This implementation of Shape 4 <shape_4> is designed for exploiting symmetric spatial resolutions.
Δkx = Δky, nx = ny
For short, we will apply the notation Δk = Δkx = Δky and n = nx = ny. It follows that
Y − jy(y) = Ȳjy(y), Z − jy, jx(z) = Zjy, jx(z), Zjx, jy(z) = Zjy, jx(z)
where Ȳjy(y) denotes the complex conjugate of Yjy(y). Consequently, we apply the following equivalent wave field formulation.
$$\phi(x,y,z,t) = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Re} \Bigl\{C_{j_y,j_x}(x, y, t)\Bigr\}\, Z_{j_y,j_x}(z)$$
$$\zeta(x,y,t) = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Re} \Bigl\{H_{ j_y,j_x}(x, y, t)\Bigr\}$$
$$\begin{aligned}
C_{j_y,j_x}(x, y, t) = &\bigl( c_{1,j_y,j_x}(t)\,Y_{j_y}(y) +
c_{2,j_y,j_x}(t)\, \bar{Y}_{j_y}(y)\bigr) X_{j_x}(x) +\\\
&\bigl( c_{3,j_y,j_x}(t)\,Y_{j_x}(y) +
c_{4,j_y,j_x}(t)\, \bar{Y}_{j_x}(y)\bigr) X_{j_y}(x)
\end{aligned}$$
$$\begin{aligned}
H_{j_y,j_x}(x, y, t) = &\bigl( h_{1,j_y,j_x}(t)\,Y_{j_y}(y) +
h_{2,j_y,j_x}(t)\, \bar{Y}_{j_y}(y)\bigr) X_{j_x}(x) +\\\
&\bigl( h_{3,j_y,j_x}(t)\,Y_{j_x}(y) +
h_{4,j_y,j_x}(t)\, \bar{Y}_{j_x}(y)\bigr) X_{j_y}(x)
\end{aligned}$$
$$\begin{aligned}
c_{1,j_y,j_x}(t) = c_{j_y,j_x}(t), \qquad
c_{2,j_y,j_x}(t) = \begin{cases}
c_{-j_y,j_x}(t), & \text{$j_y>0$}, \\\
0, & \text{$j_y=0$}
\end{cases} \\\
c_{3,j_y,j_x}(t) = \begin{cases}
c_{j_x,j_y}(t), & \text{$j_y<j_x$}, \\\
0, & \text{$j_y=j_x$}
\end{cases}, \qquad
c_{4,j_y,j_x}(t) = \begin{cases}
c_{-j_x,j_y}(t), & \text{$j_y<j_x$}, \\\
0, & \text{$j_y=j_x$}
\end{cases} \\\
h_{1,j_y,j_x}(t) = h_{j_y,j_x}(t), \qquad
h_{2,j_y,j_x}(t) = \begin{cases}
h_{-j_y,j_x}(t), & \text{$j_y>0$}, \\\
0, & \text{$j_y=0$}
\end{cases} \\\
h_{3,j_y,j_x}(t) = \begin{cases}
h_{j_x,j_y}(t), & \text{$j_y<j_x$}, \\\
0, & \text{$j_y=j_x$}
\end{cases}, \qquad
h_{4,j_y,j_x}(t) = \begin{cases}
h_{-j_x,j_y}(t), & \text{$j_y<j_x$}, \\\
0, & \text{$j_y=j_x$}
\end{cases}
\end{aligned}$$
$$k_{j_x} = j_x\cdot\Delta k, \quad k_{j_y} = j_y\cdot\Delta k, \quad
k_{j_y,j_x} = \sqrt{j_x^2+ j_y^2}\, \Delta k, \quad i = \sqrt{-1}$$
Given the definitions above we obtain the following explicit kinematics:
$$\phi(\bar{x},\bar{y},\bar{z},\bar{t})= \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Re} \Bigl\{C_{j_y,j_x}(x, y, t)\Bigr\} Z_{j_y,j_x}(z)$$
φ(x̄, ȳ, z̄, t̄) ≡ 0
$$\frac{\partial\phi}{\partial \bar{t}}(\bar{x},\bar{y},\bar{z},\bar{t}) = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Re} \Bigl\{\frac{d C_{j_y,j_x}(x, y, t)}{dt}\Bigr\} Z_{j_y,j_x}(z)$$
$$\zeta(\bar{x},\bar{y},\bar{t})= \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Re}\Bigl\{H_{j_y,j_x}(x, y, t)\Bigr\}$$
$$\frac{\partial\zeta}{\partial \bar{t}}(\bar{x},\bar{y},\bar{t}) = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Re} \Bigl\{\frac{d H_{j_y,j_x}(x, y, t)}{dt}\Bigr\}$$
$$\frac{\partial\zeta}{\partial \bar{x}}(\bar{x},\bar{y},\bar{t}) = \zeta_x\cos\beta - \zeta_y\sin\beta, \qquad
\frac{\partial\zeta}{\partial \bar{y}}(\bar{x},\bar{y},\bar{t}) = \zeta_x\sin\beta + \zeta_y\cos\beta$$
$$\zeta_x =\sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Im} \Bigl\{H_{j_y,j_x}^{1,0}(x, y, t)\Bigr\}$$
$$\zeta_y = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Im} \Bigl\{H_{j_y,j_x}^{0,1}(x, y, t)\Bigr\}$$
$$\bar{\nabla}\phi(\bar{x},\bar{y},\bar{z},\bar{t}) =
[\phi_x\cos\beta - \phi_y\sin\beta, \phi_x\sin\beta + \phi_y\cos\beta,\phi_z]^T$$
$$\phi_x = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Im} \Bigl\{C_{j_y,j_x}^{1,0}(x, y, t)\Bigr\} \, Z_{j_y,j_x}(z)$$
$$\phi_y = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Im} \Bigl\{C_{j_y,j_x}^{0,1}(x, y, t)\Bigr\} \, Z_{j_y,j_x}(z)$$
$$\phi_z = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
k_{j_y,j_x} \mathcal{Re} \Bigl\{C_{j_y,j_x}(x, y, t)\Bigr\} \, Z_{j_y,j_x}(z)$$
$$\frac{\partial\bar{\nabla}\phi}{\partial \bar{t}}(\bar{x},\bar{y},\bar{z},\bar{t}) =
[\phi_{xt}\cos\beta - \phi_{yt}\sin\beta, \phi_{xt}\sin\beta + \phi_{yt}\cos\beta,\phi_z]^T$$
$$\phi_{xt} = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Im} \Bigl\{\frac{d C_{j_y,j_x}^{1,0}(x, y, t)}{dt}\Bigr\} \, Z_{j_y,j_x}(z)$$
$$\phi_{yt} = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Im} \Bigl\{\frac{d C_{j_y,j_x}^{0,1}(x, y, t)}{dt}\Bigr\} \, Z_{j_y,j_x}(z)$$
$$\phi_{zt} = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
k_{j_y,j_x} \mathcal{Re} \Bigl\{\frac{d C_{j_y,j_x}(x, y, t)}{dt}\Bigr\} \, Z_{j_y,j_x}(z)$$
$$\frac{d\bar{\nabla}\phi}{d\bar{t}}(\bar{x},\bar{y},\bar{z},\bar{t}) =
\frac{\partial\bar{\nabla}\phi}{\partial \bar{t}} +
\bar{\nabla}\phi \cdot \bar{\nabla}\bar{\nabla}\phi$$
$$\begin{aligned}
\bar{\nabla}\bar{\nabla}\phi (\bar{x},\bar{y},\bar{z},\bar{t}) =
\begin{bmatrix}
\phi_{\bar{x},\bar{x}} & \phi_{\bar{x},\bar{y}} & \phi_{\bar{x},\bar{z}} \\\
\phi_{\bar{x},\bar{y}} & \phi_{\bar{y},\bar{y}} & \phi_{\bar{y},\bar{z}} \\\
\phi_{\bar{x},\bar{z}} & \phi_{\bar{y},\bar{z}} & \phi_{\bar{z},\bar{z}}
\end{bmatrix}
\end{aligned}$$
ϕx̄, x̄ = ϕxxcos2β − ϕxysin (2β) + ϕyysin2β
ϕx̄, ȳ = ϕxy(cos2β − sin2β) + (ϕxx − ϕyy)sin βcos β
ϕx̄, z̄ = ϕxzcos β − ϕyzsin β
ϕȳ, ȳ = ϕyycos2β + ϕxysin (2β) + ϕxxsin2β
ϕȳ, z̄ = ϕyzcos β + ϕxzsin β
ϕz̄, z̄ = ϕzz = − ϕxx − ϕyy
$$\phi_{xx} = - \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Re} \Bigl\{C_{j_y,j_x}^{2,0}(x, y, t)\Bigr\} Z_{j_y,j_x}(z)$$
$$\phi_{xy} = - \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Re} \Bigl\{C_{j_y,j_x}^{1,1}(x, y, t)\Bigr\} Z_{j_y,j_x}(z)$$
$$\phi_{xz} = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
k_{j_y,j_x} \mathcal{Im} \Bigl\{C_{j_y,j_x}^{1,0}(x, y, t)\Bigr\} Z_{j_y,j_x}(z)$$
$$\phi_{yy} = - \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Re} \Bigl\{C_{j_y,j_x}^{0,2}(x, y, t)\Bigr\} Z_{j_y,j_x}(z)$$
$$\phi_{yz} = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
k_{j_y,j_x} \mathcal{Im} \Bigl\{C_{j_y,j_x}^{0,1}(x, y, t)\Bigr\} Z_{j_y,j_x}(z)$$
$$\phi_{zz} = \sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
k_{j_y,j_x}^2 \mathcal{Re} \Bigl\{C_{j_y,j_x}(x, y, t)\Bigr\} Z_{j_y,j_x}(z)
= -\phi_{xx} - \phi_{yy}$$
$$\frac{\partial^2\zeta}{\partial \bar{x}^2}(\bar{x},\bar{y},\bar{t}) =
\zeta_{xx}\cos^2\beta - \zeta_{xy}\sin(2\beta) + \zeta_{yy}\sin^2\beta$$
$$\frac{\partial^2\zeta}{\partial\bar{x}\partial\bar{y}}(\bar{x},\bar{y},\bar{t}) =
\zeta_{xy}(\cos^2\beta - \sin^2\beta) + (\zeta_{xx} - \zeta_{yy})\sin\beta\cos\beta$$
$$\frac{\partial^2\zeta}{\partial\bar{y}^2}(\bar{x},\bar{y},\bar{t}) =
\zeta_{yy}\cos^2\beta + \zeta_{xy}\sin(2\beta) + \zeta_{xx}\sin^2\beta$$
$$\zeta_{xx} = -\sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Re} \Bigl\{H_{j_y,j_x}^{2,0}(x, y, t)\Bigr\}$$
$$\zeta_{xy} = -\sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Re} \Bigl\{H_{j_y,j_x}^{1,1}(x, y, t)\Bigr\}$$
$$\zeta_{yy} = -\sum_{j_x=0}^{n}\sum_{j_y=0}^{j_x}
\mathcal{Re} \Bigl\{H_{j_y,j_x}^{0,2}(x, y, t)\Bigr\}$$
$$p = -\rho\frac{\partial\phi}{\partial \bar{t}}
-\frac{1}{2}\rho\bar{\nabla}\phi\cdot\bar{\nabla}\phi
-\rho g \bar{z}$$
where $\bar{\nabla}$ denotes gradients with respect to x̄, ȳ and z̄. We also apply the notation
$$\begin{aligned}
\frac{\partial^{i+j}C_{j_y,j_x}(x, y, t)}{\partial x^i \partial y^j} =
(-i)^{i+j} C_{j_y,j_x}^{i,j}(x, y, t) \\\
\frac{\partial^{i+j}H_{j_y,j_x}(x, y, t)}{\partial x^i \partial y^j} =
(-i)^{i+j} H_{j_y,j_x}^{i,j}(x, y, t)
\end{aligned}$$
$$\begin{aligned}
C_{j_y,j_x}^{i,j}(x, y, t) = &k_{j_x}^ik_{j_y}^j\bigl( c_{1,j_y,j_x}(t)\,Y_{j_y}(y) +(-1)^j
c_{2,j_y,j_x}(t)\, \bar{Y}_{j_y}(y)\bigr) X_{j_x}(x) +\\\
&k_{j_y}^i k_{j_x}^j\bigl( c_{3,j_y,j_x}(t)\,Y_{j_x}(y) + (-1)^j
c_{4,j_y,j_x}(t)\, \bar{Y}_{j_x}(y)\bigr) X_{j_y}(x)
\end{aligned}$$
$$\begin{aligned}
H_{j_y,j_x}^{i,j}(x, y, t) = &k_{j_x}^ik_{j_y}^j\bigl( h_{1,j_y,j_x}(t)\,Y_{j_y}(y) +(-1)^j
h_{2,j_y,j_x}(t)\, \bar{Y}_{j_y}(y)\bigr) X_{j_x}(x) +\\\
&k_{j_y}^i k_{j_x}^j\bigl( h_{3,j_y,j_x}(t)\,Y_{j_x}(y) + (-1)^j
h_{4,j_y,j_x}(t)\, \bar{Y}_{j_x}(y)\bigr) X_{j_y}(x)
\end{aligned}$$
The particle acceleration is labeled $\frac{d\bar{\nabla}\phi}{d\bar{t}}$.
The stream function φ is not relevant for short crested seas. Hence, we apply the dummy definition φ = 0 for all locations.
Evaluation of costly transcendental functions (cos , sin , exp , ...) is significantly reduced by exploiting the following recursive relations
Xjx(x) = X1(x) Xjx − 1(x), Yjy(y) = Y1(y) Yjy − 1(y)
It should be noted that contrary to long crested seas, there are no trivial recursive relations for the z-dependent term Zjy, jx(z). This makes calculations of surface elevations significantly faster than calculations of other kinematics for short crested seas.
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.
$$Z_{j_y, j_x}(z) = 1 + \sum_{p=1}^{q-1}\frac{(k_{j_y, j_x} z)^p}{p!}, \qquad z > 0$$