Trapezoidal breakwater on reducing resonant wave amplitude on a rectangular basin

2.1 Governing equations

To encompass both scenarios, where the breakwater exhibits smooth and rough characteristics, we adapt the LSWEs model proposed by Magdalena and Jonathan (2022), Magdalena et al. (2023a) by introducing a friction factor $C_{f}u$ within the momentum balance equation (Magdalena et al., 2020a, 2022c, 2023b; Dean and Dalrymple, 1991). The friction coefficient $C_f$ encapsulates the frictional effects arising from the roughness of the structure. The modified LSWEs are presented as follows in terms of the variables wave elevation $\eta$ and horizontal velocity $u$ :

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Fig. 1

Model setup illustration.

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2.2 Numerical method

In this section, we outline the discrete formulation of our mathematical model. We employ a staggered finite volume method with an average approach. The grid arrangement discussed herein is visually represented in Fig. 2. The computational domain spans both spatial and temporal domains, designated as ${\Omega }_L=[0,L]$ and ${\Omega }_T=[0,T]$ , respectively.

We partition ${\Omega }_L$ into half and full grids, each with a step’s length of $\Delta x$ , and discretize ${\Omega }_T$ into finite time steps of $\Delta t$ . Eq. (1) is computed on cells centered at $x_i$ , ensuring that the full-grid points exclusively store information related to $\eta$ , $d$ , and $h$ . Simultaneously, Eq. (2) is calculated on cells centered at $x_{i+1/2}$ , indicating that $u(x,t)$ is stored solely in the half-grid points. This approach facilitates an effective and accurate representation of the system dynamics within the defined computational framework.

Implementing a Leapfrog Scheme, we have formulated the discrete representations of Eqs. (1) and (2). These expressions are articulated as:

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Fig. 2

Illustration of finite volume method on a staggered grid.

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In these equations, subscripts denote spatial grid points, and superscripts represent time grid points. Additionally, $\eta$ in Eq. (5) is determined implicitly to ensure stability in the numerical scheme. To address Eq. (4), an approximation of $h$ on half-grid points, denoted as $h^{\ast }$ , is necessary. The notation $h^{\ast }$ is estimated using the average value: $$h_{i+\frac{1}{2}}^{\ast }=\frac{1}{2}(h_i+h_{i+1}).$$ It is important to highlight that the friction term in Eq. (5) is computed implicitly, a strategic choice made to navigate limitations on numerical stability. Consequently, the stability condition for this scheme remains independent of the friction coefficient, as expressed by: $$\frac{\Delta t}{\Delta x}\sqrt{gd}\le 1.$$

4.1 Resonance over a flat bottom

Here, the formulated analytical formula is used to reproduce the natural period in a case where the seafloor is flat and no structure is involved. The resulted natural periods for several scenarios are then compared to the ones generated by Magdalena et al. (2020b) to validate the newly derived formula. The analytical formula for flat bottom that is formulated by Magdalena et al. (2020b) is written as $T_1=\frac{4L}{\sqrt{g{h}_1}}$ , where $L$ is the length of the observed domain. In order to compare both formulas, a few adjustments in the values of $\alpha ,\beta ,\gamma$ , and $\delta$ in Eq. (22) are needed. Table 1 listed two scenarios evaluated in the comparisons between the natural periods produced by Magdalena et al. (2020b) and by Eq. (22). Note that $T_1^1$ denotes the natural period produced by the formula from Magdalena et al. (2020b) while $T_1^2$ denotes the ones calculated from Eq. (22). The errors presented in Table 1 were the relative errors calculated using the formula: $\text{Error}\left(\text{\%}\right)=\frac{|T_1^2-T_1^1|}{T_1^1}\times 100$ . It is evident in Table 1 that the errors are all less than 2%, which is considered to be quite small. This outcome shows that the newly derived formula for determining the natural frequencies of a basin with a trapezoidal breakwater may, in fact, be applied to more generic scenarios, such as a flat-bottom basin without a breakwater or with a rectangular breakwater (since the slopes can be adjusted).

4.2 Resonance phenomena induced by the trapezoidal breakwater

In this case, we consider a trapezoidal breakwater being placed in a rectangular basin (flat topography). Both the sea floor and the breakwater are considered smooth, which means that the friction effect is neglected or that $C_f=0$ in the whole domain. The parameters used in these simulations are $h_3=5$ m, $\zeta =2$ , $\alpha =2,\beta =2,\gamma =3,\delta =4,$ and $T=100$ s. The frequency of the initial wave used is the natural frequency (natural period) obtained analytically. In this case, the natural frequency is $1.311387{\mathrm{s}}^{-1}$ (or $T_1\approx 4.7913$ s). The other parameters are the same as the ones defined previously. The resulted wave evolution near the wall can be seen in Fig. 3.

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Fig. 3

The increase in wave elevation over time, computed after it passes over a smooth trapezoidal breakwater, indicates the occurrence of resonance phenomena.

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4.3 The influence of a rough trapezoidal breakwater on resonance wave phenomena

Here, the friction effect is considered present only on the crest of the trapezoidal breakwater (domain $R_3$ ), while the structure’s slopes and the seafloor are still smooth. In each simulation, the parameters related to the breakwater’s configuration are fixed at $h_3=5$ m, $\zeta =2$ , $\alpha =2,\beta =2,\gamma =3,\text{and}\delta =4$ . The simulations were observed for $T=100\text{s}$ . The results of the simulations are presented in Fig. 4 with several different values of $C_f$ .

We now see how wave resonance responds to the change in friction coefficient if the friction-affected domain is much bigger. We consider the friction factor to impact the whole trapezoidal breakwater, including its crest and its two slopes ( $R_2,R_3,R_4$ ), while the seafloor is still considered smooth. The parameters are now $h_3=5$ m, $\zeta =2$ , $\alpha =2,\beta =2,\gamma =3,\delta =4,$ and $T=100\text{s}$ , where the values of $C_f$ are again varied. The generated wave evolution for each case is shown in Fig. 5.

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Fig. 4

The decrease in the resonant wave’s maximum amplitude, due to the increase in friction coefficient ( $C_f$ ) which represents the presence of a trapezoidal breakwater with a rough crest and smooth slopes.

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Fig. 5

The decrease in the resonant wave’s maximum amplitude, due to the increase in friction coefficient ( $C_f$ ) which represents the presence of a trapezoidal breakwater with rough crest and slopes.

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4.4 Sensitivity analysis

In this subsection, several parameters are analyzed to observe each parameter’s impact on the wave’s evolution, including the wave’s period and maximum amplitude. Those parameters are $\alpha$ which represents the changes in the distance between the breakwater and the wall, $L_2$ which represents the length of the breakwater’s slopes, $\beta$ which represents the changes in the length of the breakwater’s crest, and $\zeta$ which represents the changes in the breakwater’s height. For all of the simulations conducted in this subsection, the trapezoidal breakwater is considered rough, which means that the friction factor is considered to exist in subdomains $R_2,R_3$ , and $R_4$ , with a fixed friction coefficient $C_f=0.01$ .

The first set of simulations to be conducted is the one considering the changes in the breakwater’s distance from the wall ( $\alpha$ ). The breakwater’s distance from the wall itself is represented by the term $\alpha L_2$ , where the value of $L_2$ is fixed at $L_2=5$ m and the value of $\alpha$ is varied. The other parameters are set to be $h_3=5$ m, $\zeta =2$ , $\beta =2,\gamma =3,\delta =4,\text{and}T=100\text{s}$ . Fig. 6 shows how the changes in breakwater’s distance from the wall affect the wave behavior over time.

The next parameter to be discussed is $L_2$ which represents the length of the breakwater’s slope. Therefore, several simulations were done with variations of $L_2$ . Even though $L_2$ only represents the length of the front-slope of the breakwater in the setup, in the computation, it also rules the breakwater’s back-slope. Hence, changing $L_2$ will change both slopes simultaneously. As usual, we set the parameters to be $h_3=5$ m, $\zeta =2$ , $\beta =2,\gamma =3,\delta =4,\text{and}T=100\text{s}$ . The value of $\alpha$ is also varied following $\alpha =10/L_2$ to make the distance between the wall and the breakwater fix at $10$ m. The results of the simulations for several values of $L_2$ are presented in Fig. 7.

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Fig. 6

The decrease in the resonant wave’s maximum amplitude, following the increase in the values of $\alpha$ which represents the trapezoidal breakwater’s distance from the wall.

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Fig. 7

The decrease in the resonant wave’s maximum amplitude, following the increase in the values of $L_2$ which represents the length of the trapezoidal breakwater’s slopes.

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Now, we are examining the effect of another parameter, which is the length of the breakwater’s crest, on the wave maximum amplitude and wave period. This characteristic is represented by $\beta$ . For fixed breakwater’s slopes, varying $\beta$ means varying $\gamma$ as well. In this case, we set $\gamma =\beta +1$ . However, for a fixed length of the observation domain, the variation of $\beta$ is limited to be less than $2$ , while $\delta =4$ is required. Other than that, $h_3=5$ m, $\zeta =2$ , $\alpha =2,L_2=5\mathrm{m},\delta =4,\text{and}T=100\text{s}$ are still used. The computational results are shown in Fig. 8.

The last parameter to be examined is the breakwater’s height, represented by the water depth on the crest ( $h_3$ ). In this case, the maximum water depth is fixed at $10$ m, such that the parameter $\zeta$ also varies following $\zeta =10/h_3$ . The rest of the parameters are the same, which are $\alpha =2,L_2=5\mathrm{m},\beta =2,\gamma =3,\delta =4,\text{and}T=100\text{s}$ . Fig. 9 shows the simulation results of several breakwater’s heights.

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Fig. 8

The decrease in the resonant wave’s maximum amplitude, following the increase in the values of $\beta$ which represents the length of the trapezoidal breakwater’s crest.

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Fig. 9

The decrease in the resonant wave’s maximum amplitude, following the increase in the values of $h_3$ which represents the water depth on the trapezoidal breakwater’s crest.

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5.1 Resonant wave’s response to different cases of breakwater’s roughness

The cases depicted in Fig. 4 include friction coefficients of $C_f=0$ , 0.05, 0.125, and 0.2. These values were selected to illustrate how changes in the friction coefficient impact the evolution of resonant waves and which value can effectively halt resonance. Upon closer examination of Fig. 4, it was observed that despite the consistent increase in the value of $C_f$ in the last three cases (where $C_f$ increases by 0.075), each case had a distinct effect on the maximum amplitude of the resonant wave.

In particular, the third case ( $C_f=0.125$ ) demonstrated a reduction of approximately 37% in the maximum amplitude compared to the second case ( $C_f=0.05$ ). Conversely, the fourth case ( $C_f=0.2$ ) achieved a reduction of only about 31% from the third case. These findings indicate that as $C_f$ increases, its impact on diminishing the maximum amplitude of the resonant wave becomes less significant. Nonetheless, this parameter remains influential in significantly reducing the maximum resonant wave amplitude, with an average reduction of approximately 4.9% for every increase of 0.01 in $C_f$ .

The second piece of information that can be obtained from Fig. 4 is the value of $C_f$ that can stop resonance. A certain set of parameters is considered to be able to stop the resonance if the generated wave amplitude is not increasing infinitely, which means that the last amplitude remains the same or decreases from the previous one. These resonance-stopping parameters were found through a trial-and-error approach. In this case, the resonance is stopped when $C_f=0.125$ , which can be seen in Fig. 4 by looking in more detail at the last two amplitudes, which have similar values. This characteristic could not be found in cases where $C_f=0$ or $C_f=0.05$ , since the amplitude keeps on increasing, even with different growth rates. At $C_f\approx 0.2$ , the generated wave even seems like a standing wave, where the amplitude is maintained the same throughout the observation time, which means that the resonance barely exists. This result is confirmed by Magdalena et al. (2022c) who stated that at certain values of friction coefficient, a wave resonance is possible to prevent. However, the friction coefficient that is able to prevent resonance in this study is much bigger than the ones generated in Magdalena et al. (2022c) due to the difference in the portion of the water body affected by the friction factor. This difference also impacted the generated natural periods, which are similar for all values of $C_f$ ( $T_1\approx 4.79$ ), compared to the ones addressed by Magdalena et al. (2022c), which are varied each time the friction coefficient is changed, since Magdalena et al. (2022c) considered friction factor on the whole observed domain, not just a certain subdomain.

We observed similar results from Fig. 5, where an increase in $C_f$ generates a decrease in the wave’s maximum amplitude. In this case, the values of friction coefficients are $C_f=0,0.03,0.065,\text{and}0.1$ . It was found from the simulations that the average reduction generated by an increase of $C_f$ by 0.01 is about 10.2%. This value is larger than the previous case, which makes sense as the portion of the basin’s domain affected by the friction factor was larger in this case compared to the previous one. This means that in this case, the change in $C_f$ affects the wave’s maximum amplitude more significantly than it did in the previous case. Another difference is that in this case, the resonance is able to be stopped at $C_f=0.065$ , which is almost two times smaller than the previous case. This is again due to the portion of the basin’s domain affected by the friction factor, which was larger in this case. However, the findings regarding wave period are also applied in this case, where the changes in the friction coefficient barely affect the generated wave period since $T_1\approx 4.79$ in all cases.

5.2 Resonant wave’s response to different configurations of trapezoidal breakwater

In Fig. 6, the cases are presented with values of $\alpha$ being $\alpha =2,2.5,\text{and}3$ , with resonance being halted at $\alpha =2.5$ . This value was determined through a trial-and-error method. Although not clearly depicted in the figure, the wave for $\alpha =2.5$ exhibited constant oscillations, evidenced by the similarity in its last few amplitudes. A more distinct comparison can be made with the case where $\alpha =2$ , where the wave amplitude infinitely increases. Furthermore, Fig. 6 provides insights into how the breakwater’s location influences the resonant wave’s maximum amplitude and period. It is evident that as the value of $\alpha$ increases (indicating a farther breakwater location from the wall), the wave’s maximum amplitude decreases. An increase of $\alpha$ by 0.5 (shifting the breakwater 2.5 m further into the ocean from the original position) resulted in a reduction of approximately 43.95% in the maximum amplitude. In other words, each 1 m movement of the breakwater towards the ocean led to a reduction of about 17.58% in the maximum amplitude. However, this finding does not extend to the wave period, as simulations showed fluctuations for different $\alpha$ values. The wave periods for $\alpha =2,2.5,\text{and}3$ were found to be $T_1=4.7913,5.0577$ , and 3.2401 s, respectively.

Through a trial-and-error method with various $L_2$ values, three were chosen to represent the effect of changes in the breakwater’s slope: $L_2=5,5.38$ , and 5.7. The case of $L_2=5$ signifies wave resonance with an infinitely increasing amplitude. When $L_2=5.38$ , resonance is completely stopped, evident by the reduction in wave amplitude after approximately $t=50$ s. Unlike previous cases, the transition from resonance to non-resonance in this scenario is abrupt when $L_2$ increases by 0.01 from the resonance state (with resonance still present at $L_2=5.37$ ). Furthermore, for $L_2>5.38$ (including $L_2=5.7$ ), the wave is entirely free from resonance. The wave reduction rate due to the change in $L_2$ showed that an increase of 0.1 m in $L_2$ resulted in an average reduction of approximately 14.05% in the wave’s maximum amplitude. In practical terms, this implies that a gentler breakwater slope is more effective at reducing resonant wave amplitude and ultimately preventing resonance. Similar to the $\alpha$ cases, the resulting wave periods fluctuated for a certain range of $L_2$ , with $T_1=4.7913,3.1424$ , and 3.3293 s for $L_2=5,5.38$ , and 5.7, respectively.

Fig. 8 displays resonant wave evolution for three different values of $\beta$ : $\beta =2,1.65$ , and 1.4. The breakwater’s crest length is defined by $\beta L_2$ , meaning a smaller $\beta$ corresponds to a shorter breakwater crest. The simulations revealed that resonance could be stopped by implementing a breakwater with a crest length of 8.25 m ( $\beta =1.65$ ). Furthermore, a decrease in $\beta$ by 0.1 from resonance ( $\beta =2$ ) to non-resonance resulted in an approximately 8.06% reduction in maximum amplitude. Conversely, when $\beta >1.65$ , the same decrease in $\beta$ led to a more significant reduction (approximately 34.21%). A decrease in $\beta$ also resulted in a decrease in the wave period, with a more pronounced reduction observed when $\beta >1.65$ . The wave periods for $\beta =2,1.65$ , and 1.4 were $T_1=4.7913,4.7462$ , and 3.2878 s, respectively.

In these simulations, larger $h_3$ corresponds to a shorter breakwater, and vice versa. The results differ slightly from previous cases in terms of the parameter value at which resonance is stopped. Instead of a minimum (as seen in the cases of friction coefficient, $\alpha$ , and $L_2$ ) or maximum (as observed in the case of $\beta$ ) value stopping resonance, there are several ranges of values where resonance does not occur. Fig. 9 illustrates that resonance does not occur when $h_3\approx 5.45$ m, even though it still occurs when $h_3\approx 5.44$ m. Similar to the case of $L_2$ , the transition from resonance to non-resonance is abrupt. However, in this case, resonance also abruptly reappears when $h_3\approx 6.48$ m, with even larger wave amplitudes. This suggests a non-resonance range for $h_3$ of $5.45\mathrm{m}\le h_3\le 6.47$ m. However, this is not the sole non-resonance range for the breakwater’s height parameter. Within the observed domain of $0<h_3\le 10$ m, a non-resonance state was also observed when $3.27\mathrm{m}<h_3<4.48$ m and $7.06\mathrm{m}<h_3<7.97$ m. It is essential to consider the feasibility of constructing the breakwater, as when $3.27\mathrm{m}<h_3<4.48$ m, the breakwater would be $5.52$ m to $6.73$ m tall, requiring more material and, consequently, increasing construction costs. On the other hand, the range of $7.06\mathrm{m}<h_3<7.97$ m may be deemed suitable for design, as the breakwater would not be excessively tall, although the maximum wave amplitude ( ${\eta }_{max}\approx 1.39$ m) is larger than for $5.45\mathrm{m}\le h_3\le 6.47$ m ( ${\eta }_{max}\approx 0.28$ m). Therefore, constructing the breakwater with a height within the range $5.45\mathrm{m}\le h_3\le 6.47$ m is recommended.