Waves 1. Regular Waves Regular waves are given as a list of one or more wave components. Each component has a propagation direction, phase angle, wave period, and amplitude. 2. Irregular Waves When using Irregular waves the sea state is described using a wave spectrum function. The Spreading Type option may be used to choose between a unidirectional or short crested sea state using a cosine spreading function. 2.1. Six parameter JONSWAP spectrum The six parameter JONSWAP spectrum is given by \[S(\omega )=\frac{\alpha g^2}{\omega ^5}\exp(-\beta (\frac{\omega _p}{\omega })^4)\gamma ^{\exp(-(\displaystyle \frac{(\frac{\omega }{\omega _p}-1)^2}{2\sigma ^2}))} \\ \\ \sigma = \begin{cases} \sigma_a, & \omega \leq \omega_p \\ \\ \sigma_b, & \omega > \omega_p \end{cases} \\ \\ \omega_p = \frac{2\pi}{T_p} \label{eq-jonswap-6p}\] where the following parameters are input: \(\alpha\) spectral parameter \(T_p\) peak period \(\gamma\) peakedness parameter \(\beta\) form parameter, default value \(\beta = 1.25\) \(\sigma_a\) spectral parameter a, default value \(\sigma_a = 0.07\) \(\sigma_b\) spectral parameter b, default value \(\sigma_b = 0.09\) 2.2. Three parameter JONSWAP spectrum The spectrum function is the same as for the six parameter JONSWAP spectrum given in Equation \(\eqref{eq-jonswap-6p}\) with the following relations: \[\alpha =5.061\frac{H_s^2}{T_p^4}(1-0.287\ln(\gamma )) \\ \beta = 1.25 \\ \sigma_a = 0.07 \\ \sigma_b = 0.09 \label{eq-jonswap-3p}\] The following values are given as input: \(H_s\) significant wave height \(T_p\) peak period \(\gamma\) peakedness parameter 2.3. Two parameter JONSWAP spectrum The two parameter JONSWAP spectrum is given by Equation \(\eqref{eq-jonswap-6p}\), Equation \(\eqref{eq-jonswap-3p}\) and the following relation to determine \(\gamma\): \[\gamma = \begin{cases} 5.0, & T_p \leq 3.6 \sqrt{H_s} \\ \\ \exp\left[3.484 \left(1-0.1975~\delta \frac{T_p^4}{H_s^2}\right)\right], & 3.6 \sqrt{H_s} < T_p < 5.0 \sqrt{H_s} \\ \\ 1.0, & T_p \geq 5.0 \sqrt{H_s} \end{cases}\] where \[\delta = 0.036-0.0056\frac{T_p}{\sqrt{H_s}}\] 2.4. Double peaked JONSWAP spectrum The spectrum function is given in (Torsethaugen, 1996). 2.5. Pierson-Moskowitz spectrum The Pierson-Moskowitz spectrum is given by \[S(\omega )=\frac{a}{\omega ^5}\exp(-\frac{b}{\omega ^4}) \label{eq-pm-spec}\] where \[a = 0.0081 g^2\\ \\ b = \frac{3.11}{H_s^2}\] and where \(g\) acceleration of gravity The following parameters are given as input: \(H_s\) significant wave height 2.6. Pierson-Moskowitz ISSC spectrum The spectrum is given by Equation \(\eqref{eq-pm-spec}\) where \[a = 173.0 \frac{H_s^2}{T_1} \\ \\ b = \frac{691.0}{T_1^4} \label{eq-pm-issc-ab}\] The following parameters are given as input: \(H_s\) significant wave height \(T_1\) average period 2.7. Pierson-Moskowitz spectrum with zero-corrsing period The spectrum is given by Equation \(\eqref{eq-pm-spec}\) and Equation \(\eqref{eq-pm-issc-ab}\) where \[T_1 = 1.0864 T_z\] The following parameters are given as input: \(H_s\) significant wave height \(T_z\) zero upcrossing period 2.8. Ochi-Hubble spectrum The spectrum is given by \[\begin{align}\begin{split} S(\omega) = & E_1 G_1 \omega_{n, 1}^{4 - 0.25 \lambda_1} \exp\left[-\left(\lambda_1 + 0.25\right)\omega_{n,1}^{-4}\right] \\ & + E_2 G_2 \omega_{n, 2}^{4 - 0.25 \lambda_2} \exp\left[-\left(\lambda_2 + 0.25\right)\omega_{n,2}^{-4}\right] \end{split}\end{align}\] where \[\begin{align}\begin{split} \omega_{n,1} &= \frac{\omega}{w_{p,1}} \\ \omega_{n,2} &= \frac{\omega}{w_{p,2}} \\ E_1 &= \frac{H_{s,1}^2}{16\omega_{p,1}} \\ E_2 &= \frac{H_{s,2}^2}{16\omega_{p,2}} \\ G_1 &= \frac{4(\lambda_1 + 0.25)^{\lambda_1}}{\gamma(\lambda_1)} \\ G_2 &= \frac{4(\lambda_2 + 0.25)^{\lambda_2}}{\gamma(\lambda_2)} \\ \lambda_1 &= 3.0 \\ \lambda_2 &= 1.54 e^{-0.062 H_s} \\ \omega_{p,1} &= 0.7 e^{-0.046 H_s} \\ \omega_{p,2} &= 1.15 e^{-0.039 H_s} \\ H_{s,1} &= 0.84 H_s \\ H_{s,2} &= 0.54 H_s \end{split}\end{align}\] The following parameters are given as input: \(H_s\) significant wave height 2.9. Numerically defined spectrum This is a wave spectrum specified by the user. The spectrum is defined by spectral values and frequencies. Spectral values outside the given frequency range are assumed to be zero. Linear interpolation is used within the frequency range. The wave spectrum can be specified in an external file. 2.9.1. Numerically defined spectrum from file One input line specifying the number of frequencies and wave directions NWFRE NWDIR NWFRE: integer: Number of wave frequencies NWDIR: integer: Number of wave directions One input line specifying the wave directions WADIR(1) ... WADIR(NWDIR) WADIR: real: Wave propagation direction, \(\mathrm {[deg]}\) Wave propagation direction (NWDIR+1)/2 is used as average wave direction in DYNMOD in cases where responses are pregenerated in a range around the average direction. NWFRE lines specifying the wave frequency and wave spectrum values for all directions WAFREi SPVAL(j,i) ... SPVAL(NWDIR,i) WAFREi: real: Wave frequency no. i, \(\mathrm {[rad}/T]\) SPVAL(j,i): real: Spectrum value for wave frequency no. i and wave direction no. j, \([L^2T/\mathrm{rad]}\) Input file example Lines starting with the symbol ' is treated as a comment The specified file must contain: '--------------------------------------------------------------- ' START OF FILE '--------------------------------------------------------------- ' NWFRE = Number of wave frequencies ' NWDIR = Number of wave directions '--------------------------------------------------------------- 'nwfre nwdir 25 24 '--------------------------------------------------------------- ' WADIR_j = real: Wave propagation direction, [deg] '--------------------------------------------------------------- ' WADIR_1 ... WADIR_j ... WADIR_nwdir ' 2 17 32 47 62 77 92 107 122 137 152 167 182 197 212 227 242 257 272 287 302 317 332 347 '--------------------------------------------------------------- ' WAFRE_i = Wave frequency no. i, ' SPVAL_j,i = Spectrum value no. ji '--------------------------------------------------------------- ' WAFRE_i SPVAL_1,i ...... SPVAL_j,i ... SPVAL_nwdir,i ' 0.042 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0462 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0508 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.6E-4 2.6E-4 0.0 0.0 2.6E-4 0.0 0.0 0.0 0.0 0.0559 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 7.9E-4 0.00602 0.01047 0.00157 2.6E-4 0.00576 0.00314 7.9E-4 0.0 0.0 0.0615 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.00497 0.04294 0.14608 0.13352 0.01047 0.04424 0.02749 0.01518 0.00471 5.2E-4 0.0676 0.00157 2.6E-4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 5.2E-4 0.01257 0.09006 0.30683 0.96237 0.20211 0.10105 0.09765 0.15499 0.06807 0.01073 0.0744 0.01754 0.00209 2.6E-4 5.2E-4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.00131 0.01257 0.11153 0.70319 2.92142 1.97894 0.44558 0.26756 0.25892 0.16441 0.08666 0.0818 0.0589 0.01073 0.00105 0.00157 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.6E-4 0.00262 0.01623 0.14111 1.0506 4.81816 4.53594 2.17974 1.01814 0.38249 0.26677 0.31128 0.09 0.06178 0.01702 0.00262 7.9E-4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 5.2E-4 0.00445 0.0267 0.19268 1.39199 4.82522 5.46244 4.45975 3.85212 1.04406 0.4982 0.42987 0.099 0.08299 0.0178 0.00262 0.00105 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.00131 0.00838 0.06964 0.42019 1.04877 2.88163 4.4375 3.09421 2.93111 2.17948 1.23726 0.57779 0.1089 0.14713 0.02618 0.0034 0.00131 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.6E-4 0.00183 0.01518 0.13561 0.3961 0.68303 1.59933 2.69156 1.86715 1.50508 1.3467 1.32628 0.76917 0.1198 0.17331 0.03325 0.00445 0.00105 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.6E-4 0.00236 0.02539 0.1542 0.3139 0.50396 0.90897 1.24407 1.21213 0.9451 0.7443 0.57439 0.50527 0.1318 0.12933 0.03299 0.00419 5.2E-4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.6E-4 0.00236 0.02827 0.14582 0.26861 0.41364 0.73356 0.87389 0.83854 0.67466 0.54795 0.41888 0.29243 0.145 0.10393 0.02618 0.0034 2.6E-4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.00183 0.02644 0.12488 0.22436 0.35579 0.59952 0.68408 0.61732 0.48904 0.39139 0.32515 0.2296 0.1595 0.08849 0.02199 0.00262 2.6E-4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.00157 0.02173 0.09739 0.18404 0.31337 0.45003 0.50894 0.47019 0.39322 0.2885 0.22881 0.16781 0.1754 0.06231 0.01806 0.00183 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.00105 0.01676 0.07173 0.13561 0.23536 0.31285 0.34348 0.31023 0.26573 0.20289 0.15053 0.11074 0.193 0.04006 0.01257 0.00131 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.00105 0.01283 0.05 0.09687 0.16415 0.20918 0.22331 0.20106 0.17253 0.13614 0.09713 0.07121 0.2123 0.02513 0.00812 7.9E-4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 7.9E-4 0.00995 0.03508 0.07016 0.11153 0.13456 0.13797 0.12933 0.11336 0.09032 0.06178 0.04398 0.2335 0.01545 0.00497 5.2E-4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 5.2E-4 0.00733 0.02513 0.05105 0.07671 0.08927 0.08901 0.08482 0.07645 0.06205 0.0411 0.02723 0.2569 0.00916 0.00288 2.6E-4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 5.2E-4 0.0055 0.01806 0.03665 0.0521 0.05838 0.0576 0.0555 0.05131 0.04241 0.02775 0.01702 0.2826 0.00497 0.00157 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.6E-4 0.00393 0.01283 0.02566 0.03482 0.03744 0.03691 0.03639 0.03456 0.0288 0.01859 0.01047 0.3108 0.00262 7.9E-4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.6E-4 0.00262 0.0089 0.01728 0.02225 0.0233 0.0233 0.02304 0.02225 0.01885 0.0123 0.00628 0.3419 0.00131 2.6E-4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.6E-4 0.00183 0.00576 0.011 0.01388 0.0144 0.0144 0.01466 0.01414 0.0123 0.00785 0.00367 0.3761 5.2E-4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.00105 0.00367 0.00707 0.00838 0.0089 0.0089 0.00916 0.0089 0.00759 0.00497 0.00209 0.4137 2.6E-4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 7.9E-4 0.00236 0.00419 0.00524 0.0055 0.0055 0.00576 0.0055 0.00471 0.00314 0.00131 '--------------------------------------------------------------- ' END OF FILE '--------------------------------------------------------------- Current Wind