Multivariate

JS_BWG_HNE{K}(α,ωₚ,γ,r,ϕₘ,β,ν,σₗ,σᵣ)
JS_BWG_HNE{K}(x)

9 parameter model for vertical, northward and eastward displacement of a partical in waves with JONSWAP marginal spectra and bimodal wrapped Gaussian spreading.

Type parameter

The type parameter K is a non-negative integer representing the ammount of aliasing to be done. This is useful as some devices have high-pass filters which limit the ammount of aliasing that will be present in a recorded series. In particular, a JS_BWG_HNE{K} model for a series recorded every Δ seconds would be appropriate for a series sampled every Δ seconds and filtered with a high-pass filter at (2K-1)π/Δ.

Arguments

• α: The scale parameter.
• ωₚ: The peak angular frequency.
• γ: The peak enhancement factor.
• r: The tail decay index.
• ϕₘ: The mean direction.
• β: The limiting peak separation.
• ν: The peak separation shape.
• σₗ: The limiting angular width.
• σᵣ: The angular width shape.

Vector constructor arguments

• x: Vector of parameters [α,ωₚ,γ,r,ϕₘ,β,ν,σₗ,σᵣ].

Background

The model is a transformation of a model for the frequency-direction spectra of a wave process. Such a frequency-direction spectra, denoted $f(ω,ϕ)$ is typically decomposed by writing:

\[f(ω,ϕ) = f(ω)D(ω,ϕ)\]

where $f(ω)$ is the marginal spectral density function and $D(ω,ϕ)$ is the spreading function. This model uses a JONSWAP marginal sdf and a bimodal wrapped Gaussian spreading function. The JONSWAP spectral density function is

\[f(ω) = αω^{-r}\exp\left \{-\frac{r}{4}\left(\frac{|ω|}{ωₚ}\right)^{-4}\right \}γ^{δ(|ω|)}\]

where

\[δ(ω) = \exp\left\{-\tfrac{1}{2 (0.07+0.02\cdot\mathbb{1}_{ωₚ>|ω|})^2}\left (\tfrac{|ω|}{ωₚ}-1\right )^2\right\}.\]

The bimodal wrapped Gaussian spreading function is

\[D(ω,ϕ) = \frac{1}{2σ(ω)\sqrt{2π}}\sum\limits_{k=-∞}^{∞} \sum\limits_{i=1}^{2} \exp\left\{-\frac{1}{2}\left(\frac{ϕ-ϕ_{mi}(ω)-2\pi k}{σ(ω)}\right)^2\right\}\]

where

\[\begin{align*} ϕ_{m1}(ω) &= ϕ_m + β \exp\{-ν \min(ω_p/|ω|,1)\}/2 \\ ϕ_{m2}(ω) &= ϕ_m - β \exp\{-ν \min(ω_p/|ω|,1)\}/2 \\ σ(ω) &= σ_l - \frac{σ_r}{3}\left\{ 4\left(\frac{ω_p}{|ω|}\right)^2 - \left(\frac{ω_p}{|ω|}\right)^8 \right\} \end{align*}\]

JS_BWG_HNE_DL{K,H}(α,ωₚ,γ,r,ϕₘ,β,ν,σₗ,σᵣ)
JS_BWG_HNE_DL{K,H}(x)

Depth limited version of the JS_BWG_HNE model.

Type parameter

The additional type parameter H is the water depth (m).

More information

For more information, see the documentation for JS_BWG_HNE.