxxsin βcos β, ϕx̄, z̄ = ϕxzcos β
ϕȳ, ȳ = ϕxxsin2β, ϕȳ, z̄ = ϕxzsin β, ϕz̄, z̄ = ϕzz = − ϕxx
$$\phi_{xx} = -\sum_{j=0}^n k_j^2 \mathcal{Re} \Bigl\{c_j(t) \, X_j(x)\Bigr\} Z_j(z)$$
$$\phi_{zz} = \sum_{j=0}^n k_j^2 \mathcal{Re} \Bigl\{c_j(t) \, X_j(x)\Bigr\} Z_j(z)
= - \phi_{xx}$$
$$\phi_{xz} = \sum_{j=0}^n k_j^2 \mathcal{Im} \Bigl\{c_j(t) \, X_j(x)\Bigr\} Z_j(z)$$
$$\frac{\partial^2\zeta}{\partial \bar{x}^2}(\bar{x},\bar{y},\bar{t}) = \zeta_{xx}\cos^2\beta
\qquad
\frac{\partial^2\zeta}{\partial \bar{y}^2}(\bar{x},\bar{y},\bar{t}) = \zeta_{xx}\sin^2\beta$$
$$\frac{\partial^2\zeta}{\partial\bar{x}\partial\bar{y}}(\bar{x},\bar{y},\bar{t}) =
\zeta_{xx}\sin\beta\cos\beta$$
$$\zeta_{xx} = -\sum_{j=0}^n k_j^2 \mathcal{Re} \Bigl\{h_j(t) \, X_j(x)\Bigr\}$$
$$p = -\rho\frac{\partial\phi}{\partial \bar{t}}
-\frac{1}{2}\rho\bar{\nabla}\phi\cdot\bar{\nabla}\phi
-\rho g \bar{z}$$
where $\bar{\nabla}$ denotes gradients with respect to x̄, ȳ and z̄. The particle acceleration is labeled $\frac{d\bar{\nabla}\phi}{d\bar{t}}$.
The stream function φ is related to the velocity potential ϕ. Hence ∂ϕ/∂x = ∂φ/∂z and ∂ϕ/∂z = − ∂φ/∂x.
Evaluation of costly transcendental functions (cos , sin , exp , ...) are almost eliminated by exploiting the following recursive relations
Xj(x) = X1(x) Xj − 1(x), Zj(z) = Z1(z) Zj − 1(z), j > 1
In case the wave generator<wave-generator> applies a perturbation theory of order q we apply the following Taylor expansion above the calm free surface.
$$Z_j(z) = 1 + \sum_{p=1}^{q-1}\frac{(k_j z)^p}{p!}, \qquad z > 0$$