Dynamic Scattering Approach for Solving the Radar Cross-Section of the Warship under Complex Motion Conditions

: To obtain the electromagnetic scattering characteristics of the warship under complex motion conditions, a dynamic scattering approach (DSA) based on physical optics and physical theory of di ﬀ raction is presented. The observation angles, turret rotation, hull attitude changes and sea wave models are carefully studied and discussed. The research results show that the pitching and rolling angles have a large e ﬀ ect on the radar cross-section (RCS) of the warship. Turret movement has a greater impact on its own RCS but less impact on the warship. The RCS of the warship varies greatly at various azimuths and elevations. Di ﬀ erent sea surface models have a greater impact on the lateral RCS of the warship. The DSA is e ﬀ ective and e ﬃ cient to study the dynamic RCS of the warship under complex motion conditions.


Introduction
Since entering the new century, physical optics (PO) and diffraction theory have important and extensive applications in solving the radar cross-section of military targets [1,2]. The design of modern warships tends to be simple in appearance to facilitate stealth, but the electromagnetic scattering characteristics of ships will undergo complex changes if the variety of sport conditions, including the turret movement, attitude changes and the influence of the sea surface, are taken into account [2,3]. Therefore, the radar cross-section of warships under complex motion conditions has gradually gained significant interest among researchers.
For the solution of the target radar cross-section, scholars from various countries have conducted a lot of research work, focusing on the evaluation of the radar stealth characteristics of the target and design the target stealth. Due to the need to carry various weapons and equipment, the shape of the warship will have more narrow areas and sharp corners, which will affect its ability to deflect radar waves [4]. For targets on the water surface, the threat of a radar wave of a ship comes from the distant water surface or the air, and the maximum possible observation elevation angle is limited to a small elevation angle range (up to 1 •~2• ) [5,6]. To evaluate the fringe diffraction field, physical theory of diffraction (PTD) is developed to overcome these residual or fringe current solutions [6,7]. During the buoyantly rising of small-and medium-sized submarines, the roll instability is analyzed by using a computational fluid dynamics method based on Reynoldsaveraged Navier-Stokes equations (RANS) solver and the six degree of freedom solid body motion equations [8,9]. The scattering of water waves induced by tension leg structures over uneven bottoms is determined by using the Eigen function matching method, where the wave amplitude and the surge-motion displacement are assumed to be small [10]. In order to eliminate the strong reflection source of the ship surface,

Dynamic Simulation Method
When the gun turret starts working, it rotates around the zt axis, and its dynamic model can be updated to the following form: where M(mtur) is the grid matrix of the turret model. Transforming this matrix into the Oxyz coordinate system yields the following expression:  (2) where Xtur is distance from zt axis to z axis, noting that the xt axis coincides with the x axis, and the yt axis is parallel to the y axis. Thus the model of this warship can be expressed as: where mwars is the warship model, and mhul is the hull model of this warship after removing the turret. The grid matrix of the warship can be updated to: [ ] When the hull rolls, the model of the entire warship can be determined as:

Dynamic Simulation Method
When the gun turret starts working, it rotates around the z t axis, and its dynamic model can be updated to the following form: · M(m tur (t = 0)) Ox t y t z t (1) where M(m tur ) is the grid matrix of the turret model. Transforming this matrix into the Oxyz coordinate system yields the following expression: M(m tur ) Oxyz = M(x(m tur ) + X tur ) Ox t y t z t (2) where X tur is distance from z t axis to z axis, noting that the x t axis coincides with the x axis, and the y t axis is parallel to the y axis. Thus the model of this warship can be expressed as: where m wars is the warship model, and m hul is the hull model of this warship after removing the turret. The grid matrix of the warship can be updated to: When the entire warship is pitched, its grid matrix can be expressed as follows: When the hull rolls, the model of the entire warship can be determined as: When the warship model sways around the z-axis, its grid matrix can be updated to: The wave model of a sea area can be expressed as a function of space coordinate points and time [37], which has the following form: For a given warship model, the initially dark and illuminated areas are generated when illuminated by radar waves without considering seawater, as shown in Figure 2. Then, the model of the warship can be expressed as: When the warship model sways around the z-axis, its grid matrix can be updated to: wars  s  wars   cos  sin  0 , , , sin cos 0 , , The wave model of a sea area can be expressed as a function of space coordinate points and time [37], which has the following form: For a given warship model, the initially dark and illuminated areas are generated when illuminated by radar waves without considering seawater, as shown in Figure 2. Then, the model of the warship can be expressed as: For more information about sea wave models, please refer to Appendix A.

Electromagnetic Scattering Calculation
For the calculation of the electromagnetic scattering characteristics of the warship, the PO + PTD method is used here to determine its radar cross-section. The total RCS can be determined as: For more information about sea wave models, please refer to Appendix B and [9,15]. For more information about sea wave models, please refer to Appendix A.

Electromagnetic Scattering Calculation
For the calculation of the electromagnetic scattering characteristics of the warship, the PO + PTD method is used here to determine its radar cross-section. The total RCS can be determined as: For more information about sea wave models, please refer to Appendix B and [9,15].

Method Validation
The presented DSA is verified by the PO + MOM/MLFMM (multi-level fast multipole method) in FEldberechnung bei Korpern mit beliebiger Oberflache (FEKO), as shown in the Figure 3, where f R is the radar wave frequency, radar wave uses horizontal polarization, and the rotation rate of the turret is set at 0.785 rad/s, in addition, Figure 3a where t = 2.48 s, the RCS-α curve determined by DSA is generally consistent with the results of FEKO, and there are obvious differences in the ranges of 121.3~159.6 • , 272.2~288 • , and 297~320.3 • where the mean RCS of DSA curve is 1.39 dBm 2 smaller than that of FEKO. This is because the grid data and RCS calculation method affect the electromagnetic scattering characteristics of the complex turret model. For Figure 3b where α = 30 • , quasi-static principle (QSP) combined with the FEKO algorithm requires a large number of discrete states to obtain a more continuous RCS-t curve, while DSA can directly give closer RCS results according to time accuracy. QSP is a simulation method that separates the rotational movement of the turret into a series of discrete states. These results show that DSA is efficient and accurate in handling the dynamic RCS of the turret model.

Method Validation
The presented DSA is verified by the PO + MOM/MLFMM (multi-level fast multipole method) in FEldberechnung bei Korpern mit beliebiger Oberflache (FEKO), as shown in the Figure 3, where fR is the radar wave frequency, radar wave uses horizontal polarization, and the rotation rate of the turret is set at 0.785 rad/s, in addition, β = 0°, At = -90~90°, fR = 7 GHz. For Figure 3a where t = 2.48 s, the RCS-α curve determined by DSA is generally consistent with the results of FEKO, and there are obvious differences in the ranges of 121.3~159.6°, 272.2~288°, and 297~320.3° where the mean RCS of DSA curve is 1.39 dBm 2 smaller than that of FEKO. This is because the grid data and RCS calculation method affect the electromagnetic scattering characteristics of the complex turret model. For Figure  3b where α = 30°, quasi-static principle (QSP) combined with the FEKO algorithm requires a large number of discrete states to obtain a more continuous RCS-t curve, while DSA can directly give closer RCS results according to time accuracy. QSP is a simulation method that separates the rotational movement of the turret into a series of discrete states. These results show that DSA is efficient and accurate in handling the dynamic RCS of the turret model.

Model
Referencing common stealth battleship designs, including the Visby Class stealth frigate and the La Fayette class frigate, a stealth warship model is built, as shown in Figure 4, where Lsh is the length of the ship, Hsh is the height of the ship, Wsh is the width of the ship, and H2 is the height from the deck to the bottom of the ship. The building on the deck is designed with an integrated inclined side wall, which facilitates the deflection of the radar wave, while the barrel and turret are designed with an inclined polyhedron. The main dimensions of this battleship are shown in Table 1 and the ship's model is modeled using a 1:1 scale.

Model
Referencing common stealth battleship designs, including the Visby Class stealth frigate and the La Fayette class frigate, a stealth warship model is built, as shown in Figure 4, where L sh is the length of the ship, H sh is the height of the ship, W sh is the width of the ship, and H 2 is the height from the deck to the bottom of the ship. The building on the deck is designed with an integrated inclined side wall, which facilitates the deflection of the radar wave, while the barrel and turret are designed with an inclined polyhedron. The main dimensions of this battleship are shown in Table 1 and the ship's model is modeled using a 1:1 scale.
The entire surface of the warship model is meshed using high-precision unstructured grid technology as shown in Figure 5, where local mesh encryption is used to process smaller areas or parts, including the gun barrel, turret, deck edges, bow, stern and side ridges. Local mesh encryption is the use of denser grids in these local areas that require more accurate simulation. The grid size of each part of the surface of the warship is shown in Table 2.   The entire surface of the warship model is meshed using high-precision unstructured grid technology as shown in Figure 5, where local mesh encryption is used to process smaller areas or parts, including the gun barrel, turret, deck edges, bow, stern and side ridges. Local mesh encryption is the use of denser grids in these local areas that require more accurate simulation. The grid size of each part of the surface of the warship is shown in Table 2.     The entire surface of the warship model is meshed using high-precision unstructured grid technology as shown in Figure 5, where local mesh encryption is used to process smaller areas or parts, including the gun barrel, turret, deck edges, bow, stern and side ridges. Local mesh encryption is the use of denser grids in these local areas that require more accurate simulation. The grid size of each part of the surface of the warship is shown in Table 2.    completely covered by crimson because the tilt design of the turret at this time does not have a strong ability to deflect radar waves. At the same time, the front area of the superstructure of the warship appeared red, and the round radome even appeared black-red. For Figure 6b where α = 30 • and t = 2.967 s, the dark red on the surface of the turret is reduced because the angle between the gun tube and the radar wave is small at this time, and the inclined design of the turret is beneficial in the reduction of the strong scattering sources on its surface. The front area of the warship superstructure is lighter, but the sparse red on the right is a little darker. These results show that the DSA method used to describe the dynamic electromagnetic scattering of the warship is intuitive and efficient.

Results and Discussion
Hull side ridge 40 Warship body 50 Figure 6 demonstrates that the electromagnetic scattering characteristic on the surface of the turret and warship body changed significantly with the change in time and azimuth, where β = 0°, θ = γ = As = 0°. For Figure 6a where α = 20° and t = 0.967 s, the left surface of the turret is almost completely covered by crimson because the tilt design of the turret at this time does not have a strong ability to deflect radar waves. At the same time, the front area of the superstructure of the warship appeared red, and the round radome even appeared black-red. For Figure 6b where α = 30° and t = 2.967 s, the dark red on the surface of the turret is reduced because the angle between the gun tube and the radar wave is small at this time, and the inclined design of the turret is beneficial in the reduction of the strong scattering sources on its surface. The front area of the warship superstructure is lighter, but the sparse red on the right is a little darker. These results show that the DSA method used to describe the dynamic electromagnetic scattering of the warship is intuitive and efficient.   Figure 7 supports the notion that turret rotation causes a huge change in its radar cross-section under the given azimuths in the rotation angle range, where β = 0°, θ = γ = As = 0°, fR = 7 GHz. The fluctuation range of the RCS-time curve at α = 0°, 10° and 20° is similar, but there are large differences in the peak and curve shapes, where the peak value of the RCS-time curve at α = 0° is 5.92 dBm 2 appearing at t = 0.04 s, that of the RCS curve at α = 10° is 7.769 dBm 2 at t = 0.28 s, and that of the RCS curve at α = 20° is 7.666 dBm 2 at t = 0.5 s. The RCS-time curves at α = 70°, 80° and 90° are also very different, but the RCS variation ranges are all within [-41.31,6.434] dBm 2 . For a given azimuth, the angle between the surface element of the turret and the radar wave is continuously changed during

Effect of Turret Rotation
The fluctuation range of the RCS-time curve at α = 0 • , 10 • and 20 • is similar, but there are large differences in the peak and curve shapes, where the peak value of the RCS-time curve at α = 0 • is 5.92 dBm 2 appearing at t = 0.04 s, that of the RCS curve at α = 10 • is 7.769 dBm 2 at t = 0.28 s, and that of the RCS curve at α = 20 • is 7.666 dBm 2 at t = 0.5 s. The RCS-time curves at α = 70 • , 80 • and 90 • are also very different, but the RCS variation ranges are all within [-41.31,6.434] dBm 2 . For a given azimuth, the angle between the surface element of the turret and the radar wave is continuously changed during the rotation of the turret. As the gun barrel gradually points towards the bow, the front side of the turret is tilted at a large angle to make it easier to deflect the radar waves in the range of the head to a non-threatening direction. These results show that the turret rotation does have a dynamic effect on the electromagnetic scattering characteristics of the turret under different azimuth angles. Figure 8 indicates that turret rotation has a huge impact on its own RCS-azimuth curve where GHz, but its impact on the RCS-azimuth curve of the entire warship is limited, where the main changes are reflected in the minimum value and the small fluctuation in the RCS within the local angle of attack. At α = 169.5 • , the fluctuation in the RCS of the single turret is as high as 37.131 dBm 2 , while the fluctuation brought to the warship is only 8.511 dBm 2 . The reason for this change is that the size of the turret is much smaller than the size of the hull. Although rough stealth designs are used for both the turret and the warship, there are far stronger scattering sources on the hull surface than on the turret. These results suggest that impact of turret rotation on warship RCS cannot be ignored, while more important influencing factors on the dynamic electromagnetic scattering of warships need to be discovered and discussed. the rotation of the turret. As the gun barrel gradually points towards the bow, the front side of the turret is tilted at a large angle to make it easier to deflect the radar waves in the range of the head to a non-threatening direction. These results show that the turret rotation does have a dynamic effect on the electromagnetic scattering characteristics of the turret under different azimuth angles.   Figure 8 indicates that turret rotation has a huge impact on its own RCS-azimuth curve where β = 0°, θ = γ = As = 0°, fR = 7 GHz, but its impact on the RCS-azimuth curve of the entire warship is limited, where the main changes are reflected in the minimum value and the small fluctuation in the RCS within the local angle of attack. At α = 169.5°, the fluctuation in the RCS of the single turret is as high as 37.131 dBm 2 , while the fluctuation brought to the warship is only 8.511 dBm 2 . The reason for this change is that the size of the turret is much smaller than the size of the hull. Although rough stealth designs are used for both the turret and the warship, there are far stronger scattering sources on the hull surface than on the turret. These results suggest that impact of turret rotation on warship RCS cannot be ignored, while more important influencing factors on the dynamic electromagnetic scattering of warships need to be discovered and discussed.     Figure 8 indicates that turret rotation has a huge impact on its own RCS-azimuth curve where β = 0°, θ = γ = As = 0°, fR = 7 GHz, but its impact on the RCS-azimuth curve of the entire warship is limited, where the main changes are reflected in the minimum value and the small fluctuation in the RCS within the local angle of attack. At α = 169.5°, the fluctuation in the RCS of the single turret is as high as 37.131 dBm 2 , while the fluctuation brought to the warship is only 8.511 dBm 2 . The reason for this change is that the size of the turret is much smaller than the size of the hull. Although rough stealth designs are used for both the turret and the warship, there are far stronger scattering sources on the hull surface than on the turret. These results suggest that impact of turret rotation on warship RCS cannot be ignored, while more important influencing factors on the dynamic electromagnetic scattering of warships need to be discovered and discussed.  Generally speaking, the rotation of the turret makes its own RCS have dynamic characteristics, but the impact on the entire ship's RCS should be further discussed. This also requires detailed and practical research based on the size comparison and appearance design of the warship and turret.

Effect of Observation Angles
GHz. It can be found that the RCS value of the first half of the RCS-time curve at α = 0 • , 10 • , and 20 • is significantly higher than that of the second half; this is because the warship is restrained in the lower half in the first half, which leads to the angle between the originally inclined surface of the front superstructure and the radar wave being too large, which reduces the radar stealth characteristics of these curved surfaces. The maximum value of the three RCS curves exceeds 24.19 dBm 2 , and the minimum value is also lower than −6.097 dBm 2 . For the RCS-azimuth curve at different times, the RCS of the warship also changes greatly with the azimuth, where the maximum value of the RCS curve at t = 2.25 s is 36.71 dBm 2 , and the minimum value is lower than −25 dBm 2 . These results show that the DSA method can well explain the effect of azimuth on the dynamic RCS of the warship.
value of the first half of the RCS-time curve at α = 0°, 10°, and 20° is significantly higher than that of the second half; this is because the warship is restrained in the lower half in the first half, which leads to the angle between the originally inclined surface of the front superstructure and the radar wave being too large, which reduces the radar stealth characteristics of these curved surfaces. The maximum value of the three RCS curves exceeds 24.19 dBm 2 , and the minimum value is also lower than -6.097 dBm 2 . For the RCS-azimuth curve at different times, the RCS of the warship also changes greatly with the azimuth, where the maximum value of the RCS curve at t = 2.25 s is 36.71 dBm 2 , and the minimum value is lower than -25 dBm 2 . These results show that the DSA method can well explain the effect of azimuth on the dynamic RCS of the warship.    Figure 10b where t = 2.7 s, the increase in the elevation angle clearly improves the lateral RCS characteristics of the warship, but it reduces the stealth characteristics of the radar in the tail direction. The increase in the elevation angle corresponds to the different radar station locations, including radar observation stations of other warships or mountains, while the increase in this angle changes the angle between the radar wave and the surface area of the warship surface lighting area, subsequently affecting the ability of these facet elements to deflect radar waves. These results show that the elevation angle is a very practical factor for the electromagnetic scattering characteristics of warships and that it cannot be ignored.   Figure 10b where t = 2.7 s, the increase in the elevation angle clearly improves the lateral RCS characteristics of the warship, but it reduces the stealth characteristics of the radar in the tail direction. The increase in the elevation angle corresponds to the different radar station locations, including radar observation stations of other warships or mountains, while the increase in this angle changes the angle between the radar wave and the surface area of the warship surface lighting area, subsequently affecting the ability of these facet elements to deflect radar waves. These results show that the elevation angle is a very practical factor for the electromagnetic scattering characteristics of warships and that it cannot be ignored. In general, studying the effects of observation angles is helpful in determining the right combination of azimuth and elevation to achieve a lower warship RCS, which has practical significance and application value for the effects of warship static and the dynamic RCS. In general, studying the effects of observation angles is helpful in determining the right combination of azimuth and elevation to achieve a lower warship RCS, which has practical significance and application value for the effects of warship static and the dynamic RCS.  Figure 12 shows that the sway angle of the warship's heading has a significant dynamic effect on its RCS under the given key azimuths and omnidirectional angles, where the setting is As(°) = -8sin t, β = 2°, θ = γ = At = 0°, fR = 7 GHz. For Figure 12a, the RCS curve at α = 0° fluctuates greatly as a whole, but there are few small local fluctuations when compared with the other two RCS curves. The peak value of the RCS curve at α = 0° is 18.88 dBm 2 at t = 6.417 s, and the minimum value is only -6.664 dBm 2 at t = 1.225 s. With the increase in the azimuth, the fluctuation in the RCS-time curve gradually increases. This is mainly because the large-scale side buildings of the warship are added to the lighting area, making the ability to deflect radar waves more variable. For Figure 12b, the change in the RCS-azimuth curve of the warship over time is also very clear, with large differences occurring in the azimuth range of 26.5°~52.25°, 121.3°~152.8°, 181°~217.3° and 319°~355.8°. The peak of the RCSazimuth curve at t = 1.517 s is 29.72 dBm 2 , appearing at α = 82°, and that of the RCS curve at t = 5.6 s is 24.86 dBm 2 at α = 185°. The heading sway of the bow is similar to the rotation of the turret, but the angle of the former changes within a small range and can last a long time, which has a non-negligible impact on radar stealth under the warship's key azimuths. These results show that the DSA method can well capture the dynamic impact of bow sway on warship RCS, which is obviously of practical  Figure 12 shows that the sway angle of the warship's heading has a significant dynamic effect on its RCS under the given key azimuths and omnidirectional angles, where the setting is A s ( Figure 12a, the RCS curve at α = 0 • fluctuates greatly as a whole, but there are few small local fluctuations when compared with the other two RCS curves. The peak value of the RCS curve at α = 0 • is 18.88 dBm 2 at t = 6.417 s, and the minimum value is only −6.664 dBm 2 at t = 1.225 s. With the increase in the azimuth, the fluctuation in the RCS-time curve gradually increases. This is mainly because the large-scale side buildings of the warship are added to the lighting area, making the ability to deflect radar waves more variable. For Figure 12b of the RCS-azimuth curve at t = 1.517 s is 29.72 dBm 2 , appearing at α = 82 • , and that of the RCS curve at t = 5.6 s is 24.86 dBm 2 at α = 185 • . The heading sway of the bow is similar to the rotation of the turret, but the angle of the former changes within a small range and can last a long time, which has a non-negligible impact on radar stealth under the warship's key azimuths. These results show that the DSA method can well capture the dynamic impact of bow sway on warship RCS, which is obviously of practical significance.

Effect of Attitude Changes
whole, but there are few small local fluctuations when compared with the other two RCS curves. The peak value of the RCS curve at α = 0° is 18.88 dBm 2 at t = 6.417 s, and the minimum value is only -6.664 dBm 2 at t = 1.225 s. With the increase in the azimuth, the fluctuation in the RCS-time curve gradually increases. This is mainly because the large-scale side buildings of the warship are added to the lighting area, making the ability to deflect radar waves more variable. For Figure 12b, the change in the RCS-azimuth curve of the warship over time is also very clear, with large differences occurring in the azimuth range of 26.5°~52.25°, 121.3°~152.8°, 181°~217.3° and 319°~355.8°. The peak of the RCSazimuth curve at t = 1.517 s is 29.72 dBm 2 , appearing at α = 82°, and that of the RCS curve at t = 5.6 s is 24.86 dBm 2 at α = 185°. The heading sway of the bow is similar to the rotation of the turret, but the angle of the former changes within a small range and can last a long time, which has a non-negligible impact on radar stealth under the warship's key azimuths. These results show that the DSA method can well capture the dynamic impact of bow sway on warship RCS, which is obviously of practical significance.   Figure 13 investigates the notion that the existence of ocean waves indeed affects the electromagnetic scattering characteristics of warship surfaces under the given observation conditions, where the following settings are made: θ( • ) = −2sin t, γ( • ) = 3sin t, A s ( • ) = −1.5sin t, A t ( • ) = 90sin t, α = 330 • , β = 2 • , t = 0.642 s, f R = 7 GHz, RCS unit: dBm 2 . When the sea wave model is not considered, all surface elements in the Z I1 area of the warship surface participate in the RCS contribution, where the azimuth is equal to 330 • and the elevation angle is equal to 2 • , causing most of the warship's right and front areas to be illuminated by radar waves. Following this, a large area of deep red and red appeared on the right side of the turret, the front of the superstructure, the right front of the spherical radome and the front of the chimney. When the ideal stationary sea (ISS, as shown in Appendix A) wave model was added, where Z m = −2.1 m, x n = y n = −90 m, x m = y m = 90 m, the original Z I1 area was clearly cut out. Although there was no effect on the strong scattering area, the lighting zone changed significantly. Since there is no fluctuation in the wave model at this time, the cut in the surface electromagnetic scattering model of the warship is flat. When the trigonometric function sea (TFS, as shown in Appendix A) model is added, where A x = 0.5, A y = 0.2, ω x = ω y = 10, the notch of the warship electromagnetic scattering model is not flat, but changes with the waves. Since the TFS model is based on the trigonometric functions, taking into account linear variables, tiny random phases, and tiny random wave fluctuations, the edges of the cutout mainly show the undulating shape of the trigonometric function but with small irregular jitters. For the simplified regular sea (SRS, as shown in Appendix A) model, where k 1 = 4, χ 1 = π/3, χ 0 = π/10, ω 1 = C z = 1, U = 10 m/s, it is also based on a trigonometric function, taking into account the small random wave increments, which makes the cutout very smooth overall. Figure 14 demonstrates that the presence of sea waves has a non-negligible impact on the instant RCS under the omnidirectional angle of the warship, where β = 0 • , t = 1.692 s, f R = 7 GHz. For Figure 14a, the addition of ISS waves caused some large changes in the original RCS-azimuth curve within the azimuth ranges of 82.5 •~8 7.75 • and 165.5 •~1 78.5 • . Although the average values of the two RCS curves are similar, 16.411 dBm 2 (no sea wave) and 16.525 dBm 2 (ISS), the RCS difference can exceed which results in similar RCS averages at all angles. The front shape of the warship uses an excellent stealth design, which greatly weakens the impact of the waves. The rear design of the warship is rough, and the portion submerged by the sea waves accounts for a large proportion of the entire projection area, which causes the waves in this direction to have a great impact. For Figure 14b here, the two RCS curves are almost coincident, except for some minimum values which include the RCS at α = 161.8 • , 176.5 • , and 263 • . Both the TFS and SRS sea wave models dynamically change the submerged part of the warship, and can set sea waves with different amplitudes and phase changes.

Effect of Sea Waves
of the warship's right and front areas to be illuminated by radar waves. Following this, a large area of deep red and red appeared on the right side of the turret, the front of the superstructure, the right front of the spherical radome and the front of the chimney. When the ideal stationary sea (ISS, as shown in Appendix A) wave model was added, where Zm = -2.1 m, xn = yn = -90 m, xm = ym = 90 m, the original ZI1 area was clearly cut out. Although there was no effect on the strong scattering area, the lighting zone changed significantly. Since there is no fluctuation in the wave model at this time, the cut in the surface electromagnetic scattering model of the warship is flat. When the trigonometric function sea (TFS, as shown in Appendix A) model is added, where Ax = 0.5, Ay = 0.2, ωx = ωy = 10, the notch of the warship electromagnetic scattering model is not flat, but changes with the waves. Since the TFS model is based on the trigonometric functions, taking into account linear variables, tiny random phases, and tiny random wave fluctuations, the edges of the cutout mainly show the undulating shape of the trigonometric function but with small irregular jitters. For the simplified regular sea (SRS, as shown in Appendix A) model, where k1 = 4, χ1 = π/3, χ0 = π/10, ω1 = Cz = 1,U = 10 m/s, it is also based on a trigonometric function, taking into account the small random wave increments, which makes the cutout very smooth overall.   16.525 dBm 2 (ISS), the RCS difference can exceed 8 dBm 2 (α = 86.75°) or even 16 dBm 2 (α = 176.5°), because the warships here have a stealth design and shallow draft, while the hull below the deck can reflect some of the radar waves well into the water, which results in similar RCS averages at all angles. The front shape of the warship uses an excellent stealth design, which greatly weakens the impact of the waves. The rear design of the warship is rough, and the portion submerged by the sea waves accounts for a large proportion of the entire projection area, which causes the waves in this direction to have a great impact. For Figure 14b here, the two RCS curves are almost coincident, except for some minimum values which include the RCS at α = 161.8°, 176.5°, and 263°. Both the TFS and SRS sea wave models dynamically change the submerged part of the warship, and can set sea waves with different amplitudes and phase changes.   Figure 15a here, the two RCS curves are generally similar and have large fluctuations. The maximum value of the RCS curve without the sea wave model exceeds 28 dBm 2 at t = 3.325 s, and the minimum value is −16.34 dBm 2 at t = 1.808 s, while the change in the warship RCS mainly comes from the real-time change in its attitude, although there are no sea waves. When the ISS wave model is added, the most obvious difference in RCS is reflected in the time range of [0.7583, 2.392] s when compared with the case without waves. At this time, the peak value of the RCS curve is 28.24 dBm 2 appearing at t = 3.325 s, and the minimum value is −15.17 dBm 2 at t = 1.05 s. The addition of the ISS wave model covers the lower part of the warship absolutely and uniformly in real time, which further affects the dynamic RCS of the warship when compared with the absence of waves. For Figure 15b here, the two RCS-time curves are also similar, except for some local fluctuations which are mainly reflected in the interval [0.233, 2.917] s. The minimum value of the RCS curve with the TFS wave model is −12 dBm 2 at t = 1.925 s and the maximum value is 28.27 dBm 2 at t = 3.325 s, while the other RCS curve has a peak value of 28.32 dBm 2 and a minimum value below −18 dBm 2 . Both the TFS and SRS wave models are dynamic models that change with time and space, which causes the lighting area of warships to constantly change, while the trigonometric function structure, phase, amplitude, and random variables that make up the TFS and SRS models are different, which causes differences in their RCS-time curves. These results show that the DSA method can well capture the impact of wave models on warship RCS, including both the static and dynamic aspects.  In general, the sea wave model is a realistic factor that cannot be ignored when studying the electromagnetic scattering characteristics of warships. The presence of sea waves can change the attitude of warships, can affect the electromagnetic scattering area of warships, and, subsequently, can affect the radar cross-section of warships.

Conclusions
In this paper, we presents a dynamic scattering approach to solve the radar cross-section of the warship under complex motion conditions. Observation angle, turret rotation, attitude angle and wave model are considered and analyzed. Turret rotation brings significant changes to its average RCS and the RCS in all azimuths, but it has little effect on the average RCS of warships. The RCS-t characteristics of warships at different azimuth and elevation angles are very different. A slight increase in the elevation angle in the head-to-head direction can yield changes in the RCS average and peak position. The roll angle and pitch angle changes greatly affect the RCS-t curve of the warship at each azimuth angle. The sway angle has a greater impact on RCS-t than on RCS-α. The sea wave model has a significant impact on the RCS of warships in the lateral and tail directions. The stealth design of warship superstructure based on the dynamic RCS under important azimuth angles In general, the sea wave model is a realistic factor that cannot be ignored when studying the electromagnetic scattering characteristics of warships. The presence of sea waves can change the attitude of warships, can affect the electromagnetic scattering area of warships, and, subsequently, can affect the radar cross-section of warships.

Conclusions
In this paper, we presents a dynamic scattering approach to solve the radar cross-section of the warship under complex motion conditions. Observation angle, turret rotation, attitude angle and wave model are considered and analyzed. Turret rotation brings significant changes to its average RCS and the RCS in all azimuths, but it has little effect on the average RCS of warships. The RCS-t characteristics of warships at different azimuth and elevation angles are very different. A slight increase in the elevation angle in the head-to-head direction can yield changes in the RCS average and peak position. The roll angle and pitch angle changes greatly affect the RCS-t curve of the warship at each azimuth angle. The sway angle has a greater impact on RCS-t than on RCS-α. The sea wave model has a significant impact on the RCS of warships in the lateral and tail directions. The stealth design of warship superstructure based on the dynamic RCS under important azimuth angles is one of the key development directions of future research.

Conflicts of Interest:
The authors declare that there is no conflict of interest. When considering dynamic sea waves, the bins in Z I1 area need to be re-differentiated:

Nomenclature
F u ∈ Z I ∀F ∈ Z I1 , min(z(P 1 ), z(P 2 ), z(P 3 )) > max(z wave (t) ) (A1) where F is the facet of the warship model in Z I1 area, F u is the facet above the sea wave surface, and Z I is the illuminated area. Then, the facets below the waves can be defined as: F b ∈ Z D ∀F ∈ Z I1 , max(z(P 1 ), z(P 2 ), z(P 3 )) < min(z wave (t) ) where n is the unit normal vector of the outer normal direction of r' at the target surface. Based on the mirror principle: For the case where the incident wave is a plane wave: Then, there are the following expressions: E s (r) = j λr |E 0 |e −jk·r S r × r × (n(r ) · E 0 )k − n(r ) ·k E 0 e − jk(−r+k)·r dS (A18) Considering the characteristics of plane waves, an integral term can be written as: Then, the RCS calculation formula can be sorted into the following form: PTD is used to solve the diffraction coefficient of the edge; then, the actual scattering field is the sum result of PO and PTD: The diffraction coefficient of PTD can be expressed as: where f 1 is the diffraction coefficients of transverse magnetic waves of PTD, g 1 is the diffraction coefficients of transverse electric waves of PTD, f is the diffraction coefficients of transverse magnetic waves of GTD, and g is the diffraction coefficients of transverse electric waves of GTD.