Scattering of Oblique Water Waves by Two Unequal Surface-Piercing Vertical Thin Plates with Stepped Bottom Topography

Based on linear water-wave theory, this study investigated the scattering of oblique incident water waves by two unequal surface-piercing thin vertical rigid plates with stepped bottom topography. By using the matched eigenfunction expansion method and a least square approach, the analytical solutions are sought for the established boundary value problem. The effects of the incidence angle, location of step, depth ratio of deep to shallow waters, and column width between two plates, on the reflection coefficients, the horizontal wave forces acting on the two plates, and the mean surface elevation between the two plates, are numerically examined under a variety of wave conditions. The results show that the existence of the stepped bottom between two plates considerably impacts the hydrodynamic performances of the present system. It is found that the effect of stepped bottom on the reflection coefficient of the present two-plate structure is evident only with waves of the low dimensionless frequency. Moreover, the influence of the step location on the hydrodynamic performance of the present two-plate structure is slight if the step is placed in between the two plates.


Introduction
With the rise of world population, there is a rapid increase of developmental activities along the coast regions worldwide. To ensure the safe coastal engineering practices, such as bays, ports and harbors, the protection of these coastal infrastructures from rough wave conditions has been paid much attention. Breakwaters, as one of the most important coastal structures, are often constructed near the coastline to protect the coastal infrastructures. After interacting with waves incoming from the open sea, suitable breakwaters can effectively reduce the wave energy transmitted through them. Traditional breakwaters such as rubble mound breakwaters and concrete caissons are widely used for the construction of harbours and for shore protection works. The cost of these types of breakwaters is generally high if they are employed in deep water regions. Since the wave energy is mainly concentrated near the water free surface, breakwaters which can effectively reflect this energy are proposed, such as partially immersed single vertical barriers (Ursell and Dean, 1947;Wiegel, 1960).
To understand physical insight of these breakwaters, a number of researchers have investigated the wave scattering by a single vertical thin plate and two vertical thin plates on the free surface using either analytical, numerical, or experimental approaches. For the problem with a single fixed vertical thin plate, analytic studies were usually carried out within the framework of classical linear water-wave theory, such as Ursell and Dean (1947), Losada et al. (1992), Porter and Evans (1995), Mandal and Das (1996). Ursell and Dean (1947) derived the exact solution of the diffraction of waves past a surface-piercing vertical barrier in deep water using modified Bessel function. This problem has also been studied by Losada et al. (1992) and Porter and Evans (1995). The former used the matched eigenfunction expansion method to obtain the solutions of the boundary value problem while the latter adopted a multi-term Galerkin approximation method. The overall wave energy is well conserved before and after the wave diffraction by the single plate by introducing the evanescent waves. In addition to analytical methods, numerical methods (Liu and Abbaspour, 1982) and physical model tests (Reddy and Neelamani, 1992) were also applied to investigate the problem of wave interacting with a fixed vertical plate on the free surface. To further improve the performance of this single plate, twin plates were suggested by Isaacson et al. (1999). Studies on the wave scattering by two vertical thin surface-piercing plates were carried out by many researchers, including Newman (1974), McIver (1985), Porter and Evans (1995), Evans and Porter (1997), Neelamani and Vedagiri (2002), and Roy et al. (2016). In most of these studies, the two plates are identical except those in References (McIver, 1985;Neelamani and Vedagiri, 2002;Roy et al., 2016). McIver (1985) investigated the scattering of a normally incident wave by two unequal partially submerged plates below the free surface in water of finite depth by using the eigenfunction expansion method. It was found that total reflection or total transmission is possible for two identical plates while total transmission no longer exists for two unequal plates. Similar observations were also found by Roy et al. (2016) who investigated this problem for an obliquely incident plane wave in infinitely deep water. A one-term Gerlerkin approximation method following Morris (1975) was employed to obtain the reflection and transmission coefficients. Neelamani and Vedagiri (2002) experimentally tested on the performance of barriers configured by two unequal surface-piercing plates and found that the transmission coefficient reduces significantly with the increased relative water depth. In addition, the unequal immersion configuration was found superior to the equal configuration under the deep water condition.
In all these investigations involving wave interacting with two vertical thin surface-piercing plates, the plates are assumed to be located in either infinitely deep or uniform finite-depth water. However, since the bottom of an ocean is rarely of uniform depth throughout, it is worthwhile to study wave propagation problems over sea bed of variable depth for their possible applications in the coastal and marine engineering. Over the years, these problems have been studied by considering either undulation-type (Mandal and Gayen, 2006;Martha and Bora, 2007;Dhillon et al., 2013) or step-type (Newman, 1965;Karmakar and Sahoo, 2008;Karmakar et al., 2010;Soares, 2010, 2011;Dhillon et al., 2016) bottom profiles. The former profile is described by a continuous function such as the sinusoidal function while the latter one represents an abrupt change over water depth. Due to the change of water depth, wave reflection, refraction and shoaling may occur and eventually this change shows significant effects on the design and construction of coastal structures like breakwaters. Newman (1965) studied the problem of a normal incident wave past over a step-type bottom wherein the depth varied from finite to infinite. Later, Rhee (1997) reconsidered this problem up to the second-order with oblique waves. A Green's theorem integral equation with a finite depth Green's function was applied and the results of the first-order transmission and reflection coefficients were shown consistent with those obtained by Newman (1965) using the matched eigenfunction method. Karmakar et al. (2010) investigated the oblique flexural water wave scattering by a single and multiple steps bottom in water of finite-depth with shallow-water approximation theory. It was observed that there is a resonating pattern in the curves of the reflection coefficients varying with dimensionless wave number due to the abrupt change in water depth. Using the similar method as Karmakar et al. (2010), Karmakar and Sahoo (2008) examined the scattering of waves by a semi-infinite floating membrane considering both finite and infinite steps. Later, Karmakar and Soares (2012) studied the problem of an oblique incident wave scattered by a moored finite floating membrane with step-type bottom topography. Soares (2010, 2011) carried out research on both normal and oblique water waves scattering by a floating structure near a wall with stepped bottom typography. Rezanejad et al. (2015) investigated the effect of stepped bottom on the efficiency of the device of implementing a dual-chamber oscillating water column by using both analytical and numerical approaches.
In the above literature review, the wave scattering problem by fixed vertical thin plates on the free surface are all assumed of either infinite or uniform water depth. To the authors' knowledge, studies considered the effect of uneven bottom on the behavior of the scattered waves and the structural loads of fixed vertical thin plates are rare. In this study, the scattering of oblique incident water waves by two unequal surface-piercing thin vertical rigid plates with stepped bottom topography is investigated based on linear waterwave theory. By using the matched eigenfunction expansion method and a least square approach, analytical solutions are sought for the established boundary value problem. The reflection coefficient, the horizontal wave forces acting on the two plates, and the mean surface elevation between the two plates are obtained for various parameters, including the incident wave angle, the location of the step, the depths of deep and shallow waters, and the two-unequalplates structure (including two drafts and the spacing between these two plates). Fig. 1 shows the sketch of the problem that an oblique

The boundary value problem
incident wave is scattered by two fixed mutually-parallel thin rigid plates on the free surface. A rectangular Cartesian coordinate system is chosen with the -axis being on the calm free surface and -axis pointing vertically upwards. The origin of the coordinate system is chosen at the intersection between the calm free surface and the left plate. As shown in Fig. 1, the incident wave propagates from in the water with stepped bottom topography. The horizontal position of the stepped bottom is at , where the water depth changes abruptly from to . In this study, it is assumed that the incident wave propagates from deep water towards shallower water, i.e., . On the free surface, there are two fixed surface-piercing vertical thin rigid plates, with the left one in front of the step while the right one behind. Denoting the width between the two plates by , thus the horizontal positions of the front and rear plates can be described by and , respectively. Instead of two identical submergences, the two plates in this study are immersed with different drafts, i.e., and . Thus, there are four sub-regions occupied by the fluid which can be described by: Region 1 ( ), Region 2 ( , ), Region 3 ( , ), and Region 4 ( , ).
The fluid is assumed inviscid, incompressible and flow irrotational, thus the motion of the fluid in the j-th region can be described by a velocity potential . For small monochromatic incident waves of the angular frequency propagating in the water with a stepped bottom of uniform profile in the direction, the velocity potential can be written as: where denotes the real part of the argument to be taken, ( ), the component of the wavenumber ( ) of the incident wave, , the incident wave angle with respect to the -axis, , and , the time. By applying the potential theory, the spatial velocity potential satisfies the following equation throughout the corresponding region (2) Apart from the above equation, the spatial velocity po-tential must also satisfy the following boundary conditions (BCs), together with , the component of the wavenumber ( ) of the transmitted wave, , the gravity, , the wave height of the incident wave, and , the wavenumber of progressive transmitted waves in the water depth of . and are the complex reflection and transmission coefficients, respectively. The first far-field radiation condition in Eq. (9), states that as the fluid motion is a combination of an incident wave and an outgoing reflected wave. However, Eq. (10) states that as , the fluid motion is only composed of an outgoing transmitted wave.
x = 0 l w Aside from the above BCs, due to the continuity of the pressure and horizontal velocity on the three common interfaces between the four regions, i.e., at , , and , the spatial velocity potential must satisfy the following matching boundary conditions: (12) (14) (16)

Method of solution ϕ j
To solve the boundary value problem formulated in Eqs.
, applying the zero-normal-flux BC on the front plate (Eq. (6)) and the continuity of the horizontal velocity condition on the interface (Eq. (12)) gives Substituting the expressions in Eqs. (17)-(18) into the above equation yields x = w Similarly, at , after applying Eqs. (7) and (16), we have Substituting the expressions of Eqs. (19)- (20) into the above equation yields Consequently, and can be obtained once , , , are solved.

Least square method
In this study, we use a least square method to determine the unknown complex coefficients , , , . This method has been widely adopted by many researchers, such as Losada et al. (1992) and Liu and Li (2011). At , a new function which specifies the velocity potential along the depth is introduced by is equal to zero for . Then the least square method requires A m m = 0, 1, 2, . . .
Minimizing the above integral with respect to each of ( ) gives Here the superscript asterisk denotes the complex conjugate. Substituting Eq. (32) into Eq. (34) and truncating after terms of , yields the following linear algebraic equations where   The dynamic pressure can be obtained by using the linearized form of unsteady Bernoulli equation ρ where is the fluid density. The horizontal wave loads acting on the front and rear plates of unit width can be computed by integrating the dynamic pressure along the vertical direction, respectively, F

Validation
In order to validate the present method, it is first applied to compute the reflection and transmission coefficients of a normal incident wave scattered by a vertical step on the bottom. This case is similar to the model shown in Fig. 1, excepting that there are no plates present. The depth ratio between the deep water and the shallow water regions is . Since there are no plates involved in this problem, both of the two drafts and are zero. After tests on the number of truncation terms of eigenfunctions in Eqs. (17)-(20), is found appropriate to give converged results. Fig. 2 shows the reflection and transmission coefficients varying against the dimensionless wave number . The present results are found in good agreement with those computed by Rhee (1997). Moreover, the energy conservation relation, i.e., has been numerically checked. Here and are the group velocities for the regions of water depth and , respectively. As shown in Fig. 3, the wave energy is well conserved before and after the wave diffracted by the stepped bottom.
Apart from the above validation without surface-piercing plates, the present method is also used to compute the reflection coefficient of water waves scattered by two identical vertical surface-piercing plates in water of uniform depth throughout the fluid domain. Thus, in this computation, the water depth is and the drafts of the two plates are the same, i.e., . Compared with the model shown in Fig. 1, no step present on the bottom and the drafts of the two plates are equal. Fig. 4 shows the comparison of the reflection coefficient between the present result and that obtained by Das et al. (1997) at and . Good agreement is observed and further proves the correctness of the present method.

Numerical examples
After validation, the present method is used to compute the problem of interest shown by Fig. 1. This two-plate structure can be built as a monolithic structure along with a deck to connect the two plates. Then it can be supported by piles which transfer the wave loads acting on the structure to the seabed. In order to assess the hydrodynamic perform-ω 2 h 1 /g h 1 = 20 m ρ = 1025 kg/m 3 g = 9.8 m/s 2 ance of this structure as a floating breakwater, the reflection coefficient, the horizontal wave forces acting on the two plates, and the mean surface elevation between the two plates are computed in this study. As mentioned in Section 3.1, the reflection and transmission coefficients satisfy the energy conservation relation, hence the transmission coefficient is not presented in the following numerical results. All results are plotted against the dimensionless wave frequency, which is defined by . Throughout the present computation, the water depth of Regions 1 and 2, the density of water, and gravity are fixed at , , and , respectively. . 5 shows the variation of the reflection coefficient versus the dimensionless wave frequency with different water depth ratios . represents the flat bottom while and 1/3 denotes the stepped bottom. As seen in Fig. 5, the effect of stepped bottom on is found only significant for the low wave frequency , which corresponds to the incident wave period larger than 5 s. For the high dimensionless wave frequency , the water depth throughout all the regions of fluid can be seen as deep enough and hence the bottom effect is negligible. This explains the slight difference of between these water depth ratios when the dimensionless wave frequency is relatively high. Moreover, compared with  Fig. 4. Reflection coefficient varying against for wave scattered by two identical thin vertical plates in water of uniform depth.

Effect of water depth ratio
, and .
WANG Li-xian et al. China Ocean Eng., 2018, Vol. 32, No. 5, P. 524-535 529 the flat bottom case, the reflection coefficient with stepped bottom case becomes larger for the low frequency range. This reveals that the overall effect of the stepped bottom in the middle of the two plates weakens the shelter ability of the present breakwater, though there is possible interaction between the reflected wave by the stepped bottom and the wave confined by the two plates. Additionally, unlike the monotonic increase with the dimensionless wave frequency for the flat bottom case, there appear fluctuations of within the low frequency range for the stepped bottom . Two local extremum occur at and 1.9, which indicates that the stepped bottom could significantly affect the reflection coefficient of waves scattered by the two-plate structure. Fig. 6 illustrates the comparison of the dimensionless horizontal wave forces on the front and rear plates and with different water depth ratios . It can be seen that the dimensionless wave force on the rear plate is more sensitive to the water depth in Regions 3 and 4. This is perhaps due to the fact that the rear plate is located in the water of depth . Moreover, there appears a single peak in both wave force profiles of the two plates. This peak refers to the resonant wave motion confined by the two plates. With the small water depth ratio, i.e., , the frequency corresponding to the peak force of the rear plate occurs at while it shifts to for both and 1. This reveals that the effect of the stepped bottom starts to play a remarkable role with the water depth ratio .
In Fig. 7, the mean surface elevation between the two plates is compared with different water depth ratios . Similar to the wave forces acting on the two plates, there appears a single peak in these curves. The corresponding frequencies to the peak are , 3.2 and 3.3 for , 1/2 and 1, respectively. When the water depth ratio decreases, the peak value of the mean surface elevation also decreases. For instance, the peak value with is about 10% smaller than that for . Therefore, in case of using the present two-plate structure as oscillating water column devices, the uneven bottom is likely to reduce the energy capture.

Effect of step location
show the effect of the step location on the reflection coefficient and the dimensionless horizontal wave forces on the front and rear plates and , respectively. Three step locations, i.e., , 0.5, and 0.75 are considered in which represents that the step is located right in the middle of the two plates. As shown in Fig. 8, the effect of the step location on the reflection coefficient is prominent only for waves of the dimensionless frequency range Fig. 6. Comparison of the dimensionless horizontal wave forces on the front and rear plates under the same conditions as in Fig. 5.   Fig. 7. Dimensionless mean surface elevation in between the two plates under the same conditions as in Fig. 5. . Moreover, the influence of the step location on both the dimensionless wave forces acting on the two plates is found negligible, as shown by Fig. 9. For instance, the variation of the peak values of the horizontal wave forces on the rear plate (shown in Fig. 9b) is within 2% for the three step locations. As seen in Fig. 10, the location of the step shows a moderate effect on the mean surface elevation in between the two plates. When the step is getting closer to the rear plate, i.e. increasing the value of , the mean surface elevation becomes higher. Compared with the case of , the mean surface elevation gives a 6% increase for the case of . This reveals that the step location could slightly change the wave fluctuation in between the two plates. Figs. 11-12, the reflection coefficient and the dimensionless horizontal wave forces on the front and rear plates and are plotted as a function of the dimensionless wave frequency for different incident wave angles , respectively. As seen in Fig. 11, the reflection coefficient monotonically increases with the dimensionless wave frequency for all these incident wave angles ranging from -. Given a certain incident wave frequency, the reflection coefficient decreases with the increase of the incident wave angle. This is simply because x the present barrier can only reflect the component of the incident wave, which becomes smaller with the increase of the incident wave angle. For the same reason, the principal effect of the incident wave angle shifts the dimensionless wave frequency corresponding to the peak in the wave force profiles towards higher values with large incident wave angles, which can be observed from Fig. 12. Fig. 13 shows the mean surface elevation between the two plates with different incident wave angles. As seen in this figure, both the peak value of the mean surface elevation and the corresponding dimensionless wave frequency increase with the increase of the incident wave angle. When the incident wave angle reaches , the peak mean surface elevation is about 1.76 times of the amplitude of the incident wave. This means that the required deck level should be larger than this peak in order to avoid wave splashing in between the two plates. . This is because for this spacing, the fundamental standing wave motions could occur within the Fig. 9. Comparison of the dimensionless horizontal wave forces on the front and rear plates with different step locations, other conditions are the same as Fig. 8.   Fig. 10. Comparison of the dimensionless mean surface elevation in between the two plates with different step locations under the same conditions as in Fig. 8. . 11. Reflection coefficient as a function of for different incidence angles . In all cases, , ,

Effect of plate width
WANG Li-xian et al. China Ocean Eng., 2018, Vol. 32, No. 5, P. 524-535 531 w/h 1 = 0.75 considered range of dimensionless wave frequency. This perhaps leads to the fluctuation of the reflection coefficient with (Shin and Cho, 2016).
In Fig. 15, the dimensionless horizontal wave forces on the front and rear plates and are plotted with different plate spacings . When the plate spacing is smaller than 0.75, there appears a single peak in the wave force profiles within the present frequency range . As the plate spacing widens up to , two peaks are shown in the wave force profile of the rear plate (see Fig. 15b). Although only a single peak is presented in the wave force curve of the front plate shown in Fig. 15a, one can expect another peak by extending the w/h 1 computation range of the dimensionless wave frequency. These peaks presented in both the horizontal wave forces on the two plates are due to the resonant motions occurring at wavelengths corresponding to the existence of standing waves in between the two plates. Fig. 16 shows the comparison of the mean surface elevation between the two plates with different plate spacings ( ). It is interesting to see that the overall patterns of these curves are similar to the wave force profiles of the rear plate. Further investigation is necessary to clarify the reason behind this similarity.

Effect of plate drafts
In Fig. 17, the reflection coefficient is plotted as a function of the dimensionless wave frequency with α Fig. 12. Comparison of the dimensionless horizontal wave forces on the front and rear plates for different incidence angles under the same conditions as in Fig. 11. Fig. 13. The variation of the mean surface elevation against for different incidence angles under the same conditions as in Fig. 11. ). When the two plates are submerged with close drafts, there exists a local minimum of (McIver, 1985;Roy et al., 2016). When the difference of the two drafts and becomes pronounced, the reflection coefficient of the two-plate structure is close to that for the plate of larger draft in isolation. Similarly, a local minimum appears in the curve of the reflection coefficient with , as shown by Fig. 18. Among the three drafts of the rear plate, this draft is the closest one to the draft of the front plate ( ). Fig. 19 shows the variation of the dimensionless horizontal wave forces on the front and rear plates and with different drafts of the front plate . As seen from Fig. 19a, the horizontal wave force acting on the front plate shows a significant increase with the draft of the front plate . This is simply because there is more submerged area of the front plate to withstand the wave motions with larger . On the other hand, the draft of the front plate d 1 shows its influence on the horizontal wave force acting on the rear plate with the frequency range , as depicted in Fig. 19b. Within this high frequency range, the wave force acting on the rear plate decreases with the draft of the front plate, which is just opposite to that of the front plate. In addition, the change of the draft of the front plate tends to alter the dimensionless wave frequency corresponding to the peak of the wave force acting on the two plates. This is because that the resonant wave motions in between the two plates are affected by the drafts of the two plates. The draft of the rear plate d 2 shows a similar effect on the wave forces on the two plates as the draft of the front plate . As seen from Fig. 20a, the change of shows a slight influence on the peak wave forces on the front plate while it significantly alters the dimensionless Fig. 16. Comparison of the dimensionless mean surface elevation in between the two plates under the same conditions as in Fig. 14.  Fig. 19. Comparison of the dimensionless horizontal wave forces on the front and rear plates for different drafts of the front plate ( ) and under the same conditions as in Fig. 17. wave frequency corresponding to the peaks. As expected, the increase of substantially enlarges the horizontal wave force acting on the rear plate, which is clearly shown in Fig. 20b.
As shown in Figs. 21 and 22, the draft of the two plates leads to an obvious shift of the dimensionless wave frequency corresponding to the peak of the mean surface elevation in between the two plates. This frequency becomes smaller with the increase of the draft of the two plates. By changing the draft of the front plate from = 0.05-0.15, the peak value of the mean surface elevation in between the two plates only shows about 13% increase (see Fig. 21). Moreover, when the draft of the rear plate increases from 0.15-0.25, the peak value of the mean surface elevation gives about 22% increase, which can be observed from Fig. 22. It seems that the wave confined by the present two-unequal-plate structure seems more influenced by the rear plate.

Conclusions
The analytical study on the interaction between oblique ocean waves and two unequal surface-piercing thin vertical rigid plates, with the stepped bottom topography, is developed under the linear water-wave theory. By employing the matched eigenfunction expansion method and a least square approach, the effects of various parameters, such as the water depth ratio of deep to shallow waters, the step location, the incident wave angle, the plate spacing, and the drafts of the two plates on the reflection coefficient, the horizontal wave force acting on the plates, and the mean surface elevation between the two plates, are numerically examined under a variety of incident wave conditions. The major conclusions are made.
(1) The effect of stepped bottom on the reflection coefficient of the present two-plate structure is evident only with waves of the low dimensionless frequency.
(2) Compared with other parameters, the influence of the step location on the hydrodynamic performance of the present two-plate structure is slight.
(3) With the wide plate spacing, there may exist several peaks in the wave force profiles due to the resonant wave motions in between the two plates.
(4) When the two plates are submerged with close drafts, there exists a local minimum in the curve of the reflection coefficient varying with the dimensionless wave frequency.
(5) The mean surface elevation in between the two plates is found more influenced by the rear plate of larger submergence. ) and under the same conditions as in Fig. 18. Fig. 21. Comparison of the mean surface elevation in between the two plates for different and under the same conditions as in Fig. 17. Fig. 22. Comparison of the mean surface elevation in between the two plates for different and under the same conditions as in Fig. 18.