Study on wave dissipation of the structure combined by baffle and submerged breakwater

This paper proposes a structure combined by baffle and submerged breakwater (abbreviated to SCBSB in the following texts). Such a combined structure is conducive to the water exchange in the harbor, and has strong capability on wave dissipation. Our paper focuses on the discussion of two typical structures, i.e., the submerged baffle and rectangular breakwater combined with the upper baffle respectively, which are named as SCBSB1 and SCBSB2 for short. The eigenfunction method corrected by experimental results is used to investigate the wave dissipation characteristics. It shows that the calculated results agree well with the experimental data and the minimum value of the wave transmission coefficient can be obtained when the distance between the front and rear structures is from 1/4 to 1/2 of the incident wave length.


Introduction
The wave propagation of submerged breakwater or baffle structure for wave dissipation is a traditional research topic, and there have been some new research findings in recent years (Ting et al., 2016;Driscoll et al., 1992;Wang et al., 2011). Research and engineering practical applications show that since single structural form is affected by the fluctuation of water level, the effect of wave attenuation varies greatly, especially in the coastal area with large tidal range, which has been disturbing engineering designers all the time. The wave dissipation structure combined by baffle and submerged breakwater proposed in this paper has good adaptability to the changes in the water level. Some design parameters of this new type of structure, including the water depth of the crest, water penetration depth of the baffle and distance between two structures, exert a direct impact on the wave dissipation performance. However, there have been few application of this structure and no systematic research findings have been obtained so far.
There are many calculation methods to evaluate the wave dissipation effect for different structures. Especially with the improvement of computer technology in recent years, various discrete methods such as finite element meth-od and boundary element method, VOF method and SPH method in free surface processing, and different N-S based high-order nonlinear studies are also emerging constantly. However, most of these methods consume much computing time and occupy massive memory. Although some early analytical forms can only solve the interaction between simple structure and water wave, their physical concepts are clear and computations are fast. Kreisel (1949) firstly established a relatively complete eigenfunction theory. Newman (1965) studied the propagation of waves over an infinite step, and the wave scattering by a cylinder is also one of a few problems which can find analytic solution. Both Mei and Black (1969) and Massel (1983) have studied the submerged rectangular structure by the method of eigenfunctions. Losada et al. (1992) established a functional with mixed boundary condition by the extended function method, and solved the boundary conditional functional by the least square method. They theoretically solved the potential function of the vertical baffle in three kinds of wave dissipation structures, and eventually obtained the theoretical solution of the transmission coefficient of the vertical baffle. However, their calculation method is comparatively complicated. Kriebel and Bollmann (1996) modified the calcula-tion method of Losada et al. and proposed a new expression of mixed boundary condition, which had no clearly physical meaning. Shen et al. (2006) study the scattering of linear water waves by an infinitely long rectangular structure parallel to a vertical wall in oblique seas, the expressions for wave forces on the structure are given, and the calculated results are compared with those obtained by the boundary element method.
In terms of the SCBSB proposed in this paper, the eigenfunction method is established to solve the equations. Physical model experiments are conducted to compare the calculation results with experimental results for single upper baffle structure, single lower baffle structure, single rectangular structure and combined structure, and to put forward the calculation method of nonlinear correction eigenfunction, which calculates the transmission coefficient of the SCBSB.

Combined wave dissipation structure
Submerged breakwater is one kind of offshore structure widely used in waterway and offshore engineering. Its main functions are wave dissipation, sand barrier and river diversion, etc. The baffle type wave dissipation structure is often used in port engineering. Different from the underwater submerged breakwater, this structure usually adopts the pile foundation structure as the support, and installs the horizontal platform to construct the framed bent. Vertical baffles are arranged near the water surface to dissipate waves. These two wave attenuation structures have common features, namely, incomplete closure in space and free water exchange between two sides of the breakwater. However, these two structures also have their own shortcomings, i.e., the submerged breakwater has increasing wave transmission with the increase of the water level, meanwhile, the baffle structure has gradually weakening wave attenuation performance with the decrease of the water level.
The combined structure is aimed to utilize the advantages of submerged breakwater and baffle structure and to avoid their disadvantages. Therefore, the combined wave dissipation structure is mostly equipped with a submerged breakwater in the front and a baffle structure at the rear. The specific structure can be varied. For the convenience of the research, two representative structures are selected. The SCBSB1 is a combined structure of two baffles as shown in Fig. 1. The SCBSB2 is composed of a rectangular submerged breakwater and a baffle as shown in Fig. 2.

Calculation of the SCBSB by the eigenfunction method
According to the linear wave theory, the wave number k is the solution of the dispersion relation, including one real number solution and infinite imaginary number solutions: σ k n where, is the wave frequency, is the wave number, and h is the water depth.
Under a linear condition, the eigenfunctions of the wave equation solution are: where, g is the acceleration of gravity. As illustrated in Fig. 2, the SCBSB2 is divided into four regions, namely, Φ 1 , Φ 2 , Φ 3 and Φ 4 . The eigenfunction expression of the velocity function for each region is: x ⩽ 0 (1) Region Φ 1 : B n k n where, A 1 is the amplitude of the incident wave, is the amplitude of the reflected wave for different .
C n D n where, and are the amplitude of the transmitted and reflected wave for different k n , respectively. b E n F n where, and are the amplitude of the transmitted and reflected wave for different k n , respectively.
x > b 2 (4) Region Φ 4 : is the amplitude of the transmitted wave for different k n .
The analysis of the boundary conditions in different regions results in the following relations:  (1) Boundary of region Φ 1 and Φ 2 , i.e. x=0: ϕ 1 = ϕ 2 Use boundary condition to figure out: Use boundary condition to obtain: where, and are the coordinates of the x and z axis respectively at the right side of the rectangular as shown in Fig. 2.
(2) Boundary of region Φ 2 and Φ 3 , i.e. x=b 1 : Use boundary condition to where, and are the coordinates of the x and z axis respectively at the right side of the baffle as shown in Fig. 2.
(3) Boundary of region Φ 3 and Φ 4 , i.e. x=b 2 : Use boundary condition (12) With the orthogonality of the function I n , the boundary conditions of each group can be solved separately to obtain the coefficients B n , C n , D n , E n , F n and G n one by one. After reducing the influence of reflections from each other by multiple calculations, the final transmission coefficient of the structure can be obtained.
For the SCBSB1 shown in Fig. 1, the region Φ 2 can be eliminated and only three regions are existed, so that the above-mentioned equations can be simplified accordingly.

Physical model experiment
The eigenfunction method presented in Section 3 can calculate the wave potential function of the single dike structure and the SCBSB to further obtain the wave transmission coefficient after the structure. However, the eigenfunction method is based on the linear theory, resulting difference between the calculated and actual result. To improve the accuracy of the result calculated by the eigenfunction method, it is required to conduct physical model experiment for comparison and nonlinear correction.
The experiment is carried out in a water tank with 66 m in length, 1.8 m in width and 1.8 m in height. Fig. 3 shows the arrangement of the wave-height gauge and diagram of the physical model.
The experimental water depth h is fixed at 0.5 m. By changing the water depth on the crest of the submerged breakwater, the crest width and the underwater penetration depth of the baffle, the effect of these factors on the wave transmission coefficient is investigated. In order to study the influence of the wave length and wave height on the transmission coefficient, the experimental wave period T is assumed to be 0.637, 0.76, 0.905, 1.02, 1.22, 1.56 and 2.0 s. The experimental wave height H ranges from 0.02 to 0.10 m.
The experimental structures include single upper baffle, single lower baffle, single rectangular structure, SCBSB1 and SCBSB2.
The crest relative water depth d 1 /h for the experimental scenarios of the lower baffle structure is 0.125, 0.25, 0.375 and 0.5, respectively.
The crest relative water depth d 2 /h for the experimental scenarios of the upper baffle structure is 0.125, 0.25, 0.375 and 0.5, respectively.
The crest relative water depth d 2 /h for the experimental scenarios of the rectangular submerged breakwater is 0.125, 0.25, 0.375 and 0.5, respectively. Under the condition of d 2 /h=0.125, the crest relative width B/L=0.1-0.2.
Regarding to the SCBSB1, the experimental scenarios for the relative span b/h =0.4-5.7.

Case of upper baffle structure
The main structural parameter of the upper baffle structure is the relative underwater penetration depth d 1 /h. Fig. 4 depicts the ratio (K t ′/K t ) of the calculated wave transmission coefficient (K t ′) to the experimental one (K t ) changed with kh when d 1 /h=0.125, 0.25, 0.375 and 0.5. Mainly due to the influence of the dispersion parameter μ=h/L (L is the wave length), the experimental result is slightly smaller than the calculation result.  5 presents K t ′/K t changed with the shallow water parameter ε=H/h (H is the wave height) when kh=1.5 and 2.5, respectively. Similarly, due to the influence of ε, the experimental result is generally smaller than the calculated one.
Based on the transmission coefficient calculated value and the model experiment result of the upper baffle structure, the least squares method is used to fit the dispersion parameter μ=h/L, the shallow water parameter ε=H/h and the baffle underwater penetration depth d 1 /h respectively to obtain the following correction relations for the calculation of the transmission coefficient of the upper baffle by the ei-genfunction method: Eq. (13) shows that the influence of d 1 /h is relatively small. The comparison of the corrected transmission coefficient ( ) calculated by the eigenfunction method with the experimental value is shown in Fig. 6. The comparison result proves that, except for the case of larger transmission coefficient, in which the experimental result is slightly smaller than the calculated one, the other cases (while ) are in good agreements.

Case of lower baffle structure
The main structural parameter of the lower baffle structure is the crest relative water depth d 2 /h. Compared with the upper baffle structure, the lower baffle structure has a more obvious wave transmission phenomenon. Fig. 7 shows the value of changed with kh when d 2 /h of the lower baffle structure is 0.125, 0.25, 0.375 and 0.5, respectively. Similarly, the experimental result is slightly smaller than the calculated one due to the influence of the dispersion parameter μ.
Based on the calculated and experimental results of the transmission coefficient of the lower baffle structure, the    least squares method is used to fit the influence of the dispersion parameter μ=h/L, the shallow water parameter ε=H/h and the relative water depth on the baffle top d 2 /h to obtain the following correction relations for the calculation of the transmission coefficient of the lower baffle by the eigenfunction method (due to small influence of d 2 /h, it has been neglected): The comparison of the corrected transmission coefficient ( ) calculated by the eigenfunction method with the experimental value is presented in Fig. 8. The fitting result coincides well.

Case of the rectangular submerged breakwater
Compared with the lower baffle structure, the rectangular structure mainly aims to enhance the influence of the breakwater crest width B. In the case of shallow water depth, the nonlinear effect of wave propagation through submerged breakwater becomes more noticeable, and the energy of partially transmitted fundamental frequency wave shifts to that of the higher order frequency wave, resulting in the reduction of fundamental frequency wave energy. When B/h=1 and d 2 /h of the rectangular submerged breakwater are 0.125 and 0.25, 0.375 and 0.5, respectively, the calculation result of the rectangular submerged breakwater is corrected by Eq. (14) and compared with the experimental result as shown in Fig. 9. For the sake of comparison, the transmission coefficient of the lower baffle structure calculated by the corrected eigenfunction method with the same water depth on the breakwater crest are also added in Fig. 9.
With different water depths on the crest of the rectangular submerged breakwater, the experimental result is compared with the calculated result corrected by Eq. (14). In the case of d 2 /h=0.125, there still exist differences between the  result calculated by the corrected eigenfunction method and the experimental result. Under other conditions, these two are closed. In terms of d 2 /h=0.125, the experiment is conducted again with different crest width B from 0.1 to 1.1 m. Since the nonlinear effect of wave propagation is significant through submerged breakwater, the transmission coefficient after the breakwater corresponding to the incident frequency is analyzed and compared with the calculation result as shown in Fig. 10.
It can be seen through comparison that with the increase of the incident wavelength, the increase of the transmission coefficient after the breakwater is significantly reduced corresponding with the increase of B. The exponential relationship is used to fit the relationship between the transmission coefficient after the breakwater and B to obtain: L ′ where, is the wave length upon the rectangular breakwater, and B is the width of the rectangular.

Comparison of the corrected transmission coefficient of SCBSB with the experimental result
6.1 Correction the transmission coefficient of the SCBSB According to the correction relationship obtained from the experiment of single breakwater, waves propagate through single breakwater structure separately in the SCBSB. In our opinion, the total correction relationship is the superposition of the correction relations of each single breakwater. For the SCBSB1: where, and are the wave transmission coefficient and corrected wave transmission coefficient of the SCBSB1 calculated by the eigenfunction method, respectively; A 1 and A 2 are the correction relation of Eq. (13) and Eq. (14), respectively. Eq. (16) can be expressed as follows: For the SCBSB2: where, A 3 are the correction relation of Eq. (15). Eq. (18) can be presented as follows: in which, kh is not smaller than 0.8, H/h is smaller than 0.2, and d 2 /h is not smaller than 0.25. 6.2 Case of SCBSB1

Comparison between calculated and experimental value with different relative span b/L′
The SCBSB is composed of the upper baffle structure and the lower baffle structure, and the distance between them is an important parameter that affects the transmission coefficient. The correction formulae of the eigenfunction calculation in Sections 5.1 and 5.2 with respect to the upper baffle structure and the lower baffle structure is used to correct the calculated result of the SCBSB1. The corrected calculation result is compared with the experimental result. Under the condition of kh=1.05 or 2.0, d 1 /h=0.25 and d 2 /h=0.5, the comparison of the calculated transmission coefficients with experimental result is shown in Fig. 11 with different relative span.
The calculated and experimental results both show that the wave transmission coefficient after the breakwater fluctuates periodically with different relative span b/L. The wave transmission coefficient is small when b/L is about 0.5 or its integral multiple. The reason why b/L is not exactly equal to 0.5 when the transmission coefficient is minimum is that the phase will change after wave passing through the baffle structure. On the contrary, when the value of b/L is located in the range in which exists two minimum transmission coefficients, the transmission coefficient is the largest. Therefore, the SCBSB should be designed to avoid this situation based on characteristic wave conditions.

Comparison between calculated and experimental results with different kh
Under the following two operating conditions, the corrected eigenfunction calculation method is used to calculate the transmission coefficient of the SCBSB1 and the result is compared with the experimental one as shown in Fig. 12. During the comparison, the experimental data are backward processed by use of the shallow water correction coefficient.

Case of SCBSB2
Under the following two operating conditions, the corrected eigenfunction calculation method is used to calculate the transmission coefficient of the SCBSB2 and the result is compared with the experimental one as shown in Fig. 13. Eq. (19) is selected to correct the influence of the submerged breakwater width under the operating condition (2).
7 Analysis on the wave attenuation performance of the SCBSB Compared with single breakwater, the SCBSB has a better adaptability to the change of water level change in terms of the wave attenuation performance. Furthermore, the span of combined breakwaters has a greater impact on the wave height after the breakwater. The corrected eigenfunction method is used to calculate the transmission coefficient of the SCBSB and the influence of the span and water level change on the wave attenuation performance of the SCBSB is investigated.

Influence of the span
In the two cases of (a) d 1 /h=0.125 and d 2 /h=0.5 and (b) d 1 /h=0.25 and d 2 /h=0.5, the change rules of the wave transmission coefficient with different span of the upper and lower baffles are illustrated in Fig. 14. When the crest relative width b/h of the SCBSB2 is 1.0, the change rules of the wave transmission coefficient are presented in Fig. 15. As the span increases, the transmission coefficient increases gradually, reaches the maximum and then decreases progressively which represents the fluctuation characteristics. Hence, the influence of the span of the SCBSB on the transmission coefficient varies from the smallest to the largest for a specified wave period.

−π/4
When waves propagate through the SCBSB, they will mutually reflect among the SCBSB. If there is no phase mutation in the reflected waves, when the span between two breakwaters is L/4, the waves of secondary reflection through the SCBSB will share the same phase with the incident wave, and the transmission coefficient of mutual superposition is the largest. However, in fact, there exists the phase change when the waves are reflected through the upper baffle and the lower baffle. When the relative water depth d 1 /h on the crest of the lower baffle is increased from 0 to 1, the angle of the phase change of the reflected wave gradually changes from 0 to ; when the relative underwater penetration depth of the upper baffle structure d 2 /h is increased from 0 to 1, the angle of the phase change of the −π/4 −π/2 reflected wave changes from to 0. Therefore, there is no abrupt change in the phase of the reflected wave when d 1 /h≈0.0 and d 2 /h≈1.0. The mutation change angle is about at d 1 /h≈1.0 and d 2 /h≈0.0. Fig. 16 depicts the relationship between the wave transmission coefficient and the span for the SCBSB1 in the two extreme cases. Thus, the span is ranged from L/4 to L/2 for a specified incident wave when the transmission coefficient of the SCBSB is the minimum.
7.2 Influence of the water level Fig. 17 depicts the comparison of the transmission coefficient among the SCBSB1, the single upper baffle and the single lower baffle. The comparison conditions are as follows: d 1 /h 0 =0.25 and d 2 /h 0 =0.5 for the single upper and lower baffle, respectively; b/L=0.4 and kh 0 =2.5; h/h 0 is changed from 0.5 to 1.0. It can be seen through the comparison that the wave attenuation performance of the SCBSB will not increase monotonically with the increase or decrease of the water level, and will be significantly improved by the comparison with single structure. As a result, the SCBSB is suitable for the sea with large tidal change.

Influence of the structure type
Owing to the influence of the rectangular submerged breakwater on the nonlinear effect of wave propagation in the SCBSB2, the wave transmission coefficient decreases with the increase of the crest width. This phenomenon occurs more apparently when the water depth on the crest of  JU Lie-hong et al. China Ocean Eng., 2017, Vol. 31, No. 6, P. 674-682 681 the breakwater is small and the incident wavelength is large. The comparisons between the SCBSB1 and SCBSB2 in the wave transmission coefficient changed with kh are shown in Fig. 12 and Fig. 13, respectively. With d 1 /h=0.125, d 2 /h= 0.25, b/h=1 and the crest relative width of the rectangular submerged breakwater B/h of the SCBSB2 is 1, when kh is between 2.0 and 3.0, the transmission coefficient of the SCBSB2 is about 0.4, while the transmission coefficient of SCBSB1 is about 0.5; when kh is between 1.0 and 2.0, the transmission coefficient of the SCBSB2 and SCBSB1 are about 0.5 and 0.7, respectively. Therefore, for long-period waves, the SCBSB2 has relatively better wave attenuation performance, and in addition, the wave attenuation performance is continuously improved with the increase of the crest width of the submerged breakwater.

Conclusions
The wave dissipation performance of the structure combined by the baffle and submerged breakwater is investigated by physical model experiment and nonlinear corrected eigenfunction method. The main conclusions can be drawn as follows.
(1) As the phase of reflected wave of the SCBSB varies correspondingly with the change of the relative water depth on the breakwater crest and the relative underwater penetration depth of the baffle, the minimum value of the wave trans-mission coefficient can be obtained when the distance between the front and rear structures is from 1/4 to 1/2 of the incident wave length.
(2) Compared with the single baffle structure or submerged structure, the SCBSB has a better adaptability to the water level change and significantly improved wave attenuation capability.
(3) The wave transmission coefficient of the SCBSB2 decreases with the increase of the submerged breakwater width, and its performance of reducing wave transmission also improves continuously. Therefore, the SCBSB presents better wave attenuation performance.