Secondly, the wave height of even 2[m] (corresponding to the unity wave amplitude) is essential. The wave height of 4[m] is absolutely 'destructive' for the turning manoeuvre- the surge velocity vx goes below zero (for such a reason, Fig. 9 does not contain the drift angle β in that case). A very interesting issue is that the yaw velocity ωz, is very resistant to the wave second order impact, also in situations of higher wave heights. Notwithstanding some oscillations, the yaw velocity is generally like in calm water - even large alterations in drift angle (contributing also to hull and rudder forces) do not have a higher impact here.
Fig.7 Turning tracks in waves
Fig.8 Surge velocity during turning in waves
Fig.9 Drift angle during turning in waves
Fig.10 Sway velocity during tunring in waves
Fig.11 Yaw velocity during turning in waves
Though the model presented above is not extremely accurate (it reflects in a certain way the status of the ship manoeuvring hydrodynamics science) as relying still upon not fully validated assumptions (hypotheses), it seems to have enough properties to test the wave impact within the stated research objectives.
3. FINAL REMARKS
The regular waves, though justified from a theoretical point of view (being simple to investigate analytically and/or experimentally and enabling gathering input data for irregular wave case) are not however representative for simulating the ship manoeuvring behaviour in waves, at least with regard to the wave second order forces. Two major motives lie under such a conclusion.
The first one is that the ship manoeuvring motion is very sensitive on small wave amplitudes of order even 1[m] if associated with low wave/ship length ratios (λ/L less than e.g. 0.6). Depending of course upon the sea spectrum parameters (significant wave height and modal frequency) and additionally upon the spectrum discretisation while transforming it into harmonic (regular) components, those wave amplitudes and lengths are sometimes hardly to be achieved in, a real-world. A short wave/ship length ratio could be easier reached in a seaway by very long ships. Though the ship analysed in the paper belongs to somewhat shorter ones, and due to her freeboard 〜2[m] (as a rough criterion of manoeuvring motion sensitivity upon the wave amplitude) being subject already to small wave amplitudes, it could be considered a bit not illustrative in real-world conditions. However, a quite similar and true response seems to be experienced if the ship and the wave height are proportionally scaled up to higher dimensions - the case of e.g. a big tanker is obtained.
The second aspect concerns the wave to wind dependence. This relationship (actually there are many formulas more or less widely adopted) exists only in relation to irregular waves- e.g. the wind velocity 15[m/s] produces a fully developed random waves of the significant wave height ca. 5 [m]. Thus to have an adequate simulation under combined wind and wave conditions, the only accepted is the ship manoeuvring in irregular waves like in e.g. . However, the ready-to-use formulations (strictly lookup tables) by  for second order forces are of no advantage as they are computed for one particular sea spectrum. The latter is strongly linked to the wave significant height, which in turn is affected by the wind velocity.
The wave first or second order forces and the frequency dependence of added masses and hull derivatives are not the only aspect of the wave action. Of some importance appears the propeller, wake, and rudder performance in waves (as usual with any interactions)- a great progress has been made here so far. Moreover, a higher interest is returning lately with reference to modelling the wave forces in restricted waterways (including the bank proximity) e.g. .
|AR -rudder area
||m66 -yaw added inertia
|B - ship beam
||Mz -yaw moment
|cB -block coeff.
||n -propeller revs
|cD -rudder drag coeff.
||P/D -pitch ratio
|cL -rudder lift coeff.
||vPS -slipstream velocity
||vx -surge velocity
|cTh -thrust load coeff.
||vy -sway velocity
|D -propeller drameter
||α -rudder incidence angle
|FDR -rudder drag force
||β -ship drift angle
|FnL -Froude number
||βR -rudder local drift angle
|FLR -rudder lift force
||γWV -wave direction
|Fx -surge force
||γWVrel -wave incidence angle
|Fy -sway force
||δ -rudder angle
|g -gravity acceleration
||ζo -wave amplitude
|h -wave height
||λ -wave length
|H -ship height
||λR -rudder aspect ratio
|Jz -inertia moment
||ρ -water density
||Ψ -ship course
||ω -wave frequency
|L -ship length
||ωE -wave encountered freq.
|m -ship mass
||ωz -yaw velocity
|m11 -surge added mass
||Ωm -modified relative yaw velocity
|m22 -sway added mass
|'H'- hull, 'P'-propeller, 'R'-rudder
|'WD'-wind, 'WV'- wave
|'WV1'-(wave) first order excitations
|'WV2'-(wave) second order excitations
|'F-K'-(wave) Froude-Krylov component
|'D'-(wave) diffraction component
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Jaroslaw Artysznk, Ph.D. in sea navigation, Assistant Professor at Szczecin Maritime University (Poland), Faculty of Navigation. Multiyear service at sea on big tankers (chief officer rank). Educational experience: lectures and simulator training on ship manoeuvring and handling. Scientific interest: ship manoeuvring mathematical model identification and ship manoeuvring simulation.