Example: Second Order Nonlinear Wave

In the previous example, the Gaussian irregular wave represented by a Fourier series is studied. In this example, the second-order nonlinear wave is analyzed. More-over, the nonlinear wave propagation is considered by allowing a nonzero design lo-cation and design time. Due to the interaction among different Fourier components, the resultant wave elevation for the current nonlinear wave is non-Gaussian.

Though the second-order nonlinear wave elevation is not directly represented as a Fourier sum, the randomness of the irregular wave can still be specified by the phase sequence of its linear dominant term. Let (φ₁, · · · , φ_N) be the phase sequence of the linear dominant term. The linear elevation profile is calculated by

η⁽¹⁾ =

N

X

n=1

a_ncos(k_nx − ω_nt + φ_n) (3.50) where k_n is the wave number for the n^th Fourier component and the wave numbers and their corresponding frequencies are related by the dispersion relation

ω_n² = g|k_n| tanh(|k_n|d) (3.51) where d is the water depth and g is the gravitational constant.

Forristall (2000) gives the second-order correction η⁽²⁾ to the irregular wave eleva-tion by considering the interaceleva-tions between different Fourier frequencies. Equaeleva-tions from 3.52 to 3.60 calculate the second-order correction so that the final nonlinear elevation is evaluated as η = η⁽¹⁾+ η⁽²⁾. The correction term is calculated by

η⁽²⁾ = 1

The second-order correction makes the crest steeper and the trough flatter. More-over, the correction becomes less important at large water depth. Hence, to illustrate whether the model accounts for the nonlinearity, the water depth of 30 meters is used in the example.

Different from the linear wave example where the design time and the design

Figure 3.48: Histogram of the wave elevation at the design location and the design time normalized by RMS of the linear components

location are set to zero, this example uses a more general setting. The design location is set at x_d = 2/k_min, where k_min is the smallest wave number among all Fourier components. The design time is set at the time needed for the wave component with the smallest group velocity to reach the design location, td = xd/(ω/k)min. For the discretized spectrum used in the linear wave example, the design location and the design time are calculated to be x_d= 90.03 m, and t_d= 7.45 s. A MCS is conducted to collect the elevations at the design time and the design location. Figure 3.48 is a histogram of the collected elevations normalized by the standard deviation calculated using the linear components. As shown in the histogram, the second-order correction has tilted the histogram to be asymmetrical and non-Gaussian. Though the mean elevation is still zero, a longer tail is observed for the positive elevation (or crest) compared to the negative elevation (or trough).

The objective of the model is to generate phase vector (φ1, · · · , φN) that the user can specify at the upstream so that a large elevation greater than 2σ will be observed at the design location downstream and the design time. In other words, the model generates phase vectors from the following distribution.

Pr(φ₁, · · · , φ_N|η(t_d, x_d) > 2σ) (3.61) As before, the bootstrapping method is used to train the TEG model at a large threshold. The bootstrapping procedures are conducted 3 times to make the model generate phase sequences leading to elevations exceeding 2σ. Figures 3.49, 3.50, and 3.51 compare the marginal distribution of the phase sequences between the given dataset (left) Di and their corresponding generated dataset D⁰_i after each bootstrap-ping step. The TEG model is again successful to recognize the joint distribution of the

Figure 3.49: Compare the marginal distribution of phases between the given dataset (left) D₁ and the generated dataset (right) D₁⁰

Figure 3.50: Compare the marginal distribution of phases between the given dataset (left) D₂ and the generated dataset (right) D₂⁰

phases and present a matched marginal distribution. The marginal distribution is no longer symmetric around zero for each phase since the wave propagates downstream.

Figures 3.52, 3.53, and 3.54 compare the histograms of the resultant elevations η(t_d, x_d) between the given phases (red) and generated phases (blue) after each boot-strapping step. As before, the histograms are matched in the tail region and the mismatch at the lower boundaries comes from the discontinuity of the filtered dataset.

After 3 bootstrapping steps, the model can generate phase sequences leading to large enough elevations at the design time and the design location. The generated phase sequences with elevations smaller than 2σ are removed, and the remaining histogram is compared against the MCS result in Figure 3.55. Figure 3.56 compares the marginal distribution of phases leading to η(td, xd) greater than 2σ between the MCS (left) and the generated dataset (right) after being filtered. Again, all thirty

Figure 3.51: Compare the marginal distribution of phases between the given dataset (left) D₃ and the generated dataset (right) D₃⁰

Figure 3.52: Compare the distribution of resultant elevations between the given dataset (red) D₁ and the generated dataset (blue) D₁⁰

Figure 3.53: Compare the distribution of resultant elevations between the given dataset (red) D₂ and the generated dataset (blue) D₂⁰

Figure 3.54: Compare the distribution of resultant elevations between the given dataset (red) D₃ and the generated dataset (blue) D₃⁰

Figure 3.55: Compare tail behavior of the distribution between the generated dataset and the MCS

histograms representing the marginal distribution are colored based on their first-order amplitudes evaluated from the discrete spectrum.

One thousand phase sequences are simulated and their elevation time windows at the design location are plotted in Figure 3.57. The realizations achieve large elevation (> 2σ) at the design location t_d= 7.45 s. As the envelope shows, the nonlinear wave has flatter troughs compared to the linear Gaussian wave.

Therefore, the proposed method succeeds in generating threshold exceedance non-linear and non-Gaussian seaways.