shape_4.rst

Shape class 4

This shape class describes general short crested spectral waves propagating in infinite water depth.

$$\phi(x,y,z,t) = \sum_{j_x=0}^{n_x}\sum_{j_y=-n_y}^{n_y} \mathcal{Re} \Bigl\{c_{j_y,j_x}(t)\, X_{j_x}(x)\,Y_{j_y}(y)\Bigr\}\, Z_{j_y,j_x}(z)$$

$$\zeta(x,y,t) = \sum_{j_x=0}^{n_x}\sum_{j_y=-n_y}^{n_y} \mathcal{Re} \Bigl\{h_{j_y,j_x}(t) \,X_{j_x}(x)\,Y_{j_y}(y)\Bigr\}$$

X_jx(x) = e^− ik_jxx, Y_jy(y) = e^− ik_jyy, Z_jy, jx(z) = e^k_jy, jxz

$$k_{j_x} = j_x\cdot\Delta k_x, \quad k_{j_y} = j_y\cdot\Delta k_y, \quad k_{j_y,j_x} = \sqrt{k_{j_x}^2+k_{j_y}^2}, \quad i = \sqrt{-1}$$

The set of real constants k_jx and k_jy resemble wave numbers in the x and y directions respectively. It follows that the kinematics is periodic in space

ϕ(x + L_x, y + L_y, z, t) = ϕ(x, y, z, t),  ζ(x + L_x, y + L_y, t) = ζ(x, y, t)

$$L_x = \frac{2\pi}{\Delta k_x}, \qquad L_y = \frac{2\pi}{\Delta k_y}, \qquad \lambda_{\min} = \frac{2\pi}{\sqrt{(n_x \Delta k_x)^2+(n_y \Delta k_y)^2}}$$

where λ_min is the shortest wave lengths resolved. The actual set of shape functions is uniquely defined by the five input parameters Δk_x, Δk_y, n_x, n_y and d.

Note

The fields related to j_x = j_y = 0 are uniform in space (DC bias). Non-zero values of h_0, 0(t) violates mass conservation. The amplitude c_0, 0(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_x = n_x and j_y =  ± n_y are expected to correspond to the Nyquist frequencies of the physical resolution applied in the wave generator<wave-generator>. Hence, typical n_x = ⌊n_x, fft/2⌋ and n_y = ⌊n_y, fft/2⌋ where n_x, fft and n_y, fft are the physical spatial resolutions applied in the wave generator<wave-generator>, in the x and y directions respectively.

Evaluation of kinematics in short-crested seas is in general computational demanding. Consequently, this API provides several alternative implementations in order to exploit eventual symmetric properties or numerical approximations.

• Shape 4, impl 1: <shape_4_impl_1> A general implementation.
• Shape 4, impl 2: <shape_4_impl_2> Optimized for symmetric resolution Δk_x = Δk_y and n_x = n_y.

shape_4_impl_1 shape_4_impl_2