Jonswap Spectra¶

GODA (1985) in terms of \(H_{S}\) and \(T_{p}\)

\[S(f)=\alpha\cdot H_{s}^{2} \cdot T_{p}^{-4} \cdot f^{-5} \cdot e^{-1.25 \cdot (T_{p} \cdot f)^{-4}} \cdot \gamma^{e^{-(T_{p} \cdot f - 1)^{2}/(2 \cdot \sigma^{2})}}\]

\[\alpha \approxeq \frac{0.0624}{0.230+0.0336 \cdot \gamma - 0.185 \cdot (1.9 + \gamma)^{-1}}\]

\[\begin{split} \sigma=\begin{cases} \sigma_{a}; f \leq f_{p} \\ \sigma_{b}; f \geq f_{p} \end{cases} \end{split}\]

\(\gamma\) = 1 to 7 (mean 3.3)

\(\sigma_{a}\approxeq0.07, \sigma_{b}\approxeq 0.09\)

# arrays
import numpy as np

# plots
import matplotlib.pyplot as plt

# panel interaction
import panel as pn
pn.extension() # this is very important for notebook display

def plotJonswap(f, Sf):
    
    fig = plt.figure(figsize=(7,7))
    plt.plot(f, Sf, c='b')
    plt.fill_between(f, np.full(len(f), 0), Sf, color='b', alpha=0.3)
    plt.xlim(1/30, 1)
    plt.ylim(0, 100)
    plt.title('JONSWAP Spectra')
    plt.xlabel('$f$ (1/s)')
    plt.ylabel('$S(f)$ ($m^{2}\cdot s$)')
    plt.close()
    return(fig)

def JonswapSpec(Hs, Tp, gamma):
    'Evaluate the JONSWAP-gamma parameter of the wave spectrum defined by PSD=f(f, E)'
    
    f = np.linspace(1/30, 1, 200)
    fp = 1 / Tp
    sigma = np.where(f < fp, np.ones(len(f))*0.07, np.ones(len(f))*0.09) 

    # Calculate spectral shape
    Sf1 = (0.0624/(0.230 + 0.0336*gamma - 0.185 * (1.9 + gamma)**(-1))) * Hs**2 * Tp**(-4) * f**(-5)
    Sf2 = np.exp(-1.25 * (Tp * f)**(-4)) * gamma**(np.exp((-(Tp*f - 1)**2)/(2 * sigma)))
    Sf = Sf1 * Sf2
    
    # call figure to plot
    fig = plotJonswap(f, Sf)
    
    return(fig)

kw = dict(
    Hs=(0, 10, 0.5, 1.5), Tp=(0, 25, 1, 3), gamma=(1, 20, 1, 3)
)

i = pn.interact(JonswapSpec, **kw)

panel = pn.Column(
    '**Custom JONSWAP spectrum from Hs, Tp, and gamma parameters**',
    pn.Row(
        i[0][0],i[0][1],i[0][2]
    ),
    pn.Row(i[1])
)
panel.embed()

Waves Book

Jonswap Spectra¶