How to model a regular wave using irregular wave input

1. Input

Time series generation parameters
- Simulation length, Tdur[s]
- Time step, dt [s]
Numerical wave spectrum
- Wave height, H [m]
- Wave period, T[s]

The number of time step must be 2^n
To estimate the actual Tdur that will be used in the time series generation
Number of time steps in FFT:
- \(\ln{\frac{T_{dur}}{dt}}=\ln{2^n}= n \ln{2} \Rightarrow n_{actual} = \lceil{{\frac{\ln{\frac{T_{dur}}{dt}}}{\ln{2}}}}\rceil\)
Actual simulation length:
- \(T_{dur,actual} = n_{actual} \cdot dt\)
Wave frequency in FFT:
- \(\Delta f = \frac{1}{T_{dur,actual}}, \Delta\omega = 2\pi\cdot df\)

Regular wave frequency
- \(\omega_{reg} = \frac{2\pi}{T}\)
Wave energy
- \(\frac{1}{2} (\frac{H}{2})^2 = S(\omega) \Delta\omega \rightarrow S(\omega) = \frac{1}{2} (\frac{H}{2})^2 \frac{1}{\Delta \omega}\)

Input frequencies
- \(\omega_1 = (\omega_{reg}- \Delta\omega/2)(1-fac)\)
- \(\omega_2 = \omega_{reg} - \Delta\omega/2\)
- \(\omega_3 = \omega_{reg}+ \Delta\omega/2\)
- \(\omega_4 = \omega_{reg}+ \Delta\omega/2(1+fac)\)
Input wave energy
- \(S(\omega_1) = 0.0\)
- \(S(\omega_2) = \frac{1}{2} (\frac{H}{2})^2 \frac{1}{\Delta \omega}\)
- \(S(\omega_3) = \frac{1}{2} (\frac{H}{2})^2 \frac{1}{\Delta \omega}\)
- \(S(\omega_4) = 0.0\)
Fac is a small value ensuring that the wave frequencies are increasing in value, example: fac=0.00005