# Numerical Modeling of Wave Disturbances in the Process of Ship Movement Control
## 2 Wave Disturbances Generated by Wind  
Surface layers of water in the sea and the air above are rarely calm. Statistical data on weather conditions prevailing in the North Atlantic show that, on average, the weather is windless only for about seven days a year (H  
In the automatic course control system, the value of deviation from a preset course depends on wave energy, as well as a ship's speed and wave angle (Figure 2). In this approach, we can consider the following situations:  
* Waves coming at a specific angle from the direction opposite to ship movement (diagonal waves);
* waves coming at an angle, but in the direction of ship movement (quartering waves);
* waves coming from the bow (head waves);
* waves perpendicular to the ship side (beam waves);
* waves striking the stern (following waves). The greatest changes in the heading angle of the ship at low speeds occur in case of transverse waves, whereas the smallest impact comes from diagonal and quartering waves; in case of diagonal and head waves, heading deviations (yawing) get larger as the ship speed increases. Taking into  
Figure 1: Boundary relation between the parameter beta and wave height. Figure 2: Definition of the wave angle. account the ship speed and wave angle, we can transform spectrum (1) to this form (determined by research based on real measurements) [14]:  
\[S(\omega)=\frac{2D\alpha_{k}\omega^{2}}{\omega^{4}+2({\alpha_{k}}^{2}-{\beta_{k}} ^{2})\omega^{2}+\left({\alpha_{k}}^{2}+{\beta_{k}}^{2}\right)^{2}} \tag{2}\]  
where  
\(\alpha_{k}\)--parameter dependent on parameter \(\beta_{k}(\alpha_{k}=0.21\beta_{k})\);  
\(\beta_{k}\)--parameter calculated from the formula \(|\beta+Vg^{-1}\cos(\xi)\beta^{2}|\);  
\(V\)--ship's speed;  
G--gravitational acceleration;  
\(\xi\)--wave angle (Figure 2). The effect of wave action on the process of ship movement control (in various control problems) can easily be determined by measuring a ship's angular velocity. The signal of angular speed is a sum of two components: One formed due to rudder deflection, and the other due to waves. The spectrum of angular velocity is created by transforming the spectrum of wave ordinates (2) and takes this form (appointed by research based on real measurements) [14]:  
\[S_{r}(\omega)=\frac{2D\alpha_{k}\alpha_{r}{{}^{2}}x_{T}{{}^{2}}cos^{2}(\xi)g ^{-2}\omega^{4}}{\omega^{4}+2({\alpha_{k}}^{2}-{\beta_{k}}^{2})\omega^{2}+ \left({\alpha_{k}}^{2}+{\beta_{k}}^{2}\right)^{2}} \tag{3}\]  
where  
\(S_{r}(\omega)\)--spectral density of wave angular velocity;  
\(x_{r},x_{T}\)--dimensionless reducing parameters (Figure 3) dependent on the length of the ship (\(L\)), wavelength (\(\lambda\)), and the ship's maximum draught marked by the waterplane (\(T\)). It should be noted that spectral density (3), due to the expression in the numerator \(\cos(\xi)\), faithfully represents the power of the wave signal for angles differing from \(90^{\circ}\) by at least \(\pm 15^{\circ}\) (for diagonal, quartering, head, and following waves). In order to use the spectrum of wave angular velocity (3) acting as disturbances affecting the ship hull due to waves, the forming filter:  
\[H_{r}(s)=\frac{\sqrt{2D\alpha_{k}}x_{r}x_{T}|cos(\xi)|g^{-1}s^{2}}{s^{2}+2 \alpha_{k}s+{\alpha_{k}}^{2}+{\beta_{k}}^{2}} \tag{4}\]  
Figure 3: Boundary dependences of reducing parameters. should be used to pass white noise through it (stochastic process with zero expected value and steady spectral density); then, the formed signal \(r_{w}\) should be added to--or, for positive wave angles, subtracted from--the ship's angular speed caused by rudder deflection. Filter (4) is obtained from this relation:  
\[S_{r}(\omega)=(|H_{r}(s)|_{s=\omega i})^{2}. \tag{5}\]  
Using Lagrange interpolation (Algorithm 1), where average values of parameters are adopted in the nodes \(\beta,x_{r},x_{T}\), we get:  
\[