The physical model experiments were carried out in the Dutch wave flume of the hydraulic laboratory at Thuyloi University in Hanoi, Viet Nam. The wave flume is 45 m long with a working section 1.0 m wide and 1.2 m high. The piston-type wave maker is equipped with an Active Reflection Compensation (ARC, Deltares) that enables the suppression of the reflected wave from the riprap wave absorber. The wave maker is capable of generating both regular and random waves with a maximum wave height of 0.30 m and a peak period of 3.0 s. The capacitive wave probes are used to measure water surface fluctuations at a sampling frequency up to 100 Hz with high accuracy (error of ±0.1 mm).

The geometrical parameters of the submerged reef, according to bathymetry surveys, in the original condition are summarized in Figure 1 . The cross-section of the submerged reef in the direction of wave propagation had a complex fore-reef slope. The lower fore-reef slope, which ranged from the seabed to the –40 m contour line, had a steep slope of about 1/1 to 1/2. The fore-reef slope between the –40 m and –15 m contour lines had a gentler slope, ranging from 1/5 to 1/15, where the interactions between waves and submerged reef occurred. The surface of the submerged coral reef was relatively flat. The width of the reef varied from hundreds of meters to several kilometers. The water depth on the reef ranged from 5 m to 20 m (Dinh et al., 2014). Therefore, the breaking wave area was on the fore-reef slope during storms (see Figure 1 ). The corals were distributed on the reef flat with an average absolute roughness (average height) of r < 0.50 m (Dinh et al., 2014). Based on the characteristics of the submerged reef in its original condition, the capability of the wave flume, and the experimental requirements, the physical model length scale of N_L = 40 was determined following Froude’s scale law. The time scale of the model was therefore N_T = √N_L = 6.32. The general layout for the experiments is shown in Figure 2 , in which the model of a submerged reef had a height of 0.50 m to ensure that the incoming waves were deep water waves (unbroken waves). According to previous studies (Tuan and Cuong, 2019a), the slope of a fore-reef slope, within the consideration range i = tanα = 1/5–1/10, has a negligible influence on the regime wave hydrodynamics on the submerged reef. Therefore, this study considered only a representative seaward slope i = 1/5. The width of the reef flat was B = 15.0 m (corresponding to 600 m in prototype) and was covered by a smooth concrete mortar. The rough bed was made separately in the form of plates and fixed onto the reef flat within 8 m of the outer edge of the reef. The wave propagation experiment across the submerged reef was performed with 04 bottom models with different roughness levels (see Table 1 ). At the end of the flume, there was a passive absorbing boundary of reflected waves, which was composed of a tangled rock roof with a gentle slope of 1/6. A high-definition (HD) video camera was arranged perpendicular to the flume wall at the top of the outer shelf to record the entire image of waves in the reef-edge surfzone for all experimental cases.

Note that it was not desirable in this study to replicate roughness conditions of the coral bed. At present, it is technically impossible to reliably describe the effect of the actual coral bed in scale models owing to viscous effects. With the aim of studying the effect of bottom roughness on across-reef wave propagation, the rough reef bed was introduced through schematizing different conditions roughness elements, as shown in Table 1 . The rough bottom parameters in Table 1 include r_k (roughness height), N (the density of roughness elements per unit area), l_h and l_v (the plane and vertical dimensions of rough element, respectively), and λ_f and λ_p (respectively the total front area and total plan area of rough elements per unit plan area), which were estimated as follows:

In order to achieve the research objectives, wave measurements on the submerged reef were divided into three areas: the deep water section, reef-edge surfzone (wave breaking area), and the wave propagation area (see Figure 1 ). Six capacitive wave gauges with an accuracy of ±1 mm were used to measure incident waves in deep water, breaking waves in the reef-edge surfzone, and propagation waves in the transmission zone. In this study, with the main objective of evaluating the effects of bottom roughness on wave propagation, the wave probes were arranged mainly behind the surfzone area (i.e., the transmission zone) on the rough bottom model section. For a further analysis of the effect of bottom friction, the bottom flow velocity was synchronously measured with the wave probe at position WG5 by a Vectrino-II 3-D flowmeter ( Figure 2 ). Synchronous measurements of wave and velocity at WG5 were also used for the evaluation of wave reflection in accordance with the wave energy flux method by Sheremet et al. (2002). Figure 3 shows some snapshots taken during the physical model tests.

In this section, we consider the behavior of waves as they propagate across the submerged reef with different bottom roughness.

First, it follows from wave reflection analyses that the bulk reflection coefficient at WG5 was less than 10% in all test cases, and thus wave reflection can be neglected in the determination of the incident wave height at locations WG2–WG6 on the reef flat.

Experimental results show that the further inside the reef, the more the energy of the IG-waves prevails over that of the SS-waves. The comparison of the results obtained in the cases of smooth bottom and rough bottom surfaces also shows that the bottom roughness has almost a negligible influence on the change of wave spectrum shape, but has a significant effect on the wave energy attenuation (wave height) when wave propagating across the reef, especially for SS-waves.

To analyze wave propagation across the submerged reef with different bottom roughness, this paper introduces the concept of the relative submergence of the reef χ (χ is a factor that represents relative submergence of the reef flat or relative reef shallowness), as follows (see Hofland et al., 2017; Tuan and Cuong, 2019a; Tuan and Cuong, 2019b):

where D is the submerged water depth of the reef, and H_m0,0 and s_m0,0 are the wave height and wave slope in the deeper section in front of the submerged reef, respectively. The wave slope, s_m0,0 , was calculated based on the characteristic spectrum period T_m-1.0 .

Note that, in the rough bottom case, the water depth on the reef needs to be adjusted depending on the characteristic geometrical dimensions of the coral bottom (Lowe et al., 2005b; Buckley et al., 2016):

The wave transmission coefficient at a wave gauge location (WGs2–6) on the reef can be estimated as:

where H_{m 0, i} and H_{m 0, t} are the incident and transmitted wave heights, respectively.

The relationship between the wave transmission coefficients (K_t , K_t-SS , and K_t-IG ) and the relative submergence of the reef χ is shown in Figures 4 – 6 for the total waves, SS-waves, and IG-waves, respectively. The results show that the wave transmission coefficients (K_t , K_t-SS , and K_t-IG ) increased with the increase in the relative submergence. Note that the coefficients K_t , K_t-SS , and K_t-IG are all equal to 1 at boundary location WG2 (at x = 1.0 m). Along the reef (from x = 2.5 m to x = 7 m), the transmission coefficient K_t for the total wave is markedly decreased, especially in the cases of the rough bottom with R ₁, R ₂, and R ₃ ( Figures 4A–D ). The effect of bottom friction on the wave transmission coefficient K_t-SS is most pronounced for the SS-waves (see K_t-SS in Figure 5 ). However, for the IG-wave, the transmission coefficient K_t-IG tends to increase (K_t-IG > 1.0) in areas inside the reef shelf ( Figure 6 ). This is because of the development of the IG-wave on the transmission zone (behind the reef-edge surfzone) as a result of wave–wave non-linear interaction, which has converted a significant part of the energy of the SS-wave into the IG-wave. In addition, in the process of spreading through the reef, both the SS- and IG-waves’ energy are reduced owing to the influence of the bottom friction. If wave energy dissipation prevails over the wave energy conversion to the IG-wave, then the IG-wave height will be attenuated, as can be observed in Figure 6 at the submergence of about 0.25. In addition, in the process of propagation, part of the IG-wave energy can also be transferred to the SS-wave, depending on the relative submergence of the reef.

Figures 7 , 8 show the variations in the wave transmission coefficients K_t-SS and K_t-IG with respect to the relative distance x/L_m-1,0-SS and x/L_m-1,0-IG for all bottom roughness cases. In these figures, L_m-1,0-SS and L_m-1,0-IG are the length of the SS- and IG-waves, respectively; note that x = 0 at WG2. It is shown that, along the wave propagation direction, the SS-wave transmission coefficient K_t-SS always decreases quickly, especially in the case of rough bottom R ₁, R ₂, and R ₃ ( Figure 7 ). On the other hand, the transmission coefficient K_t-IG of the IG-wave increases slightly along the reef, depending on the non-linear wave–wave interaction and the energy exchange between the SS- and IG-waves, as well as the energy dissipation due to the bottom friction ( Figure 8 ).

To analyze the energy exchange processes between the IG- and SS-waves, the equation of transformation of wave propagation energy at a location on the submerged reef is as follows:

in which F⁺ and F are the incoming and total across-shore wave energy fluxes, respectively; D_b is the energy dissipation due to wave breaking (negative); D_in is the possible non-linear transfer of energy between the SS- and IG-waves due to the wave–wave interaction (positive or negative); D_f is the energy dissipated by the bottom friction (always negative); F_i and F_i+1 are the energy fluxes at two adjacent cross-shore locations i and i+1, respectively; and Δx = 1.5 m (see Figure 2 ).

In Equation (5), it is considered that ΔF ⁺/Δx ≈ ΔF/Δx with an assumption that the reflected wave (seaward) varies insignificantly over short intervals of one to several Δx (Tuan and Cuong, 2019a). By applying Equation (5) separately to the SS- and IG-waves, the incoming wave energy fluxes are determined as follows:

where F S S + and F I G + are the incoming SS- and IG-wave energy fluxes, respectively; E S S + and E I G + are the total SS- and IG-wave energies (J/m²), respectively; H_rms-SS and H_rms-IG are the root-mean-square SS- and IG-wave heights, respectively (m); c_g-SS and c_g-IG are the SS- and IG-wave group velocities, respectively (m/s); and g is the gravity acceleration (m/s²).

The wave group velocity c_g-SS of the SS-wave is defined as follows:

where c_SS is the SS-wave celerity (m/s); T_m-1.0–0 is the characteristic wave spectral period at the deep water boundary in front of the reef (assuming that the variation of T_m-1,0 of the SS-waves is negligibly small across the reef), L_m-1,0 is the shallow water wave length based on T_m-1.0–0 ; and k_m is the wavenumber calculated based on the wave length L_m-1,0 .

The wave group velocity c_g-IG of the IG-waves can be estimated as follows:

in which c_IG is the IG-wave celerity (m/s).

Using Equation (5) for the smooth bottom case (D_f ≈ 0), and if D_b = 0 (no more breaking waves), then the energy transference between the SS- and IG-waves can be determined as follows:

Figure 9 shows the experimental results of the relationship between the energy transferences (D_in-SS and D_in-IG in Equation (10)) and the relative reef submergence for the SS- and IG-waves in four sections: WG2–WG3, WG4–WG3, WG5–WG4, and WG6–WG5. For each type of wave considered here, if D_in-SS > 0 and/or D_in-IG > 0, then the wave receives the energy converted from the other wave type and vice versa. The SS-wave energy has a general tendency to be lost (D_{in- SS} < 0), and this lost energy of the SS-wave has been converted to the IG-wave ( Figure 9A ). In contrast, the IG-wave mainly tends to receive the energy from the SS-wave during propagation (D_in-IG > 0). However, the development of the IG-wave depends on the relative submergence of the reef χ. It can be seen that the energy of the IG-wave tends to shift strongly to the SS-wave in the range χ = 0.15–0.35, especially at χ ≈ 0.25 (see Figure 9 ). This result is similar to the one shown in Figure 6 . In general, the SS-wave energy is mostly consumed in the section WG3–WG2 and, correspondingly, the IG-wave has primarily received the energy in this section. In addition, the significant difference in value between D_in-SS and D_in-IG in the WG3–WG2 section shows that the SS-wave continues to lose its energy owing to the wave breaking (i.e. D_b > 0). In the next sections (WG4–WG3, WG5–WG4, and WG6–WG5), the changes in both the SS- and IG-waves are clearly reduced ( Figure 9 ).

Figure 10 shows the relationship between the average energy dissipation determined by Equation (5) and the relative reef submergence χ for the SS-waves in sections WG5–WG3 (2Δx) and WG6–WG3 (3Δx), with WG3 as a base point ( Figure 10A ), and for the IG-waves on sections WG4–WG2 (2Δx), WG5–WG2 (3Δx), and WG6–WG2 (4Δx, with a base point at WG2 ( Figure 10B ). It can be seen that the magnitude of the average dissipated energies D_in of the SS- and IG-wave is quite similar, and they tend to be symmetrical about the averaged D_in-SS , which equals the averaged D_in-IG = 0 axis (i.e. the averaged D_{in -SS} ≈ the averaged (-D_{in -IG})). Assuming that the wave-to-wave energy conversion interaction between the SS- and IG-waves is not significantly influenced by the bottom friction (Herbers et al., 2000; Henderson and Bowen, 2002), these average dissipated energies D_in-SS and D_in-IG can be used for the smooth bottom case and Equation (5) can be applied for the rough bottom cases (R ₁, R ₂, and R ₃) to determine the wave energy dissipation rate of the SS- and IG-waves due to the bottom friction, as follows:

where D i n − S S and D i n − I G are the wave energy dissipation (or receipt) owing to the energy conversion interaction between the SS- and the IG-waves.

The wave energy dissipation rate of the SS- and IG-waves will be used to estimate the coefficient of energy dissipation due to friction in the next sections.

In order to be able to build a method to calculate wave propagation across a submerged reef with bottom friction, the theoretical basis has been considered for the calculation of wave energy dissipation across the reef with large bottom roughness. The measurement results of wave energy dissipation due to friction will be evaluated and compared with results calculated using theoretical formulas. The calculated wave energy dissipation in the previous section was applied to estimate the wave energy dissipation coefficient based on the theoretical formulae. In the general case, the influence of the rough bottom on wave energy dissipation is mainly due to frictional and drag forces (ignoring the effect of inertia). However, for the submerged reef considered here, the height of the rough particles (coral) is very small in comparison with the water depth, so the effect of drag can be ignored (Lowe et al., 2005a). As analyzed in the previous section of this paper, there is a difference energy dissipation owing to the friction of the reef bottom between the SS- and IG-waves. The SS-wave has a large orbital velocity, which is affected by bottom friction more than that of the IG-wave. Due to the IG-wave has a long period the effect of bottom friction on this type of wave is similar to that of a current. The SS-wave energy dissipation due to bottom friction D_f-SS can be fundamentally calculated as a function of the orbital velocity near the reef bottom and the coefficient of energy dissipation due to friction or the coefficient of bottom friction (Jonsson, 1966):

where f_w-SS is the coefficient of energy dissipation due to friction under the SS-wave; u_b and u_{b -SS} are the amplitudes of horizontal velocity near the reef bottom due to the total wave and the SS-wave, respectively. These velocities can be determined according to the linear wave theory, as follows:

where L_{m -,0} is the length of shallow water wave corresponding to the wave period T _m-1.0 at the water depth on the reef, D.

Nielsen (1992) proposed a formula to determine the coefficient of energy dissipation due to the bottom friction for a single wave:

where k_w = (1 – 2)*r_k is the bottom roughness.

Madsen (1994) and Mathisen and Madsen (1999) extend the application of Equation (14) to irregular waves with c ₁ = 7.02, c ₂ = −0.078, and c ₃ = −8.82.

Pomeroy et al. (2012) proposed a method of determining the coefficient of energy dissipation due to friction under the IG-wave, which is similar to the method developed by Burchard et al. (2011) for currents. In this method, the coefficient of energy dissipation due to bottom friction under the IG-wave depends only on the water depth and the bottom roughness, and is not affected by the near-bottom orbital velocity of the IG-wave (Equation (15)):

in which κ = 0.40 is the Karman number; f_{w -IG} is the coefficient of energy dissipation due to friction under the IG-wave; and k_w (or r_k ) is the bottom roughness sensed by the IG-wave.

There have been many previous studies providing methods to determine the Nikuradse roughness (k_w ) on a beach with different morphological characteristics of the seabed. The Nikuradse roughness k_w = (1 to 2)*d ₅₀ is commonly used for a flat seabed, where d ₅₀ is the average diameter of the sand/gravel particles at the seabed. When the seabed is a coral bottom with a high degree of roughness, the relationship between roughness (k_w ) and the absolute roughness height (σ_r ) of the coral bottom is considered as k_w = 4*σ_r (Lowe et al., 2005a,b). In this study, the bottom roughness k_w = 0.75*r_k was applied for the IG-wave.

The energy dissipation due to bottom friction under the IG-wave can be determined by the method for long waves (Henderson and Bowen, 2002; van Dongeren et al., 2007):

where H_rms and H_rms-IG are the root-mean-square wave heights of the total and IG-waves, respectively.

By applying Equations (11), (12), and (16) the energy dissipation coefficients f_w-SS and f_w-IG can be determined based on the measured data, and from now on are called the “measured” energy dissipation coefficients f_w-SS and f_w-IG . The measured energy dissipation coefficients are presented in Figure 11 . The figure shows that the measured energy dissipation coefficients under the SS-waves (with an average measured value of f_w-SS = 0.15; dashed lines in Figures 11A–C ) are much higher than those under the IG-waves (with an average value of f_w-IG = 0.05; dashed lines in Figures 11D–F ). These results are consistent with the findings of previous studies (see Lowe et al., 2005a; Hench et al., 2008; Péquignet et al., 2011; Lowe et al., 2008; Lowe et al., 2009). This significant difference between the measured coefficients f_w-SS and f_w-IG is due to the bottom friction and is dependent on the ratio of the horizontal velocity amplitude to the bottom roughness (the ratio is large under SS-waves but small under IG-waves).

The “calculated” energy dissipation coefficients from Equations (14) and (15) are presented in Figure 12 , in which the bottom roughness k_w = 2r_k is applied for the SS-waves and k_w = 0.75r_k is applied for the IG-waves. The tested values of the roughness height r_k are presented in Table 2 . Similar to the measured energy dissipation coefficients, the calculated coefficients under the SS-waves are much higher than those under the IG-waves. It is shown in Figure 11 and Figure 12 that the average values of the measured and calculated energy dissipation coefficients are similar.

Theoretically, the energy conversion in Equation (5) between the SS- and IG-waves due to the non-linear wave–wave interaction can be approximated by an analysis of the correlated spectral pair (bispectral) method presented in Herbers and Burton (1997); Henderson et al. (2006), and Pomeroy et al. (2012). However, this method is difficult to use because of the complexity in constructing the correlated spectral pair of wave energy. In this study, an experimental method is proposed to determine the energy conversion from the experimental data. Here, the energy conversion D*_{in -IG} is directly related to the IG-wave as follows:

where f_{in -IG} is the energy conversion coefficient between the SS- and IG-waves (f_{in -IG} > 0 if the IG-wave receives energy from the SS-wave, and vice versa).

The energy conversion coefficient f_{in -IG} can be determined from Equation (18), in which D in -IG * can be determined from Equation (10) for the smooth bottom case (D_{f -IG} = 0), and from Equation (11) for the rough bottom cases (after deducting the energy dissipation due to friction D_f-IG calculated by Equation (16)). Figure 13 shows the energy conversion coefficient f_{in -IG} plotted against the relative submergence of the reef χ. It shows that the energy conversion from the SS-wave to the IG-wave on the reef depends on the relative submergence χ, and that it can be divided into three modes as follows (see more results in Figure 10B ): (i) f_{in -IG} > 0 and almost constant if 0 < χ < 0.20; (ii) f_{in -IG} < 0 (the SS-wave gets energy from the IG-wave) if 0.20 ≤ χ ≤ 0.30; (iii) f_{in -IG} > 0 and increases with increasing the relative submergence χ if χ > 0.30. The solid line in Figure 13 represents the fitted curve with functions as shown in Equation (19). The fitted curve has a correlation coefficient R ² of 0.70:

where a ₀, a ₁, b ₀, b ₁, b ₂, and c₀ are the empirical coefficients found by a least-squares fit of the experimental data: a ₀ = −0.25; a ₁ = 10.0; b ₀ = −0.25; b ₁ = 1.96; b ₂ = 3.75; and c ₀ = 0.25.

To determine the wave height attenuation when a wave is propagating across the submerged reef, taking into account the effect of bottom friction, the input wave height characteristics at the seaward reef-edge boundary must be determined. There are important hydrodynamic processes that occur at this boundary of the submerged reef, such as a wave breaking and dissipating most of the incident wave energy, a wave forming infinity gravity waves (IG-waves), and non-linear wave–wave interactions between the SS- and IG-waves ( Figure 1 ).

In this section, several empirical formulae used to determine the across-reef wave heights are presented. The application range of the empirical formulae in this study is the variation range of the dominant parameters in the physical model test, as presented in Table 4 .

The procedure used to determine across-reef wave heights affected by the bottom roughness can be summarized into the following steps:

Step 1: Prepare input data setup.

Step 2: Determination of wave characteristics at the seaward edge of the reef.

• The width of the breaking zone at the seaward edge x_{b ,90%}: this can be estimated from the formula found in Tuan and Cuong (2019b), as follows:

where tanα is the slope of fore-reef slope.

The height of the IG wave H_{m 0-IG}:

Note that in Equations (30) and (31), the point x coordinate is taken from the breaking boundary (x_{b ,90%}).