Submerged breakwaters formed by natural rocks dissipate the incident wave energy. Continuous quarrying of rocks has resulted in its depletion, leading researchers to look for alternate materials for the formation of such barriers. Thus, the concept of artificial reef units has evolved which has been gaining importance owing to the flexibility in molding to any desired shapes and sizes, workability and also due to the fact that it creates a habitable environment to marine flora and fauna. From the hydrodynamic performance perspective, artificial reef units are proven to be more efficient in reducing wave transmission on the lee side (e.g., Southern Caribbean shore of Dominican Republic and Vaan Island, Tuticorin, India). A comprehensive experimental investigation to examine the effect of trench width on hydrodynamic characteristics of the submerged reef system was carried out. The trench width was varied by incorporating single, double, and multiple perforated submerged trapezoidal artificial reef units. The focus of the present study is mainly on the influence of the number of reef units, relative crest width, and relative trench width of the submerged reef system on its transmission and reflection characteristics. The relative trench width was found to be an influential factor on wave transmission past the structure, and the least wave transmission was achieved for the reef configuration with eight units. The details of the experimental investigation, results and discussion are reported in this paper.
The dynamic changes in the shoreline are due to the erosion and accretion of sediments which may be a short- or a long-term phenomenon. The erosion around islands and along the coast is caused by the natural processes such as waves, tides, currents and storms, as well as due to human intervention in the form of installation of coastal structures and excessive mining of coral reefs. The choice of soft or hard measures to protect the coast/island from erosion depends on parameters such as wave climate, sediment dynamics, structural suitability, landscape, and ecology at the site of interest [1,2]. Seawalls, bulkheads, revetments, breakwaters, and groynes are a few predominantly adopted hard measures. Natural rocks, concrete, and steel are commonly used as construction materials for the aforesaid structures. The scarce availability of natural rocks and the difficulty associated with transporting the same to remote locations has inspired researchers to seek alternative materials. Therefore, eco-friendly materials such as geo-textile bags, geo-textile, sand containers, Reef Ball units and prefabricated reef units are being adopted in submerged structures as soft measures against coastal erosion [3–5]. These structures are used in maintaining tranquility as well as in reducing siltation inside ports and harbors, thereby providing safe navigation and berthing. They also facilitate the exchange of water mass between the offshore and onshore regions, thus reducing environmental and beach pollution [6–8]. Artificial reefs are man-made structures installed on the seabed below the still water level to serve as wave attenuators. The efficiency of such reef units depends on their shape, size, and material. The installation of such reef units was demonstrated along the Southern Caribbean shore of Dominican Republic  and Vaan Island, Tuticorin, India . A series of artificial reefs were arranged in eight numbers one behind the other along the stretch of Vaan island to form a detached breakwater system which reduces the erosion of sand and also works for the rehabilitation of coral reef in order to enhance the marine biodiversity.
The laboratory studies by Ref.  demonstrated the influence of relative crest width (crest width/wavelength, B/L), relative water depth (water depth/wavelength, d/L), relative crest height (submerged breakwater height/water depth, h/d), wave steepness, shape, slope, and porosity of the structure on the hydrodynamic performance of submerged breakwaters. The influence of parameters such as breakwater dimensions, its orientation and placement with reference to the shore, approaching wave angle and wave climate dictates in deciding the extent of coastal protection .
Considering parameters such as B/L and relative depth of the crest level of breakwater (depth of submergence of breakwater/incident wave height, ds/Hi), a set of design curves was proposed for wave transmission past submerged breakwaters subjected to regular waves [10–13]. It was concluded that transmission coefficient Kt (wave height past the structure/incident wave height) less than 60% was accomplished for B/L greater than 0.6 and ds/Hsi greater than −1.3. The field observations and experimental studies conducted by Refs. [14–16] on submerged breakwater, and artificial reef reported that Kt < 60% was achieved for B/Lp of 0.5–0.9 and ds/Hsi greater than −1.25 (where Lp is the peak wave period and Hsi is the significant wave height).
The hydrodynamic characteristics of semicircular submerged breakwater subjected to regular and random waves were reported by Ref. . The tests were conducted for three different water depth (d) with a degree of submergence (water depth/model height, d/h) of 1, 1.2, and 1.4 and porosities with 7%, 11%, and 17% of the surface area of the model. For a better hydrodynamic performance of semicircular breakwater, the percentage of perforations and d/h recommended were 11% and 1.2, respectively. A significant reduction in the incident wave energy was reported by Ref.  due to a submerged breakwater with d/h range between 1.25 and 1.42.
The wave transmission and reflection characteristics of twin submerged structures were reported by Refs. [19–22]. The parameters considered in their studies include, spacing between the breakwater varied in terms of B/d (total crest width/water depth) and S/L (spacing between the breakwater/wavelength). It was inferred that Kt decreased and Kr increased for the twin breakwaters compared to a single breakwater. It was stated that the reflection observed for single solid barrier and double slotted barrier was found be lesser compared to a double solid barrier .
The experiments were carried out by Ref.  on a submerged reef by changing the shape of the crest as conventional type, saw blade type, wave profile type, and vertical type to evaluate the wave transmission. It was concluded that Kt for a submerged reef with wave profile type and saw blade type were found to be less compared to the conventional type. It was reported that the wave steepness, depth of submergence, and reef geometry influence the reduction in the incident wave height . An analytical solution was developed by Ref.  to study the hydrodynamic parameters of submerged porous bars placed in front of a vertical wall. It was found that an increase in the number of submerged bars significantly reduced the load on the vertical wall. The gap between the submerged bars, reefs placed on the submerged breakwater, submerged reef with a crest shape of saw blade type, wave profile type and vertical type correspond to the characteristics of trench between the submerged structures.
From the review of literature, it is evident that considerable researches on hydrodynamic characteristics of a submerged structure in isolation or in combination of another in its vicinity have been reported. However, the effect of trench width between the submerged structures has not been extensively studied in the past. The selection of an optimum configuration of artificial reef units and trench width between the reefs would be the key factors in reducing the wave transmission significantly through wave damping. Thus, a comprehensive experimental study on multiple artificial reef units is taken up to examine the hydrodynamic characteristics of the reef sections. The effect of trench width is studied by varying the number of units to optimize the reef configuration. An isometric view of eight reef units considered for the present study is shown in Fig. 1.
2 Experimental Procedure
2.1 Testing Facility and Model Fabrication.
A series of experiments were conducted in the shallow water wave flume at the Department of Ocean Engineering, Indian Institute of Technology Madras, India. The wave flume is 72.5 m long, 2 m wide, and 2.7 m deep with a wave maker at one end and rubble mound wave absorber at the other end. The wave maker can be operated with a piston or hinge mode, which allows the generation of shallow and deep water waves, respectively. A row of slotted and perforated pipes were installed behind the wave maker to absorb the wave energy. The maximum operating water depths for piston and hinge mode are 1 m and 2 m, respectively. The wave maker being computer controlled is able to generate regular and random waves with different wave heights, wave periods, and pre-defined spectral characteristics. The desired wave maker control signal is generated on a personal computer and sent to the hydraulic servo actuator which in turn governs the motion of the wave paddles. The longitudinal plan and sectional view of the experimental model setup along with the location of the wave gauges are shown in Fig. 2.
The physical scale hydraulic model should represent the behavior of the prototype to a maximum possible extent. The Froude number is the major scaling criterion for free surface flow problems in coastal engineering. Froude similarity is considered for open channel flows and hydraulic structures, since gravity plays an important role in free surface flows . The uncertainty involved in the scaled hydraulic models depends on the similitude criteria, scale factor, roughness, and dimensional analysis carried out in arriving the non-dimensional parameters. The largest model was chosen based on the scale factor which would possibly reproduce the highest degree of accuracy in the results obtained. The scale factor was selected based on the size of the prototype, size of the available testing facility and the limiting values of modelled wave parameters. An undistorted model scale of 1:5 is adopted based on dimensional limitations of the available testing facility. From the literature, the relative crest width (B/L) between 0.6 and 0.7, the degree of submergence (d/h) 1.25 and the porosity of 11% of the surface area of model are adopted for the present study to achieve better hydraulic performance [10–18]. An individual artificial reef unit (Fig. 3) is 0.4 m in height (h) with top and bottom widths (B and Bb) of 0.2 m and 0.5 m, respectively . The artificial reef units were fabricated using polypropylene sheets. A total of eight reef units were fabricated, and each of the fabricated reef unit was mounted on a rigid frame.
2.2 Model Setup and Instrumentation.
The reef model was set up over a flat, rigid false bottom, with a slope of 1:30 made-up of a steel frame and marine plywood. The perforated artificial reef unit models were placed at a distance of 36.65 m from the wave maker. The number of reef units is varied to arrive four sets of combinations such as a single unit, two units, four units, and eight units. The placement and arrangement of perforated eight artificial reef units in the wave flume using the crane mounted on the rails are projected in Fig. 4.
Four conductive-type wave gauges were employed in the experimental setup to register the instantaneous water surface elevations along the flume. The first three wave gauges (WG1, WG2, and WG3) were placed on the seaside with respect to the toe of the model at a distance of 10.5 m, 8.21 m, and 7.5 m, respectively. The signals from gauges WG2 and WG3 were used to decompose the composite wave elevation into the reflected and incident wave components using the two probe method of Ref. . The wave gauge WG4 was placed on the lee side of the model at a distance of 5 m from the toe of the model, to record the transmitted wave elevations. The transmission coefficient is obtained as the ratio of incident to transmitted wave height.
2.3 Wave Characteristics.
The perforated submerged single, double, and multiple trapezoidal artificial reef units in three different water depths of 0.5 m, 0.55 m, and 0.6 m were subjected to both regular and random waves. The models were exposed to the action of regular waves of period (T) ranging from 1.6 s to 5.8 s and random waves (Pierson–Moskowitz spectrum) with a peak wave period (Tp) ranging from 1.6 s to 3 s. For each wave period, two wave heights were considered in the testing. The wave heights (Hi) between 0.05 m and 0.3 m for regular waves and significant wave heights (Hsi) between 0.05 m and 0.2 m are adopted for the tests.
2.4 Dimensional Analysis.
Dimensional analysis has been used as an alternate solution, where the theoretical approach is associated with more difficulty in addressing the physical problem [27,29]. The hydrodynamic characteristics of the artificial reef units are governed by important parameters such as wave characteristics, water depth, and geometry of the reef. A typical section of two perforated reef units depicting the different physical parameters is shown in Fig. 5. The plots with typical time series of wave elevations recorded at different locations on the seaside and the lee side of the model are shown in Figs. 6(a) and 6(b) for regular and random waves, respectively.
The parameters which substantially affect the wave transmission and reflection are incident wave height (Hi), wavelength (L), crest width (B), trench width (Bt), reef height (h), water depth (d), depth of the crest level of reef below the still water level (ds), mass density of water (ρ), and acceleration due to gravity (g).
The non-dimensional parameters obtained from the dimensional analysis are relative crest width (B/L), relative water depth (d/L), degree of submergence (d/h), relative trench width (Bt/B), wave steepness (Hi/L), and relative depth of the crest level of reef (ds/Hi), whereas for the random waves, the non-dimensional parameters are expressed in terms of wavelength (Lp) corresponding to peak period (Tp) and significant wave height (Hsi). The range of non-dimensional parameters considered in the present study for both regular and random waves is given in the Tables 1 and 2, respectively.
3 Results and Discussion
Although, the present study covers a wide range of wave steepness, only typical results on Kt and Kr for a prescribed range of Hi/L and Hsi/Lp are discussed in this paper owing to the similarity in the trend observed for the entire range of the wave steepness adopted. However, the results for all the test conditions are included in arriving the percentage reduction in Kt, selection of number of reef units and for deriving an empirical relationship between different non-dimensional parameters.
3.2 Influence of Relative Crest Width on
Kt and Kr
3.2.1 Regular Waves
The variations of Kt and Kr with the relative crest width (B/L) have been obtained for three different degrees of submergences (d/h) of 1.25, 1.38, and 1.5. In general, the results on Kt and Kr with B/L for Hi/L = 0.003 to 0.033 show that an increase in the water depth increases Kt and decreases Kr for a particular reef height. The reduction in d/h initiates more dissipation, and as a result, a sudden change in the particle velocity is anticipated. The higher velocity particle engages in creating the voids and forms vortexes, building turbulence [4,13,30,31]. Further, the turbulence-generated aids in dissipating more energy leading to the reduction in Kt particularly for short period waves, as the long waves pass the submerged reef without being attenuated. The energy transmitted over the crest of the submerged reef is more compared with the energy transmitted through the reef, and as a result, Kt is found to be more for a higher value of d/h. Hence, d/h has a significant influence on deciding the hydrodynamic performance of the submerged reef.
22.214.171.124 Single and two reef units.
The trend in variations of Kt and Kr with B/L for Hi/L = 0.003–0.033 is shown in Figs. 7(a) and 7(b) for single and two reef units, respectively. For single reef unit with a relative trench width (Bt/B) of 0, it is observed that Kt decreases to about 0.5 with an increase in B/L of up to 0.04 and a further increase in B/L shows a marginal decrease in Kt. The variation in Kt with B/L up to 0.025 for d/h of 1.25 and 1.5 are found to be similar. This implies that d/h has a lesser influence on Kt for longer period waves compared with that of the shorter period waves. The range of Kt for d/h of 1.25, 1.38, and 1.5 are 0.51–0.85, 0.61–0.92, and 0.69–0.95, respectively. It is observed that for d/h of 1.38 and 1.5, Kr linearly increases up to 0.25, with an increase in B/L up to 0.03 and a further increase in B/L results in a sudden shift in the trend of Kr. Thus, it is inferred that d/h has a larger influence on Kr for shorter period waves compared to longer period waves. The range of Kr for d/h of 1.25, 1.38, and 1.5 are 0.46–0.08, 0.32–0.04, and 0.26–0.03, respectively.
The wave propagating through the perforated reef creates vortexes on the seaside, lee side as well as inside the reef as explained by Ref. . Due to the obstruction of the reef, a clockwise vortex was formed on the seaside. The wave passing through the pores of the reef creates two vortexes, one on the top in the clockwise direction and another at the bottom in the anticlockwise direction. This helps in absorbing a considerable amount of wave energy due to turbulence occurring inside the reef. The waves coming through the pores gain a higher velocity at the exit point resulting in the formation of two upwelling clockwise vortexes along with a few smaller ones which could disturb the flow pattern. As a result, an additional dissipation of wave energy is expected to take place.
The trend in variations of Kt being same for single and two reef units; however, the magnitude of Kt for the two reef units is found to be less compared to that for a single reef for d/h of 1.25–1.5. The range of Kt for d/h of 1.25, 1.38, and 1.5 are 0.32–0.7, 0.45–0.8, and 0.52–0.89, respectively. The increase in B/L up to 0.12 increases Kr to about 0.2, for d/h of 1.38 and 1.5, and a further increase in B/L leads to a slight increase in Kr and whereas, for d/h of 1.25, Kr is found to increase significantly up to 0.47. The increase in the reflection is observed due to the introduction of a second reef, and a similar phenomenon has been reported with an array of submerged breakwaters by Ref. . The range of Kr for d/h of 1.25, 1.38, and 1.5 are 0.47–0.14, 0.33–0.05, and 0.27–0.01, respectively. The phenomenon of wave breaking was experienced by Ref. , induced due to a pair of submerged barriers. The wave breaking induced by the first seaside reef was interfered by the second reef, and as a result, two waves involved in breaking were observed in the opposite direction to the incident wave. The wave on the lee side breaks prior to the wave on the seaside and further combines to form reflected waves toward the seaside.
126.96.36.199 Four and eight reef units.
The variations of Kt and Kr with B/L for Hi/L range between 0.003 and 0.033 with four and eight reef units are projected in Fig. 8, which exhibits a trend similar to that of a single and two reef units. Kt for a four reef unit system is found to be less compared to that of single and two reef units. Furthermore, the least Kt is found to be for eight reef units among the different setups of reef units considered in the study. Prior research on submerged breakwaters by Refs. [10–13] concluded that Kt < 60% was achieved for B/L greater than 0.6 and ds/Hi greater than −1.3. For the present study, the reef system with eight units was able to achieve Kt < 27%. The range of Kt for four reef units with d/h of 1.25, 1.38, and 1.5 are 0.25–0.67, 0.34–0.72, and 0.44–0.8, respectively. For the eight reef units, the range of Kt with d/h of 1.25, 1.38, and 1.5 are 0.1–0.45, 0.28–0.6, and 0.32–0.69, respectively.
It is noticed that, for four and eight reef units corresponding to all values of d/h, Kr is found to increase gradually with an increase in B/L. The range of Kr for four reef units with d/h of 1.25, 1.38, and 1.5 are 0.44–0.12, 0.38–0.07, and 0.31–0.04, respectively. Similarly, for eight reef units with d/h of 1.25, 1.38, and 1.5, Kr is found to be in the range of 0.47–0.2, 0.38–0.09, and 0.33–0.02, respectively.
The width of the crest and trench of multiple reef units has a significant influence on the wave transmission and play a vital role in evaluating its hydrodynamic characteristics. The variations of Kt, and Kr, as a function of d/L superposed for the different setups, (Bt/B) and Hi/L = 0.003–0.033, are shown in Fig. 9, which prominently reflects the effect of the number of units. Herein, the trench width between the units Bt is normalized by dividing with total width of the structure. This indicates the clear understanding of the influence of Bt/B on Kt.
The Kt is found to decrease with an increase in the Bt/B for all d/h tested. The increase in Bt/B creates turbulence and leads to wave damping thereby dissipating wave energy. The effect of Bt/B on Kr is found to be less significant for all d/h ratios considered. Among all the test cases, Kt is found to be less than 50% for d/h = 1.25 and Bt/B = 0.57, whereas Kr < 26% for d/h = 1.5 and Bt/B = 0. The increase in the Bt/B from 0 to 0.57 accentuates the friction and turbulence that is expected to occur between the crest widths associated with trenches when the wave propagates over the submerged perforated reefs. An increased area of contact associated with friction and turbulence enhances the dissipation of energy to a greater extent resulting in a reduction in the wave transmission significantly. The observations made by Ref.  states that the undulations (wave profile type and saw blade type) along the crest width of a submerged breakwater resulted in a reduced Kt compared to a conventional breakwater with the same crest width. It can be concluded that the surface undulations caused a similar effect to that of a trench width as in the present investigation.
3.2.2 Random Waves.
The plots of Kt and Kr under random wave incidence are presented similar to those of the regular waves as discussed earlier. The variations of Kt and Kr with B/Lp for Hsi/Lp = 0.007 to 0.033 for a single, two, four, and eight reef units are shown in Figs. 10(a)–10(d). The influence of d/h on Kt and Kr is analogous with the results of regular wave tests for Bt/B = 0–0.57 as discussed earlier. The effect of B/Lp on Kt is found to be less significant, whereas Kr is found to increase marginally with an increase in B/Lp for all values of d/h and Bt/B. The maximum Kt for a single, two, four, and eight reef units are found to be 0.72, 0.69, 0.58, and 0.39, respectively. Similarly, the minimum Kt is found to be 0.5, 0.46, 0.32, and 0.16 for a single, two, four, and eight reef units, respectively. The experimental and field investigations on submerged structures by Refs. [14–16] reported that Kt < 60% was obtained for B/Lp > 0.6 and ds/Hsi > −1.25, whereas the reef system with eight units due to random waves was able to attain Kt < 22% in the present study. The maximum and minimum Kr for single and double reef units are found to be 0.49 and 0.29, respectively. The maximum Kr for four and eight reef units is found to be 0.52, whereas the minimum Kr is found to be about 0.20. In random wave tests, for all values of Bt/B and d/h, Kt is found to be on the lower side due to an increase in the wave-damping action at various wave heights compared to that observed for regular waves. The high-frequency components of random waves contribute more energy dissipation as a result of which the Kr is observed to be greater than that of regular waves.
The variations of Kt and Kr with d/Lp for Hsi/Lp = 0.007–0.033 is shown in Fig. 11. The influence of Bt/B on Kt due to random waves for all the values of d/h considered is analogous with the regular wave test results. Furthermore, Kt is less than 50% for d/h ≤ 1.38 and Bt/B ≥ 0.53, whereas Kr < 36% is achieved for d/h = 1.5 and Bt/B = 0.
3.3 Overall Percentage Reduction in
The percentage reduction in Kt through its variation with d/L for Hi/L = 0.003–0.097 and with d/Lp for Hsi/Lp = 0.007–0.055 are presented in Figs. 12 and 13, respectively. The percentage reduction in Kt is evaluated for Bt/B = 0.43, 0.53, and 0.57 with reference to Bt/B = 0. In general, it is observed that the percentage reduction in Kt is lesser for long period waves, particularly for lower d/L in regular waves, whereas, in the case of random waves, this is observed to be almost uniform for the entire range of d/L tested. The percentage reduction in Kt for regular waves for Bt/B of 0.57 (eight units) is found to be maximum from about 45% to 80%, followed by that for a four unit reef system varying from 25% to 50%. For the random wave tests, the eight unit reef system with Bt/B of 0.57 is found to experience reduction to an extent of about 60% over the entire d/Lp, whereas for a four unit reef system the said reduction is only about 25%.
3.4 Selection of Number of Reef Units.
The selection of number of reef units is governed by the geometrical parameters of the reef as well as the environmental parameters. In the present study, a wave period range of 1.6–5.8 s (3.5–13 s in the field) and water depths from 0.5 m to 0.6 m (2.5–3 m in the field) are covered. It is essential to investigate the variations in Kt and Kr of the reef system, exposed to the prescribed wave climate. The deviations in Kt and Kr with Bt/B for the above-said wave periods, subjected to regular waves, are projected in Fig. 14 for a Hi/L range of 0.003–0.097. It is evident from the results that the magnitude of Kt drastically reduces with an increase in Bt/B, whereas only a marginal reduction in Kr is observed. For the longest wave period tested, Kt lesser than 67% could be achieved with a d/h = 1.25 and Bt/B ≥ 0.53. This phenomenon is attributed to the trenches lying between each of the perforated reef units that act as buffers in dissipating the wave energy to a larger extent which is not the same for the conventional type of submerged breakwater. The influence of the trench width between two submerged breakwaters has been reported earlier by Refs. [19–23,26,34]. From the aforementioned literature, it was concluded that Kt decreases and Kr increases for two or more submerged breakwaters with a specific spacing between them compared to that for a single submerged breakwater. Bt/B = 0.53 can be assessed by having four number reef units placed one behind the other with a total crest width (B) of 1.7 m. It is also noticed that for Bt/B = 0.57, Kt ≤ 50% can be achieved with d/h = 1.25. The combination of eight reef unit refers to an overall crest width (B) of 3.7 m for which Bt/B = 0.57. For the longer wave period, Kr is found to be less than 10% for Bt/B = 0.53 (four units) and Bt/B = 0.57 (eight units) with d/h = 1.5. Similarly, the variation of Kt and Kr with Bt/B due to random waves for the extreme wave periods (1.6 s and 3 s) tested are projected in Fig. 15 for Hsi/Lp = 0.007–0.055. For the longer wave period of 3 s corresponds to 6.7 s in the field, it is observed that with d/h = 1.25, Kt ≤ 40% and ≤ 22% are arrived for Bt/B = 0.53 (four units) and for Bt/B = 0.57 (eight units), respectively. It is also noticed that Kr ≤ 30% and ≤ 23% are obtained with d/h = 1.5 for Bt/B = 0.53 and Bt/B = 0.57, respectively. The reef configuration with eight units is found to be best among all based on its hydrodynamic performance.
3.5 Application of Submerged Reef System to the Field Conditions
The artificial reef system is functional to combat erosion problems around islands, with water depths varying from 3 m to 5 m, wave height less than 1.5 m and wave periods ranging from 5 s to 10 s. The design crest width, crest height, and allowable wave transmission are arrived for above-mentioned field conditions based on the present study and are given in the Table 3. In order to achieve Kt < 50% for the water depths of 3–5 m and periods 5–10 s, the design crest width and crest height of the reef system should be in the range of 6.5–47.5 m and 2.4–4 m, respectively. Similarly, to achieve Kt < 25%, for the above-stated design parameters, the crest width and height are found to vary in the range of 17.5– 77 m and 2.4–4 m, respectively.
3.5.2 A Submerged Reef System Implemented at Vaan Island, Tuticorn, India.
The artificial reef units adopted for the present study were deployed on the coasts of Vaan Island, Tuticorn, India . These units were placed to prevent the island degradation due to erosion while assisting the growth of corals on its surface. The deployed reef units near the Island successfully proved the efficiency of dissipating wave energy resulting in accretion of sand to a greater extent on the northern part of the island. The marine growth around the reef units has been observed within 5 months of its deployment.
3.6 Regression Analysis.
The measured wave transmission coefficients under the both regular and random waves were subjected to the multiple regression analysis based on the least square method . The equations for the dependent parameter Kt as a function of independent variables such as B, d, h, ds, Bt, Hi, L, Hsi, and Lp for four and eight reefs alone are projected below. The equations for prediction of Kt for breakwaters can also be expressed in logarithmic and exponentials forms 
3.6.1 Regular Waves.
The test ranges for the above equations are
3.6.2 Random Waves.
The test ranges for the above equations are
The wave transmission coefficients are predicted from Eqs. (3)–(8) and compared with the measured transmission coefficients. The measured and predicted Kt for four and eight reef unit model are projected in Figs. 16 and 17 for regular and random waves, respectively. Similarly, for reef units up to eight numbers, the measured and predicted Kt for regular and random waves are shown in Fig. 18. Based on the line of equality proposed, the measured and predicted Kt exhibits a good agreement. The correlation coefficients for the four and eight reef units exposed to regular waves are 0.95 and 0.93, respectively. Similarly, for random waves, the correlation coefficients are found to be 0.97 and 0.98 for four and eight reef units, respectively. The correlation coefficients for the reef units up to eight numbers for regular and random waves are found to be 0.95 and 0.94, respectively. The scatter plot (Fig. 18) of predicted and measured Kt for the entire reef system (one–eight reef units) shows a good correlation coefficient for the parameters log(B/L) and exp(BtB) compared to B/L and Bt/B.
4 Summary and Conclusions
The submerged artificial reef models with 11% perforation on the total surface area were subjected to regular and random waves in a flume. The regular waves of wave height ranging from 0.05 m to 0.3 m and wave period in the range of 1.6–1.5 s were tested, whereas, for random waves, the significant wave height ranging from 0.05 m to 0.18 m and peak wave period in the range of 1.6–3 s were tested. The range of wave steepness covered were 0.003–0.097 and 0.007–0.055 for regular and random waves, respectively. The relative crest width, relative trench width, and degree of submergence vary from 0.014–1.2, 0–0.57, and 1.25–1.5, respectively. The hydrodynamic performance of the submerged reef unit is evaluated experimentally using the transmission and reflection coefficients. The following salient conclusions are drawn from this study:
For a constant reef height, an increase in water depth increases Kt and decreases Kr for the entire range of relative crest width subjected to regular and random waves.
In general, the regular waves with longer wave periods have less influence on Kt compared to shorter wave periods for entire d/h ratio and vice versa in the case of Kr. However, the same effect is not seen with random waves.
For a particular d/h, Kt decreases to a larger extent with an increase in Bt/B and variation in Kr is found to be negligible.
Among all the regular wave tests, Kt lesser than 50% is achieved for d/h = 1.25 with eight reef units, whereas Kr < 26% is obtained for d/h = 1.5 for single reef unit.
For random wave tests, Kt lesser than 50% is derived for d/h ≤ 1.38 with four reef units, whereas Kr < 36% is achieved for d/h = 1.5 for single reef unit.
The percentage reduction in Kt is more than 45% for eight reef units subjected to regular waves with d/h = 1.25 and for random waves with d/h ≤ 1.5.
The empirical equations for regular and random waves with four and eight reef units are proposed.
The relative trench width is identified as an important parameter greatly influencing Kt of the submerged reef system.
From the experimental investigations, the reef configuration with eight units is recommended based on their hydrodynamic performance.