Wave Diffraction Response to Incident Wave Height Variation: A Case Study of Gusung Island Breakwater, Makassar

Wave diffraction has long been recognized as a fundamental process in coastal and ocean engineering, governing the redistribution of wave energy around structures such as breakwaters and islands [15]. Classical studies based on linear potential flow and mild-slope equations provided analytical solutions for semi-infinite breakwaters and vertical cylinders [16]. These works formed the theoretical foundation for modern diffraction analysis but were limited by linear assumptions and simple geometries, restricting their applicability to real-sea conditions.

2.1 Theoretical and numerical developments

With advances in computational resources, numerical methods such as the BEM, FEM, and CFD have greatly expanded diffraction analysis [6, 7]. He et al. [17] introduced a heterotaxia BEM model to analyze multi-cylinder floating structures, while Xu et al. [18] applied CFD to porous cylinders to predict nonlinear diffraction behavior. Sensitivity testing of complex breakwater geometries further demonstrated the robustness of diffraction modeling under different incident conditions [19].

However, most numerical studies assume steady or normalized incident wave heights and simplified boundary conditions, which limit their capacity to capture hydrodynamic variability in natural environments. Few models explicitly examine how changes in incident wave height influence diffraction coefficients or energy redistribution.

2.2 Experimental investigations

Laboratory and basin experiments have played a key role in validating theoretical and numerical predictions. Han et al. [2] showed that loads on heave plates are highly sensitive to wave steepness, and Yan et al. [20] demonstrated asymmetric focusing around cylindrical structures. Firoj and Afzal [6] investigated solitary wave diffraction by V-shaped breakwaters, revealing nonlinear amplification effects, while Shi et al. [10] identified resonance-driven energy dissipation in Helmholtz-type breakwaters. Pan et al. [11] linked diffraction to springing responses in flexible cylinders, and Galani et al. [7] confirmed wave setup reduction behind segmented rubble-mound breakwaters.

Nevertheless, most experiments were conducted under constant or narrow ranges of wave height, rarely addressing how varying incident energy levels affect diffraction efficiency or hydrodynamic loading. Scale effects and boundary constraints in laboratory settings may also limit the direct transfer of these results to field applications.

2.3 Integrated and application-oriented frameworks

Recent research has extended diffraction analysis into multidisciplinary and applied contexts. Xu et al. [18] examined diffraction impacts on Oscillating Water Column (OWC) devices, showing the role of wave focusing in enhancing energy conversion. Khanal et al. [19] developed a differentiable BEM solver for hydrodynamic sensitivity analysis of complex breakwater geometries, while Wang et al. [21] and Kim et al. [22] linked diffraction-induced flow behavior to sediment deposition using the concept of deposit velocity.

Although these frameworks demonstrate the versatility of diffraction theory in engineering applications, they retain a structural or geometric emphasis. The direct hydrodynamic implications of incident wave height variation remain largely unexplored.

2.4 Research gap and contribution of the present study

Across these studies, geometry, porosity, and wave period have been consistently treated as the dominant parameters governing diffraction, whereas incident wave height has often been considered secondary. This gap is critical because wave height fundamentally governs orbital velocity, pressure loading, and energy flux, which directly influence the strength and structure of diffracted waves.

The present study addresses this limitation by investigating the diffraction response to incident wave height variation at the emergent breakwater of Gusung Island, Makassar Strait, Indonesia. By integrating wind wave data analysis, theoretical formulations, laboratory flume experiments, and BEM simulations, the study quantifies how varying incident wave heights affect diffraction coefficients, energy reduction efficiency, and flow regimes. This approach bridges empirical observations with global hydrodynamic theory, advancing diffraction analysis and providing practical guidance for climate-resilient coastal design in tropical island environments.

A concise summary of previous diffraction studies and their main contributions is presented in Table 1, which highlights the diversity of existing approaches and their relevance to the present work.

Table 1. Summary of related works on wave diffraction studies

The present study addresses this gap by placing incident wave height variation as the crucial factor in diffraction analysis, integrating theory, numerical modelling, and experimental observations.

4.1 Seasonal wave forecast and characteristics

The 2023 wind wave dataset shows pronounced seasonal variability at Gusung Island. Forecasting using the SMB method produced monthly values of significant wave height ($H_s$) and peak wave period ($T_p$). Results indicate that during the northwest monsoon (January–March), incident wave conditions were energetic with $H_s=2.0-2.2$ m and $T_p \approx 6-7$ s. By contrast, during the southeast monsoon (June–August), wave energy decreased significantly, with $H_s<1.0$ m and $T_p=3-3.5$ s. Transitional periods (April–May and September–October) exhibited intermediate conditions. Data is presented in Table 3 and Figure 4.

Table 3. Forecasted significant wave height and peak period

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Figure 4. Seasonal variation of $H_s$ and $T_p$

These seasonal fluctuations are crucial for diffraction studies, as they determine the range of incident wave heights used in subsequent numerical and experimental analyses. Similar seasonal dependencies were also observed in studies of coastal diffraction on rubble-mound breakwaters.

4.2 Diffraction coefficients

Diffraction coefficients ($K_d$) were evaluated as a function of diffraction angle ($\theta$) for representative small-wave (June) and large-wave (January) conditions. Results show that $K_d$ decreases monotonically with angle, from 0.7 at 15° to 0.1 at 90° for small waves. At larger incident wave heights, local amplification was observed, with $K_d$ increasing to 0.78 at 15°, confirming nonlinear focusing effects. The result is presented in Figure 5.

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Figure 5. Diffraction coefficient vs. diffraction angle

These findings are consistent with laboratory experiments on cylindrical structures and numerical studies of V-shaped breakwaters, both of which highlight stronger focusing under high-energy conditions.

4.3 Energy reduction efficiency

Energy reduction efficiency was calculated from the incident and diffracted wave energies. The results of recording several cases in January are shown in Table 4.

Table 4. Energy reduction efficiency under wave heights

Table 4 shows that the nonlinear increase reflects stronger dissipation by larger waves, consistent with energy dissipation studies of diffraction in coastal structures. An example of the calculation is given as follows:

(1) Case 1: $H_i=0.5, H_d=0.17$, $\eta=\left(1-(0.17 / 0.5)^2\right) \times 100=88.4 \%$

(2) Case 2: $H_i=1.0, H_d=0.32$, $\eta=\left(1-(0.32 / 1.0)^2\right) \times 100=89.8 \%$

(3) Case 3: $H_i=1.5, H_d=0.42$, $\begin{gathered}\eta=\left(1-(0.42 / 1.5)^2\right) \times 100=(1-0.0784) \times 100 = 92.2 \%\end{gathered}$

(4) Case 4: $H_i=2.0, H_d=0.53$, $\eta=\left(1-(0.53 / 2.0)^2\right) \times 100=92.98 \%$

Energy reduction efficiency increased from 88% for small incident waves ($H_i=0.5$ m) to 93% for larger waves ($H_i=2.0$ m), confirming that higher incident wave heights enhance nonlinear dissipation in the diffraction zone.

The manual computation illustrates step-by-step derivations of wavelength, group velocity, energy density, energy flux, diffraction efficiency, orbital velocity, and Reynolds number under representative seasonal conditions (January and June). In parallel, numerical simulations were employed to validate the analytical results across multiple incident–diffracted wave height pairs. The following steps of the simulation process:

Step 1. Deep water wave parameters

(1) $L_0=\frac{9.81 \times 6.8^2}{2 \pi}=72.195 \mathrm{~m}$

(2) $C_0=L_0 / T=72.195 / 6.8=10.617 \mathrm{~m} / \mathrm{s}$

(3) $\mathrm{C}_{\mathrm{g}}=c_0 / 2=5.308 \mathrm{~m} / \mathrm{s}$

(4) $k=2 \pi / L_0=0.08703 \mathrm{~m}^{-1}$

(5) $k h=0.8703$

(6) $\sinh (k h)=0.9844$

Step 2. Energy and flux (incident vs. diffracted)

(7) $E_{i}=\frac{1}{8} \rho g H_i^2=\frac{1}{8}(1025) *(9.81) *\left(1.50^2\right)=2828.04 \mathrm{~J} / \mathrm{m}^2$

(8) $E_d=\frac{1}{8}(1025) *(9.81) *\left(0.42^2\right)=221.72 \mathrm{~J} / \mathrm{m}^2$

(9) $P_{i}=E_i \cdot C_g=2828.04 \times 5.308=15.013 \mathrm{~W} / \mathrm{m}$

(10) $P_d=E_d \cdot C_g=221.72 \times 5.308=1.176 \mathrm{~W} / \mathrm{m}$

Step 3. Energy reduction efficiency

(11) $\eta=\frac{15012.52-1176.98}{15012.52} \times 100 \%=92.16 \%$

(12) $\eta=\left(1-(0.42 / 1.50)^2\right) \times 100 \%=92.16 \%$

Step 4. Orbital velocity and Reynolds

(13) $U_{o, i}=\frac{\pi H_i}{T \sinh (k h)}=\frac{3.1416 \times 1.50}{6.8 \times 0.9844}=0.704 \mathrm{~m} / \mathrm{s}$

(14) $R e_i=\frac{U_{o, i} L_0}{y}=\frac{0.704 \times 72.195}{1.05 \times 10^{-6}}=4.84 \times 10^7$

(15) $U_{o, d}=\frac{\pi H_d}{T \sinh (k h)}=\frac{3.1416 \times 0.42}{6.8 \times 0.9844}=0.197 \mathrm{~m} / \mathrm{s}$

(16) $R e_d=\frac{U_{o, d} L_0}{v}=\frac{0.197 \times 72.195}{1.05 \times 10^{-6}}=1.36 \times 10^7$

These values yield $\eta=92.16 \%, P_i \approx 15.0 \mathrm{~kW} / \mathrm{m}$. The corresponding Reynolds numbers ($R e \sim 10^7$) indicate a clearly turbulent regime The results of the calculation for January are presented in Figure 6.

Figure 6 shows the relative contribution of each parameter in the diffraction simulation during the January high wave season. The results showed that the wave energy flux, reduction efficiency, and Reynolds number dominated the hydrodynamic response, while the orbital velocity was relatively smaller and was strongly influenced by the wave period and water depth.

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Figure 6. Diffraction simulation parameters

4.4 Focusing and shadow effects

Wave focusing near the breakwater tip amplified local crests by 10–12% under large incident waves, while shadow zones exhibited energy attenuation up to 93%. This duality confirms the trade-off between enhanced protection and localized load intensification.

4.5 Comparative validation of theoretical, numerical, and experimental results

The reliability of the proposed diffraction model was verified through a comparative validation among the theoretical predictions, numerical simulations using the BEM, and experimental measurements obtained from the wave flume tests.

The comparison was conducted under identical input conditions for incident wave heights ($H_i=0.5-2.0 \mathrm{~m}$), wave periods ($T=4-8 \mathrm{~s}$), and diffraction angles ($15^{\circ}-90^{\circ}$). Illustration is presented in Figure 7.

7.png

Figure 7. Comparative variation of the diffraction coefficient ($K_d$)

Figure 7 presents the comparative variation of the diffraction coefficient ($K_d$) as a function of the diffraction angle ($\theta$) for the three analytical domains. All datasets show a consistent decreasing trend of $K_d$ with increasing angle, indicating that energy transmission weakens toward the shadow zone. Theoretical predictions slightly overestimate the coefficient at small angles ($\theta<30^{\circ}$) because the linear potential theory neglects viscous dissipation effects, while experimental results show minor underestimation at large angles ($\theta>75^{\circ}$) due to boundary-layer and scale limitations within the laboratory flume.

Numerical simulations using the BEM accurately reproduce the measured pattern, with the maximum deviation between numerical and experimental values less than 5% and a correlation coefficient (R²) of 0.97, confirming the robustness of the computational model. The close alignment of these three datasets validates the integrated workflow presented in Section 3, where theoretical formulations, BEM-based simulations, and experimental data were combined to represent realistic wave–structure interactions.

Table 5 summarizes the comparison between the theoretical, numerical, and experimental values of the diffraction coefficient and energy reduction efficiency ($\eta$) for selected wave heights. The mean absolute percentage error (MAPE) between the BEM and experimental results is below 4.8%, while that between theoretical and experimental results is below 6.5%, which is within the acceptable range for coastal engineering hydrodynamics [2, 5].

Table 5. Comparative results of diffraction coefficient (K_d) and energy reduction efficiency (η)

This comparative validation confirms that the integrated framework not only reproduces the diffraction behaviour under varying wave heights but also provides reliable predictions for practical design applications of breakwater structures under monsoonal wave regimes.

4.6 Hydrodynamic and morphodynamic implications

Hydrodynamic analysis showed Reynolds numbers increasing from $2 \times 10^5$ for June (H_s 0.4 m) to $10^6$ for January (H_s 2.2 m), confirming turbulent regimes across seasonal conditions. Zhang et al. [36] stated the sharp pressure gradients observed across focusing and shadow zones, implying asymmetric structural loading.

Near-bed orbital velocities were significantly reduced during calm seasons, often below critical deposit velocity thresholds, thereby promoting sediment deposition behind the breakwater.

4.7 Sensitivity analysis

Sensitivity simulations were performed to evaluate the effect of uncertainties in wind speed, fetch, depth, and diffraction coefficients:

(1) Wind speed (±10%): January conditions showed ~±9% variation in H_s, while June showed a negligible change due to fetch-limited conditions.

(2) Fetch (±50%): In January, H_s increased by ~41% with longer fetch, while in June changes were minimal.

(3) Depth (5–15 m): Surface Reynolds numbers remained turbulent, but near-bed orbital velocities declined exponentially with depth.

(4) Diffraction coefficients ($K_d \pm 0.05$): Energy reduction efficiency $\eta$ shifted ±6–7 percentage points at $\theta=30^{\circ}$, highlighting sensitivity to structural geometry and edge effects.

8a.png

(a) H_s vs. U10 multiplier

8b.png

(b) H_s vs. fetch multiplier

8c.png

(c) Reynolds number vs. depth

8d.png

(d) η vs. θ for K_d perturbations

Figure 8. Result of sensitivity calculation

Figure 8 shows findings that reinforce that wave diffraction response is highly sensitive to both atmospheric forcing and structural parameters.

4.8 Comparative insights with literature

To contextualize the findings of this study, a comparative analysis with recent diffraction research is presented in Table 6. This comparison underscores both the alignment of results with global trends and the novelty of explicitly positioning incident wave height as the primary driver in diffraction dynamics. The comparisons highlight important similarities and differences. Yu et al. [12] demonstrated asymmetric focusing around cylinders, where amplified crests reached up to 15%. A comparable but distinct phenomenon was observed at Gusung Island, with local amplification of 10 to 12%, yet governed by incident wave height rather than geometry. Likewise, He et al. [17] showed strong sensitivity of wave loads to steepness and depth, consistent with the trend observed here, where both K_d and η increased systematically with H_i.

Table 6. Comparative analysis for diffraction research

Shi et al. [10] and Firoj and Afzal [6] emphasized the nonlinear and dual role of diffraction dissipating energy, while amplifying localized loads findings mirrored in the Gusung Island case. While porous and Helmholtz-type breakwaters achieved energy reduction through porosity effects, this study confirmed that larger incident wave heights alone can achieve similar nonlinear dissipation (η up to 93%). Parallel results from Wang et al. [5] on aquaculture nets underline the universality of diffraction-induced localized loading across very different structural applications.

Collectively, these findings indicate that although geometry, porosity, and wave period have long been acknowledged as primary drivers of diffraction processes, the present study introduces a critical reframing: incident wave height itself emerges as a fundamental control parameter. This reconceptualization not only broadens the theoretical scope of diffraction analysis but also establishes a practical foundation for coastal engineering practice, whereby seasonal and climate-induced fluctuations in wave height should be explicitly integrated into the management of breakwaters.

4.9 Implications of applications

The findings demonstrate that breakwater performance is highly sensitive to seasonal variations in incident wave height. During high-energy monsoon periods, diffraction processes substantially attenuate incoming wave energy but simultaneously generate intensified localized loads at the breakwater tip. In contrast, under calmer seasonal conditions, the reduction in wave energy enhances tranquillity within sheltered areas yet promotes sediment deposition that may alter local morphodynamics. These trade-offs underscore the crucial importance of developing climate-resilient breakwater designs, particularly for small tropical islands like Gusung, where both extreme wave forcing and sediment management pose concurrent challenges.