Control of Floating Body Waves Due to an Airplane Takeoff from a Very Large Floating Airport

: Numerical simulations were generated to investigate the response of a very large floating airport to an airplane takeoff, using the set of nonlinear shallow water equations of velocity potential for water waves interacting with a floating thin plate. We have proposed two methods to reduce persistent airport vibration: reflectance reduction by decreasing the flexural rigidity in airport edge parts and amplification reduction by decreasing the still water depth partially under airport runways. First, when the flexural rigidity is uniformly decreased in an airport edge part, the reflectance of the floating body waves due to a B737 was reduced because of the multiple reflections. However, the wave reflectance for a B747 increased, depending on the conditions. A too-long edge part was not effective in reducing the wave reflectance. Conversely, when the flexural rigidity is linearly decreased in an airport edge part, the wave reflectance was reduced for both airplanes. Second, when the still water depth under an airport runway is partially reduced at the location where floating body waves are amplified, the wave heights of floating body waves tended to decrease as the still water depth in the shallower area decreased.


Introduction
A very large floating structure, namely VLFS, can be used for various purposes-an offshore airport, power plant, storage facility, evacuation area, etc.When designing a VLFS with hydroelasticity, the interaction between structure oscillation and fluid motion should be considered.Hydroelastic platforms can also be seen in nature, such as wide ice plates floating at the sea surface [1].The response of an ice plate to a moving load on it has been investigated [2-4], and these results are useful in VLFS design.In addition, the hydroelasticity of a floating body can be used to obtain sustainable energy by converting water wave energy (e.g., [5,6]).
Regarding the response of a VLFS to long waves including tsunamis, the Boussinesqtype equations for surface waves were solved numerically using a finite difference method (FDM), to examine the relationship between the bending moment and flexural rigidity of a thin plate floating on a progressing solitary wave [7].The interaction of a thin plate with an incident solitary wave was also investigated by coupling a finite element method (FEM) and boundary element method (BEM) in the vertical two dimensions [8].This interaction was reproduced in the hydraulic experiments, in which the solitary waves were disintegrated by the floating thin plate, as their nonlinearity was strong [8].The results-that the wave heights of the incident waves decreased because of the generation of floating body waves-suggest that the wave height of a huge tsunami decreases after propagating through an offshore VLFS.Such tsunami height reduction using a VLFS was discussed based on the numerical simulation of water waves interacting with a floating thin plate using an FDM [9].In addition, when density stratification is developed under a hydroelastic structure, a two-dimensional problem in the vertical section was formulated with the framework of a linear potential theory [10], and the surface and internal waves due to a moving load on a VLFS in the vertical two-dimensions were examined with an FDM, considering both the nonlinearity and dispersion of water waves [11].If a load on a VLFS moves at a speed close to the internal-mode speed, internal waves can be generated in a pycnocline, etc., which will particularly affect nearshore environments.
One of the artificial loads on a VLFS is an airplane moving on a very large floating airport.For example, the drag against an airplane taking off from a floating airport of infinite length was evaluated numerically, using the Fourier transform theory for different flexural rigidities of the airport [12].Conversely, the transient response of a floating airport subjected to a landing airplane load was studied using an FEM [13].A BEM was also applied to simulate the response of a hydroelastic plate to a moving weight in the coexistence field of linear waves and a current [14].Under the combined loads of water waves and an airplane landing or taking off, the drag induced on the airplane by the deformed runway, as well as the time variation of the airport profile, was obtained using both an FEM-scheme-based method and Wilson's θ method [15].To study the hydroelastic response of floating composite plates subjected to moving loads, a combination of a BEM and moving element method (MEM) was utilized [16].Moreover, the horizontally twodimensional responses of a floating airport to the landing and takeoff of an airplane were investigated with the time-domain mode-expansion method [17], and also simulated using an FDM [18].
Thus, an airplane running on a very large floating airport generates floating body waves, that is, floating airport vibration.If airplanes land or take off while floating airport vibration remains, unexpected large floating body waves may occur because of wave superposition.Furthermore, if large floating body vibration remains for a long time because of the wave reflections at the ends of the floating airport, airplane operations will be hindered, accelerating the airport deterioration.To mitigate the hydroelastic vibration of a VLFS under sea wave action, several methods have been proposed: for example, the introduction of floating breakwaters [19], aircushions [20], and member connectors [21].The reduction of the resonance phenomena due to the presence of a breakwater near a VLFS was also investigated [22].
In the present study, to reduce the persistence of large floating body waves due to an airplane taking off from a very large floating airport, we propose two methods as follows: 1.
Reducing the flexural rigidity in a floating airport edge; 2.
Reducing the still water depth under a floating airport runway.
We have examined the effectiveness of these methods based on one-dimensional numerical calculations with an FDM.In the computation, an offshore airport was assumed to be a floating thin plate without viscous or structural damping, as well as fluid damping, and was installed in a calm water without wind waves, for simplicity.The governing equations were the nonlinear shallow water equations on velocity potential, which were obtained by reducing the equations derived based on a variational principle for water waves, interacting with horizontal flexural thin plates [23].In the numerical calculations, the flexural rigidity was given at the location of a thin plate, to express the thin plate covering part of the water area.With this method, it is possible to consider both the reflection and transmission of floating body waves at the ends of the thin plate, which were not discussed for infinitely long and wide airports.The oscillations of floating airports due to a takeoff of two types of jetliners are discussed.

Governing Equations
The illustration in Figure 1 is our schematic for a system consisting of a fluid and thin plate.The x-axis is the horizontal axis, in the direction of which surface waves propagate.In the present study, an airplane moves in the positive direction of the x-axis, so the airport end at the larger x is called the front end, and the other the rear end, to distinguish between them.Conversely, the z-axis is the vertical axis, the origin of which is located at the surface in the stationary state, and the positive direction of z is vertically upward.The floating airport is assumed to be a thin plate, where the horizontal length scale is assumed to be much larger than the thickness, so that the differences of curvature between the upper surface, neutral plane, and lower surface of the thin plate are ignored.The energy attenuation inside the airport is not considered, as described above.Therefore, the governing equation of motion for the floating airport is the following classical equation to describe the oscillation of an elastic thin plate as where η is the surface displacement, B is the flexural rigidity of the thin plate, and ∇ is the horizontal partial differential operator when also considering the y-axis, perpendicular to the x-axis in the horizontal plane.Although both the plate density m and vertical width δ are assumed to be constant throughout the thin plate, the flexural rigidity B can be distributed along the thin plate.The pressures at the upper and lower faces of the thin plate are p 0 and p 1 , respectively.
When the representative values of wave height, wavelength, fluid depth, and density are H, l, d, and ρ, respectively, the dimensionless quantities are where e = 0 and 1.We substitute Equation (2) into Equation (1), and obtain where ε = H/d and σ = d/l are the representative ratio of wave height to water depth, and that of water depth to wavelength, respectively.In the manner similar to that of [7], the water area is assumed to be relatively shallow, so the orders of the parameters are O(ε) = O(σ 2 ) ≪ 1.Thus, the first term on the left-hand side of Equation (3) can be ignored.Without this term, we obtain the plate equation for the dimensional quantities as Conversely, a set of water wave equations were derived for multi-layer fluids [24], by applying a variational principle that referenced the functional for water surface waves [25] based on the functional with rotation [26].In the derivation, the fluids were inviscid and incompressible, and the fluid motion was irrotational, resulting in the existence of velocity potentials.In the present study, we adopt just the first term of the expanded velocity potential for one-layer cases with one-dimensional wave propagation, i.e., N = 1 and i = 1 in [23].Therefore, using Equation (4), the governing equations are reduced to where b and φ are the seabed position and velocity potential, respectively, and the sea water density ρ is 1030 kg/m 3 .The gravitational acceleration g is 9.8 m/s 2 , and the load of an airplane per unit length is given as p 0 .Equation ( 5) is the equation of continuity, whereas Equation ( 6) is the nonlinear shallow water equation on velocity potential, that is, the Bernoulli equation at the surface, considering the flexural rigidity of a floating thin plate.Both surface tension and capillary action were ignored, and friction was also ignored for simplicity.

Numerical Method
The governing equations-Equations ( 5) and ( 6)-were solved numerically using a finite difference method with central and forward difference schemes for space and time, respectively.The initial value of velocity potential, ϕ(x, 0 s), was 0.0 m 2 /s at any location.In this paper, the values are written without considering significant digits, although the calculations were conducted using 64-bit floating point numbers.
To verify the accuracy of the governing equations, the reproducibility of the response of a floating thin plate was examined by comparing the results of the existing hydraulic experiments [8] and those of numerical computation [18].In the present study, the grid size ∆x was 20 m and the time interval ∆t was 0.01 s, after verification.
We performed numerical calculations for the one-dimensional propagations of surface waves generated by two sizes of airplanes with different weights.When the hydroelastic runway is not so wide compared to the spacing of the airplane's left and right landing gears, 1D wave propagation would be dominant.

Calculation Conditions
An airport with length L of 5 km or 15 km was floating at the sea surface in 0.5 km ≤ x ≤ 5.5 km or 0.5 km ≤ x ≤ 15.5 km, respectively, and an airplane ran on the floating airport in the positive direction of the x-axis from x = 1 km, whereafter it took off.The airport length of 15 km was an unrealistic value which was set to ignore wave reflections at the front end of the airport.
We considered two sizes of airplanes, i.e., B747-400 and B737-800, manufactured by the Boeing Company, which we call B747 and B737, respectively, in this paper.The former is a large airplane, known as "jumbo jet", which is no longer operated for passenger transport because it is not in keeping with the current economic situation, but is used to transport an airplane after renovation.Conversely, the latter is operated on many routes.The weights of a B747 and B737 were set at 397,000 kgs and 79,000 kgs, respectively, referring to the maximum takeoff weights [27].The unit "kgs" is often used for the mass of airplanes in aviation industry and is the same as "kg" in physics.In the present calculations, the total tire contact distance of both the B747 and B737 was assumed to be 9.8 m, considering the unit width of 1 m.
The running distances of the B747 and B737 were 3 km and 2 km, respectively, on the floating airport.We assumed that the airplane runs at a constant running acceleration on the airport when taking off.We considered the cases in which an airplane plays a rolling start, i.e., the airplane starts running to take off immediately after arriving at the starting point.Therefore, at the starting time, both the running speed and load of the airplane were assumed to be zero.In addition, we assumed that the point load due to the airplanes is constant while running on the airport.The calculation conditions of the airplanes are listed in Table 1.Regarding the airports, the values of flexural rigidity B were determined with reference to those obtained during the prototype test using the Mega-Float airport [28].The flexural rigidity B was given in the area covered by the airport, which made it possible to consider both the reflection and transmission of waves at the ends of the floating airports, as described above.
When the flexural rigidity is uniformly reduced in an edge part of an airport, i.e., in 5.5 km − λ ≤ x ≤ 5.5 km, where λ is the length of the edge part, the calculation conditions of the flexural rigidity are described in Table 2.The flexural rigidities of the main and edge parts of the airport are B main and B edge , respectively.When B edge = B main = 1.0 × 10 11 Nm, the flexural rigidity is uniform throughout the airport.In the case names, "E" stands for edge related, and "S" and "L" indicate that the edge part is short and long, respectively.The still water depth h is uniformly 50 m.Conversely, when the flexural rigidity is linearly reduced in an edge part of an airport, i.e., in 5.5 km − λ ≤ x ≤ 5.5 km, the calculation conditions of the flexural rigidity are described in Table 3, in which the flexural rigidities of the main part and front end of the airports are B main and B end , respectively.The "C" of the case names represents a continuous reduction of flexural rigidity in an edge part of the airport.
Table 3. Calculation conditions when the flexural rigidity is linearly reduced from B main to B end in an edge part of the airports, i.e., in 5.5 km − λ ≤ x ≤ 5.5 km, where the flexural rigidities of the main part and front end of the airports are B main and B end , respectively.Moreover, when the still water depth is uniformly reduced at 1.5 km ≤ x ≤ 1.5 km + D, where D is the length of the shallower area, the calculation conditions of the still water depth are described in Table 4.The length L and flexural rigidity B of the airport are 15 km and 1.0 × 10 10 Nm, respectively.In the case names, "D" stands for water depth related, and "S" and "L" indicate that the shallower area is short and long, respectively.The case names are expressed by combining two case names described above, i.e., "A" or "B" in Table 1 and one of the names in Tables 2-4.For example, in Case ADS, a B747 takes off from the floating airport with a flexural rigidity of 1.0 × 10 10 Nm and a length of 15 km.

Floating Body Waves Due to an Airplane Takeoff without Reflection Control
First, we numerically simulated the motion of a finite-length floating airport without any reflection control over a flat seabed.Figure 2 depicts the time variation of the floating airport and water surface profiles when a B747 takes off from the floating airport with a constant flexural rigidity of 1.0 × 10 11 Nm in Case AES.In the figure, the black dotted line indicates the location of the airplane running on the floating airport.The calculation conditions in this case are listed in Tables 1 and 2.
Eng 2024, 5, FOR PEER REVIEW 6 Table 4. Calculation conditions when the still water depth is uniformly reduced at 1.5 km ≤ x < 1.5 km + D, in which the still water depth is hs.The case names are expressed by combining two case names described above, i.e., "A" or "B" in Table 1 and the names in Tables 2-4.For example, in Case ADS, B747 takes off from the floating airport with a flexural rigidity of 1 × 10 10 Nm and a length of 15 km.

Floating Body Waves Due to an Airplane Takeoff without Reflection Control
First, we numerically simulated the motion of a finite-length floating airport without any reflection control on a flat seabed.Figure 2 depicts the time variation of the floating airport and water surface profiles when a B747 takes off from the floating airport with a constant flexural rigidity of 1.0 × 10 11 Nm, Case AES, in which the black dotted line indicates the location of the airplane running on the floating airport.The calculation conditions in this case are listed in Tables 1 and 2.  1 and 2. The black dotted line indicates the location of the airplane running on the floating airport.
After the airplane departed, both the wave height and wavelength of the floating body waves generated by the running airplane increased as the airplane moved faster.As detailed in [11,18], floating body waves are significantly amplified when the moving speed of a point load on a floating thin plate is roughly close to the phase velocity of the linear shallow water waves, i.e., gh, in shallow water conditions with a still water depth of h.This is due to the resonance similar to that occurring in tsunami generation due to air pressure waves, e.g., [29], based on the Proudman resonance [30].Such resonance phe-  1 and 2. The black dotted line indicates the location of the airplane running on the floating airport.
After the airplane began to take off, both the wave height and wavelength of the floating body waves generated by the running airplane increased as the airplane moved faster.As detailed in [11,18], floating body waves are significantly amplified when the moving speed of a point load on a floating thin plate is roughly close to the phase velocity of the linear shallow water waves, i.e., gh, in shallow water conditions with a still water depth of h.This is due to the resonance similar to that occurring in tsunami generation due to air pressure waves, e.g., [29], based on the Proudman resonance [30].Such resonance phenomena are also known in other transient waves, e.g., [31][32][33][34][35][36].In Case AES, gh was approximately 22 m/s, which corresponded to the running speed of the airplane at t ≃ 19 s.Based on Figure 2, the amplification of the floating body waves is noticeable approximately in the time period of 20 s ≤ t ≤ 40 s, and in the area of 1.2 km ≤ x ≤ 1.9 km.It should be noted that in order for waves to grow, the airplane needs to be traveling at a speed roughly close to gh for a sufficient period of time.During this time period, because of the energy being supplied by the running airplane, a forced wave was generated, and in parallel with this, many free waves also continued to occur so as to satisfy the dispersion relation of floating body waves.When the draft of a floating thin plate is assumed to be zero, the linear dispersion relation of floating body waves is expressed by where θ and k are the angular frequency and wavenumber of floating body waves, respectively, and B and m are the flexural rigidity and density of the floating body, respectively [18,37].The generated free waves propagated in front of the airplane because the propagation speeds of the floating body waves were larger than the running speed of the airplane during the time period.When t > 40 s, the airplane traveled faster, and the resonance effect was reduced, so the distance over which the amplified wave group, or wave clump, with relatively large wave heights existed was limited.Thereafter, when t ≥ 70 s, at the front end of the floating airport, part of the wave energy was reflected and the rest was transmitted in the x-axis direction, causing the continued airport vibration even after the airplane took off.The reflected components of the above-described localized wave group propagated in the negative direction of the x-axis, overlapped with the waves traveling in the positive direction of the x-axis, and then were reflected again at the rear end of the floating airport.Thus, the generated and amplified waves are repeatedly reflected and overlapped, causing complex vibrations at a floating airport.If this continues, it is necessary to take a long time interval between landings and takeoffs, so it disrupts airplane operations at the airport and may also accelerate the deterioration of the structure.
Conversely, Figure 3 depicts the time variation of the floating airport and water surface profiles when a B737 takes off from the same floating airport with a constant flexural rigidity of 1.0 × 10 11 Nm in Case BES, where the still water depth is 50 m throughout the water area.The calculation conditions in this case are also listed in Tables 1 and 2.
In Case BES, the phase velocity of linear shallow water waves, i.e., gh, is also approximately 22 m/s, which corresponds to the running speed of the airplane at t ≃ 15 s.Based on Figure 3, the amplification of the floating body waves is noticeable approximately in the time period of 20 s ≤ t ≤ 30 s, and in the area of 1.3 km ≤ x ≤ 1.7 km.Although the wave height was not as large as in the B747 case because a B737 is lighter than a B747, the wave behavior was similar to that seen when the B747 takes off.Even if a B737 runs when the airport is still vibrating, newly generated waves can overlap the existing floating body waves, forming unexpectedly large waves that could disrupt the airplane's takeoff and landing.
Moreover, Figure 4 depicts the time variation of the floating airport and water surface profiles when a B737 takes off from a floating airport with the same airport length of 5 km but with a lower flexural rigidity of 1.0 × 10 10 Nm.As indicated in the figure, the decrease in the flexural rigidity of the floating airport resulted in the larger wave heights of floating body waves.When the flexural rigidity is decreased, the difference in speed between floating body waves and water waves decreases based on the dispersion relation represented by Equation (7), so the wave transmittance at an airport end increases [18].However, the floating body waves were highly amplified in the present case and the wave heights of the reflected waves increased.

When Reducing the Flexural Rigidity Uniformly in a Floating Airport Edge
Second, we examine the effectiveness of the reflection reduction techniques under various conditions.One of the methods is to reduce the flexural rigidity in an edge part of an airport uniformly.In Cases AES, AEL, BES, and BEL, the conditions of which are listed in Tables 1 and 2, the flexural rigidity is uniformly reduced to B edge in 5.5 km − λ ≤ x ≤ 5.5 km, where λ is the length of the edge part of the airport, and the flexural rigidity of the main part in 0.5 km ≤ x < 5.5 km − λ, i.e., B main , is 1.0 × 10 11 Nm. Figure 5 depicts the time variation of the floating airport and water surface profiles when the flexural rigidity of the edge part, B edge , is 1.0 × 10 9 Nm in Case AES.In comparison with Figure 2, the wave heights of the reflected waves are significantly reduced in Figure 5, which is especially clear for the floating body waves at t ≥ 90 s.This is because the multiple reflections of waves occurred at both the boundary between the main and edge parts and the front end of the airport.At the boundary between the main and edge parts, the difference in flexural rigidity was decreased, so the wave reflectance decreased compared to when the main part is directly connected to the free water surface area, without providing the edge part.Through this boundary, part of the generated floating body waves entered the edge part and were then reflected at the front end of the floating airport, but since the difference in flexural rigidity between the airport and free water surface was also decreased, the wave reflectance at the front end of the airport was also suppressed.Moreover, the multiple reflections at both the part boundary and front end of the airport occurred with a time difference, which also contributed to reducing the wave reflectance.
Eng 2024, 5, FOR PEER REVIEW 9 rigidity was decreased, so the wave reflectance decreased compared to when the main part is directly connected to the free water surface area, without providing the edge part.Through this boundary, part of the generated floating body waves entered the edge part and were then reflected at the front end of the floating airport, but since the difference in flexural rigidity between the airport and free water surface was also decreased, the wave reflectance at the front end of the airport was also suppressed.Moreover, the multiple reflections at both the part boundary and front end of the airport occurred with a time difference, which also contributed to reducing the wave reflectance.Figure 6 depicts the time variation of the floating airport and water surface profiles when the flexural rigidity of the edge part, Bedge, is 1 × 10 9 Nm, Case AEL.In this case, the wave heights of the reflected waves were also reduced, as in the case depicted in Figure 5.However, in Figure 6, comparing in detail, the wave heights of the reflected waves are slightly larger than those in Figure 5.The reason is that in Case AEL, the edge part length was longer and the starting location of the edge part was closer to the airplane runway, so still larger waves that were in the process of deforming to satisfy the dispersion relation, expressed by Equation (7), were reflected at the boundary between the main and edge parts, leading to the slightly larger wave heights of the reflected waves.
The difference in elevation between the maximum and minimum surface displacements-ηmax and ηmin, respectively-at a location of x is represented as H, i.e., H = ηmax -ηmin. (8) Figure 7 depicts the ratio of the H values between with and without the reflection control, i.e., RE = H/H0, at x = 2 km, where H0 is the H value when the flexural rigidity in the edge part of the airport, Bedge, is not reduced and the flexural rigidity is uniformly 1 × Figure 6 displays the time variation of the floating airport and water surface profiles when the flexural rigidity of the edge part, B edge , is 1.0 × 10 9 Nm in Case AEL.In this case, the wave heights of the reflected waves were also reduced, as in the case depicted in Figure 5.However, in Figure 6, comparing in detail, the wave heights of the reflected waves are slightly larger than those in Figure 5.The reason is that in Case AEL, the edge part length was longer and the starting location of the edge part was closer to the airplane runway, so still larger waves that were in the deformation process to satisfy the dispersion relation, expressed by Equation (7), were reflected at the boundary between the main and edge parts, leading to the slightly larger wave heights of the reflected waves.The difference in elevation between the maximum and minimum surface displacementsη max and η min , respectively-at a location of x is represented as H, i.e., H = η maxη min . (8) Figure 7 depicts the ratio of the H values between with and without the reflection control, i.e., R E = H/H 0 , at x = 2 km, where H 0 is the H value when the flexural rigidity in the edge part of the airport, B edge , is not reduced and the flexural rigidity is uniformly 1.0 × 10 11 Nm throughout the airport.It should be noted that when using the ratio R E , reflected waves are not distinguished from newly generated waves, but the H values are compared with and without measures.1 and 2.
As indicated in Figure 7, when Bedge is close to Bmain, RE is not decreased much especially in Case AES.As the difference between Bedge and Bmain increases, RE decreases more, and when Bedge is approximately 1.0 × 10 9 Nm, RE is reduced most effectively.However, when Bedge is reduced too much, RE is not decreased much, and RE is larger than one when 1.0 × 10 6 Nm ≤ Bedge ≤ 1.0 × 10 7.2 Nm in Case AES and when 1.0 × 10 6 Nm ≤ Bedge ≤ 1.0 × 10 7. 5  Nm in Case AEL.Thus, if there is a large difference in flexural rigidity between the main and edge parts of a floating airport, the wave reflectance at this part boundary will be large, which is counterproductive.Therefore, it is necessary to appropriately allow part of the waves to transmit into the edge part and cause above-described multiple reflections at Ratio R E between the difference in the maximum and minimum surface displacements, H, and that without flexural rigidity reduction in the edge part, H 0 , at x = 2 km in Cases AES, AEL, BES, and BEL.The calculation conditions are described in Tables 1 and 2.
As indicated in Figure 7, when B edge is close to B main , R E is not decreased much especially in Case AES.As the difference between B edge and B main increases, R E decreases more, and when B edge is approximately 1.0 × 10 9 Nm, R E is reduced most effectively.However, when B edge is reduced too much, R E is not decreased much, and R E is larger than one when 1.0 × 10 6 Nm ≤ B edge ≤ 1.0 × 10 7.2 Nm in Case AES and when 1.0 × 10 6 Nm ≤ B edge ≤ 1.0 × 10 7. 5 Nm in Case AEL.Thus, if there is a large difference in flexural rigidity between the main and edge parts of a floating airport, the wave reflectance at this part boundary will be large, which is counterproductive.Therefore, it is necessary to appropriately allow part of the waves to transmit into the edge part and cause abovedescribed multiple reflections at both the part boundary and the end of the airport.It is desirable that the floating body waves are repeatedly reflected and part of the wave energy is trapped in the edge part of the airport, to avoid vibrating the main part of the airport as much as possible.
Conversely, when a B737 takes off in Cases BES and BEL, R E is reduced even when B edge ≤ 1.0 × 10 8 Nm, because the wave height of the generated floating body waves is not large and the wave slope is not steep, resulting in the low reflectance at the boundary between the main and edge parts of the airport.Thus, for recently popular economical airplanes such as a B737, reducing the flexural rigidity of the edge parts uniformly is beneficial to reduce the wave reflectance; however, because there are conditions under which the wave reflectance reaches a minimal value, case studies are required for the design of the edge parts under actual conditions.

Reflection Control of Floating Body Waves by Reducing the Flexural Rigidity Linearly in a Floating Airport Edge
In Cases ACS, ACL, BCS, and BCL, the flexural rigidity in the edge part of an airport is linearly reduced, where the conditions are listed in Tables 1 and 3.The flexural rigidity linearly decreases from B main at x = 5.5 km − λ to B end at x = 5.5 km, where λ is the length of the edge part.Figure 8 depicts the ratio R C between the maximum surface displacements with and without the reflection control, at x = 3.4 km, where the ratio R C is defined as η max /η max, 0 , and η max is the maximum surface displacement at a location of x, whereas η max, 0 is η max at the same location when the flexural rigidity in the edge part of the airport, B edge , is not reduced and the flexural rigidity is uniformly 1.0 × 10 11 Nm throughout the airport.
Eng 2024, 5, FOR PEER REVIEW 11 reflectance reaches a minimal value, case studies are required for the design of the edge parts under actual conditions.

Reflection Control of Floating Body Waves by Reducing the Flexural Rigidity Linearly in a Floating Airport Edge
In Cases ACS, ACL, BCS, and BCL, the flexural rigidity in the edge part of an airport is linearly reduced, where the conditions are listed in Tables 1 and 3.The flexural rigidity linearly decreases from Bmain at x = 5.5 km − λ to Bend at x = 5.5 km, where λ is the length of the edge part.Figure 8 depicts the ratio RC between the maximum surface displacements with and without the reflection control at x = 3.4 km, where the ratio RC is defined as ηmax/ηmax, 0, and ηmax is the maximum surface displacement at a location of x, whereas ηmax, 0 is ηmax at the same location when the flexural rigidity in the edge part of the airport, Bedge, is not reduced and the flexural rigidity is uniformly 1 × 10 11 Nm throughout the airport.  1 and 3.
Figure 8 indicates that in all these cases, there exists a minimal value of RC, as in the cases in which the flexural rigidity of the edge part is reduced uniformly.It is noteworthy that RC < 1 for Bend ≤ 1.0 × 10 7.5 Nm even when B747 takes off, unlike in Cases AES and AEL.When the flexural rigidity is gradually reduced, wave reflection occurs gradually, so even if the still larger floating body waves enter an edge part before deformation is completed to satisfy the dispersion relation, i.e., Equation (7), the wave reflectance does not increase significantly.Therefore, the method of reducing the flexural rigidity in edge parts of an airport linearly is effective regardless of the air plane size.
Moreover, for example, in Case ACL, the ratio of the H values, defined in Equation (8), at x = 4 km between with and without the reflection control was approximately 0.73 when 0 Nm ≤ Bend ≤ 1.0 × 10 7.5 Nm.Therefore, if the flexural rigidity at airport ends can be lowered further, this method is more effective and does not have a negative effect even for B747.Even when a larger airplane runs on the floating airport, a stable effect can be obtained by gradually and sufficiently lowering the flexural rigidity within the edge part.However, there is a minimal  1 and 3.
Figure 8 indicates that in all these cases, there exists a minimal value of R C , as in the cases in which the flexural rigidity of the edge part is reduced uniformly.It is noteworthy that R C < 1.0 for B end ≤ 1.0 × 10 7.5 Nm even when a B747 takes off, unlike in Cases AES and AEL.When the flexural rigidity is gradually reduced, wave reflection occurs gradually, so even if the still larger floating body waves enter an edge part before the deformation is completed to satisfy the dispersion relation, i.e., Equation (7), the wave reflectance does not increase significantly.Therefore, the method of reducing the flexural rigidity in edge parts of an airport linearly is effective regardless of the airplane size.
Moreover, for example, in Case ACL, the ratio of the H values, defined in Equation (8), at x = 4 km between with and without the reflection control was approximately 0.73 when 0.0 Nm ≤ B end ≤ 1.0 × 10 7.5 Nm.Therefore, if the flexural rigidity at airport ends can be lowered further, this method is more effective and does not have a negative effect even for B747.That is, even when a larger airplane runs on the floating airport, a stable effect can be obtained by gradually and sufficiently lowering the flexural rigidity within the edge part.However, there is a minimal value of the wave reflectance, so to achieve optimal flexural rigidity conditions, such numerical analyses should be performed at the design stage of a floating airport.
Based on Figure 8, in Case BCS, the minimal value of R C is achieved when the flexural rigidity at the front end of the floating airport, B end , is 1.0 × 10 8 Nm. Figure 9 displays the time variation of the floating airport and water surface profiles under this condition.Compared to Figure 3 without partial reduction of flexural rigidity, the wave heights of the reflected waves are obviously reduced.Even if the flexural rigidity is structurally or economically fixed for most of a floating airport, the wave reflectance can be reduced by modifying the structure or installing accessories to lower the flexural rigidity only near the airport ends, leading to increase in the calmness of the floating airport.Furthermore, in Case BCL, the minimal value of R C is achieved when B end is 1.0 × 10 9 Nm. Figure 10 depicts the time variation of the floating airport and water surface profiles under this condition.In this case, the wave heights of the reflected waves were reduced almost to the same extent as in the case depicted in Figure 9, and the waveforms showed no significant difference between the two cases.
To understand the difference in the effect of uniformly reducing the flexural rigidity of the edge part versus linearly reducing it, we compare the distributions of H, defined in Equation (8), between these cases.When the H values in Cases AEL and ACL are represented as H E and H C , respectively, Figure 11 depicts the distributions of H E and H C , where B edge = 1.0 × 10 9 Nm and B end = 1.0 × 10 9 Nm, respectively.This figure indicates that in Case AEL, the reduction in H E began immediately after the floating body waves entered the edge part, and the energy of the waves reaching the front end of the floating airport decreased.In contrast, in Case ACL, the energy of the waves reaching the airport's front end was larger.Thus, if the flexural rigidity B is decreased spatially suddenly, the floating wave reflection at the part boundary will be large.In addition, as described in the comparison between Figures 3 and 4, the wave heights of floating body waves increase as B decreases, so the longer distance with lower B will reduce the reflection control effect.
Figure 10.Profiles of the floating airport and water surface at every 5 s when the flexural rigidity at the front end of the airport is 1.0 × 10 9 Nm in Case BCL, the conditions of which are described in Tables 1 and 3.The black dotted line indicates the location of the airplane running on the floating airport.
Eng 2024, 5, FOR PEER REVIEW 13 in Case ACL, the energy of the waves reaching the airport end was larger.Thus, if the flexural rigidity B is decreased spatially suddenly, the floating wave reflection at the boundary will be large.In addition, as described in the comparison between Figures 3 and  4, the wave heights of floating body waves will increase as B decreases, so the longer distance with lower B may reduce the reflection control effect.When the maximum surface displacements ηmax at a location of x in Cases AEL and ACL are represented as ηE, max and ηC, max, respectively, Figure 12 depicts the distributions of ηE, max and ηC, max, where Bedge = 1.0 × 10 9 Nm and Bend = 1.0 × 10 9 Nm, respectively.In Case AEL, ηE, max began to decrease immediately after the floating body waves entered the airport similar to HE, and gradually decreased in the edge part, although there were ups and downs.Conversely, in Case ACL, ηC, max increased once immediately after the waves entered the edge part, and then the average height was maintained inside the edge part, which indicates that part of the wave energy was trapped within the edge part.In both cases, the distributions express vibration modes with nodes and antinodes, and future work is required to investigate the mechanism of mode generation through analyses considering the wave dispersion.When the maximum surface displacements η max at a location of x in Cases AEL and ACL are represented as η E, max and η C, max , respectively, Figure 12 depicts the distributions of η E, max and η C, max , where B edge = 1.0 × 10 9 Nm and B end = 1.0 × 10 9 Nm, respectively.In Case AEL, η E, max began to decrease immediately after the floating body waves entered the edge part, similarly to H E , and gradually decreased in the edge part, although there were ups and downs.Conversely, in Case ACL, η C, max increased once immediately after the waves entered the edge part, and then the average height was maintained inside the edge part, which indicates that part of the wave energy was trapped within the edge part.In both cases, the distributions express vibration modes with nodes and antinodes, and future work is required to investigate the mechanism of mode generation through analyses considering the wave dispersion.
of ηE, max and ηC, max, where Bedge = 1.0 × 10 9 Nm and Bend = 1.0 × 10 9 Nm, respectively.In Case AEL, ηE, max began to decrease immediately after the floating body waves entered the airport similar to HE, and gradually decreased in the edge part, although there were ups and downs.Conversely, in Case ACL, ηC, max increased once immediately after the waves entered the edge part, and then the average height was maintained inside the edge part, which indicates that part of the wave energy was trapped within the edge part.In both cases, the distributions express vibration modes with nodes and antinodes, and future work is required to investigate the mechanism of mode generation through analyses considering the wave dispersion.

Wave Height Control of Floating Body Waves by Partially Reducing the Still Water Depth under a Floating Airport Runway
As described above, resonance occurs when the running speed of an airplane on a floating airport is close to the phase velocity of water waves.In this section, we propose a method to reduce the amplification of floating body waves by decreasing the still water depth under an airport runway over a certain distance to change the wave speeds.In order not to consider both the reflection and transmission of floating body waves at the front end of a floating airport, the airport length is set to be 15 km. Figure 13 displays the time variation of the floating airport and water surface profiles when the still water depth is uniformly 50 m in Cases ADS and ADL.Conversely, Figure 14 depicts the time variation of the floating airport and water surface profiles when the still water depth is partially reduced to 10 m at 1.5 km ≤ x ≤ 2 km in Case ADS.In comparison with Figure 13, the wave amplification due to the resonance is suppressed in Figure 14, and the wave height of the floating body waves propagating in the positive direction of the x-axis is reduced.As discussed in Section 4.1, the airplane's running speed and the water wave propagation speed become closer at t ≃ 19 s, and larger waves are generated at 20 s ≤ t ≤ 40 s, i.e., at 1.2 km ≤ x ≤ 1.9 km, when the still water depth is uniformly 50 m throughout the water area.In the case depicted in Figure 14, by reducing the still water depth where the airplane's speed gradually increased, the difference between the airplane and water wave speeds increased, so the distance at which the resonance effect was larger decreased, resulting in the wave height reduction.Regarding changing still water depth, there was concern about wave reflection due to the abrupt change in still water depth.At 0.5 km ≤ x ≤ 1.5 km in Figure 14, it is certainly observed that floating body waves are traveling back and forth between the rear end of the floating airport and the starting point of the shallower water area, repeating reflections at these two locations, but the wave heights of these floating body waves are not significant.
Figure 15 depicts the time variation of the floating airport and water surface profiles when the still water depth is partially reduced to 10 m at 1.5 km ≤ x ≤ 4 km in Case ADL.In this case, the wave heights of the floating body waves were reduced more remarkably, decreasing not only the wave heights of the localized wave group with larger wave heights, but also those of the waves following them.As previously described, in order for waves to be amplified, it is necessary that the distance at which the resonance effect is large is sufficient, so making the shallower area longer leads to a more effective reduction in wave height.In addition, the wavelengths in the shallower area are shorter than the corresponding results indicated in Figures 13 and 14.This is because, based on the dispersion relation expressed by Equation (7), as the still water depth is decreased, the wave propagation speed increases and the wavelength decreases [18].Moreover, regarding the takeoff of a B737, Figure 16 depicts the time variation of the floating airport and water surface profiles when the still water depth is uniformly 50 m in Cases BDS and BDL.Conversely, when the still water depth is partially reduced to 10 m at 1.5 km ≤ x ≤ 2 km in Case BDS and at 1.5 km ≤ x ≤ 4 km in Case BDL, the results are displayed in Figures 17 and 18, respectively.A similar effect was obtained with B737s as with B747s.Although it is better to build a floating airport in a place where the still water depth is not too deep, if it is built in a deeper place, the amplification of floating body waves can be suppressed by reducing the still water depth under the runways using a natural seabed topography, artificial reefs, etc.Furthermore, in the case of landing, the running speed of an airplane gradually slows after its touchdown, so the amplification effect increases as the airplane approaches a stop position when the still water depth is, for example, 50 m [18].Therefore, it would be favorable for airport operations if the airplane stops at a place where the still water depth is reduced.
Figure 19 depicts the ratio of the H values between with and without the partial reduction in the still water depth, i.e., R D = H/H 0 , at x = 4 km, where H is evaluated by Equation ( 8) and H 0 is the H value when the still water depth is uniformly 50 m throughout the water area.This figure indicates that as the still water depth in the shallower area is decreased, the ratio R D tends to decrease.In both Cases ADL and BDL with a longer shallower area, R D decreases approximately linearly with decreasing the still water depth, where the gradient R D /h s is approximately 0.016 m −1 .Conversely, in Cases ADS and BDS, when h s < 30 m, the reduction rate of R D with respect to the decrease in h s decreases, from which it can be concluded that when the shallower area is short, reducing h s beyond a certain level does not significantly increase the wave height reduction effect.In summary, adjusting the still water depth under airport runways over a sufficient distance so that the speeds of airplanes and water waves do not come close to each other, as well as gradually changing the flexural rigidity in the edge parts of the airport, will help to reduce the floating body waves excited by airplanes.seabed topography, artificial reefs, etc.Furthermore, in the case of landing, the running speed of an airplane gradually slows after its touchdown, so the amplification effect in creases as the airplane approaches a stop position when the still water depth is, for exam ple, 50 m [18].Therefore, it would be favorable for airport operations if the airplane stops at a place where the still water depth is reduced.   1 and 4 The black dotted line indicates the location of the airplane running on the floating airport.    1 and 4.

Conclusions
To design a large floating structure that exhibit hydroelastic behavior at the water surface, it is necessary to understand the interaction between the vibration of the floating structure and the motion of the fluid.In this study, the numerical simulations were generated to investigate the response of a very large floating airport to an airplane takeoff, using the set of nonlinear shallow water equations of velocity potential for water waves interacting with a floating thin plate.
We simulated the excitation of floating body waves due to an airplane takeoff.When the running speed of the airplane approached the phase velocity of the water waves, resonance occurred, and a forced wave was excited, generating free waves to satisfy the dispersion relation.As the airplane speed increased more, the resonance effect decreased and a localized wave group with larger wave heights was formed.Thereafter, these floating body waves were reflected and transmitted at the front end of the floating airport, and the vibration of the airport remained even after the airplane took off.Therefore, to reduce such prolonged vibrations, which may disrupt airport operations, we have proposed two methods to reduce remaining floating body waves caused by an airplane takeoff: reflectance reduction by decreasing the flexural rigidity in the edge parts of an airport and amplification reduction by decreasing the still water depth partially under airport runways.
First, when the flexural rigidity is uniformly decreased in an edge part of the floating airports, the reflectance of the floating body waves due to the B737 was reduced because of the multiple reflections.However, the reflectance of the floating body waves caused by the B747 increased, depending on the conditions.In both cases of the B737 and B747, when the edge part is too long, the still larger waves reached the edge part, so the wave reflectance was not reduced much.There were conditions under which the wave reflectance reached a minimal value, so numerical simulations such as those generated here will be required for optimal design of airport edge parts.
Moreover, when the flexural rigidity is linearly decreased in an edge part of the floating airports, although a minimal value appeared in the wave reflectance, the wave reflectance decreased, never increased under the conditions investigated.
Second, when the still water depth under the airport runways is partially decreased in the section where larger floating body waves are generated, the wave heights of floating body waves tended to decrease as the still water depth in the shallower area decreased.This is because the water wave speed decreased in the shallower area, and the distance with larger resonance effect was shortened, thereby suppressing the amplification of floating body waves.
In conclusion, considering the still water depth under the runways of a floating airport to reduce the distance at which airplane speeds approach water wave speeds, and further gradually decreasing the flexural rigidity in the edge parts of the airport are effective in reducing the persistent airport vibration due to airplane movements.In the future, the behavior of airports with significant frequency dispersion, particularly in a deeper sea, and the vibration modes generated at airports should be investigated with higher-order effects of the velocity potential to consider the dispersion of floating body waves due to both running airplanes and incident ocean waves, including swells.Furthermore, when mooring lines are installed near the ends of a floating airport, they may affect the vibration of the airport ends, changing the reflectance of the floating body waves.It will also be necessary to examine such effects of mooring items, in determining the appropriate mooring methods and locations.

Figure 1 Figure 1 .
Figure 1Figure 1.Schematic for a water area with a hydroelastic airport floating at the water surface.

Figure 2 .
Figure 2. Profiles of the floating airport and water surface at every 5 s when a B747 takes off from an airport with a constant flexural rigidity of 1 × 10 11 Nm in Case AES, the conditions of which are described in Tables1 and 2. The black dotted line indicates the location of the airplane running on the floating airport.

Figure 2 .
Figure 2. Profiles of the floating airport and water surface at every 5 s when a B747 takes off from an airport with a constant flexural rigidity of 1.0 × 10 11 Nm in Case AES, the conditions of which are described in Tables1 and 2. The black dotted line indicates the location of the airplane running on the floating airport.

Figure 3 Figure 3 .
Figure 3 Figure 3. Profiles of the floating airport and water surface at every 5 s when a B737 takes off from an airport with a constant flexural rigidity of 1.0 × 10 11 Nm in Case BES, the conditions of which are described in Tables1 and 2. The black dotted line indicates the location of the airplane running on the floating airport.

Figure 4 .
Figure 4. Profiles of the floating airport and water surface at every 5 s when a B737 takes off from an airport with an airport length of 5 km and a constant flexural rigidity of 1.0 × 10 10 Nm.The still water depth was 50 m, and the airplane conditions are described in Table 1.The black dotted line indicates the location of the airplane running on the floating airport.

Figure 5 .
Figure 5. Profiles of the floating airport and water surface at every 5 s when the flexural rigidity of the edge part, Bedge, is 1 × 10 9 Nm in Case AES, the conditions of which are described in Tables1 and 2. The black dotted line indicates the location of the airplane running on the floating airport.

Figure 5 .
Figure 5. Profiles of the floating airport and water surface at every 5 s when the flexural rigidity of the edge part, B edge , is 1.0 × 10 9 Nm in Case AES, the conditions of which are described in Tables1 and 2. The black dotted line indicates the location of the airplane running on the floating airport.

4 Figure 6 Figure 6 .
Figure 6 Figure 6.Profiles of the floating airport and water surface at every 5 s when the flexural rigidity of the edge part is 1.0 × 10 9 Nm in Case AEL, the conditions of which are described in Tables1 and 2. The black dotted line indicates the location of the airplane running on the floating airport.

Eng 2024, 5 , 10 Figure 6 .
Figure 6.Profiles of the floating airport and water surface at every 5 s when the flexural rigidity of the edge part is 1 × 10 9 Nm in Case AEL, the conditions of which are described in Tables1 and 2. The black dotted line indicates the location of the airplane running on the floating airport.

Figure 7 .
Figure 7. Ratio RE between the difference in the maximum and minimum surface displacements, H, and that without flexural rigidity reduction in the edge part, H0, at x = 2 km in Cases AES, AEL, BES, and BEL.The calculation conditions are described in Tables1 and 2.

Figure 7 .
Figure7.Ratio R E between the difference in the maximum and minimum surface displacements, H, and that without flexural rigidity reduction in the edge part, H 0 , at x = 2 km in Cases AES, AEL, BES, and BEL.The calculation conditions are described in Tables1 and 2.

Figure 8 .
Figure 8. Ratio RC between the maximum surface displacement ηmax and that without flexural rigidity reduction, ηmax, 0, at x = 3.4 km in Cases ACS, ACL, BCS, and BCL.The calculation conditions are described in Tables1 and 3.

Figure 8 .
Figure 8. Ratio R C between the maximum surface displacement η max and that without flexural rigidity reduction, η max, 0 , at x = 3.4 km in Cases ACS, ACL, BCS, and BCL.The calculation conditions are described in Tables1 and 3.

Figure 9 Figure 9 .
Figure 9 Figure 9. of the floating airport and water surface at every 5 s when the flexural rigidity at the front end of the airport is 1.0 × 10 8 Nm in Case BCS, the conditions of which are described in Tables 1 and 3.The black dotted line indicates the location of the airplane running on the floating airport.

Figure 11 .
Figure 11.Distributions of the differences between the maximum and minimum surface displacements, HE and HC, where Bedge = 1.0 × 10 9 Nm in Case AEL and Bend = 1.0 × 10 9 Nm in Case ACL, respectively.The calculation conditions are described in Tables 1-3.

Figure 11 .
Figure 11.Distributions of the differences between the maximum and minimum surface displacements, H E and H C , where B edge = 1.0 × 10 9 Nm in Case AEL and B end = 1.0 × 10 9 Nm in Case ACL, respectively.The calculation conditions are described in Tables 1-3.

Figure 12 .
Figure 12.Distributions of the maximum surface displacements η E, max and η C, max , where B edge = 1.0 × 10 9 Nm in Case AEL and B end = 1.0 × 10 9 Nm in Case ACL, respectively.The calculation conditions are described in Tables 1-3.

Figure 13 .
Figure13.Profiles of the floating airport and water surface at every 5 s when the still water depth is uniformly 50 m throughout the water area in Cases ADS and ADL, the conditions of which are described in Tables1 and 4. The black dotted line indicates the location of the airplane running on the floating airport.

Figure 14 .
Figure14.Profiles of the floating airport and water surface at every 5 s when the still water depth in the shallower area is 10 m in Case ADS, the conditions of which are described in Tables1 and 4. The black dotted line indicates the location of the airplane running on the floating airport.

Figure 14 .
Figure14.Profiles of the floating airport and water surface at every 5 s when the still water depth in the shallower area is 10 m in Case ADS, the conditions of which are described in Tables1 and 4. The black dotted line indicates the location of the airplane running on the floating airport.

Figure 15 .
Figure15.Profiles of the floating airport and water surface at every 5 s when the still water depth in the shallower area is 10 m in Case ADL, the conditions of which are described in Tables1 and 4. The black dotted line indicates the location of the airplane running on the floating airport.Moreover, regarding the takeoff of a B737, Figure16depicts the time variation of the floating airport and water surface profiles when the still water depth is uniformly 50 m in Cases BDS and BDL.Conversely, when the still water depth is partially reduced to 10 m at 1.5 km ≤ x ≤ 2 km in Case BDS and at 1.5 km ≤ x ≤ 4 km in Case BDL, the results are

Figure 15 .
Figure15.Profiles of the floating airport and water surface at every 5 s when the still water depth in the shallower area is 10 m in Case ADL, the conditions of which are described in Tables1 and 4. The black dotted line indicates the location of the airplane running on the floating airport.

9 Figure 16 Figure 16 .
Figure 16Figure16.Profiles of the floating airport and water surface at every 5 s when the still water depth is uniformly 50 m throughout the water area in Cases BDS and BDL, the conditions of which are described in Tables1 and 4. The black dotted line indicates the location of the airplane running on the floating airport.

Figure 16 .
Figure16.Profiles of the floating airport and water surface at every 5 s when the still water depth is uniformly 50 m throughout the water area in Cases BDS and BDL, the conditions of which are de scribed in Tables1 and 4. The black dotted line indicates the location of the airplane running on the floating airport.

Figure 17 .
Figure17.Profiles of the floating airport and water surface at every 5 s when the still water depth in the shallower area is 10 m in Case BDS, the conditions of which are described in Tables1 and 4The black dotted line indicates the location of the airplane running on the floating airport.

Figure 17 . 10 Figure 18 Figure 18 .
Figure17.Profiles of the floating airport and water surface at every 5 s when the still water depth in the shallower area is 10 m in Case BDS, the conditions of which are described in Tables1 and 4. The black dotted line indicates the location of the airplane running on the floating airport.

Figure 19 .
Figure19.Ratio R D between the difference in the maximum and minimum surface displacements, H, and that without partial reduction in still water depth, H 0 , at x = 4 km in Cases ADS, ADL, BDS, and BDL.The calculation conditions are described in Tables1 and 4.

Table 1 .
Calculation conditions of the airplanes.The takeoff speed is the airplane speed when the airplane leaves the airport.

Table 2 .
Calculation conditions when the flexural rigidity is uniformly reduced in an edge part of the airports, i.e., in 5.5 km − λ ≤ x ≤ 5.5 km.The flexural rigidities of the main and edge parts of the airports are B main and B edge , respectively.

Table 4 .
Calculation conditions when the still water depth is uniformly reduced at 1.5 km ≤ x ≤ 1.5 km + D, at which the still water depth is h s .