The study, published in Results in Engineering, was motivated by a growing exposure problem. Climate change is intensifying tropical-cyclone wave hazard and the risk of coastal flooding, compounding the long-term vulnerability of waterfront infrastructure. Coastal cities bordering the South China Sea face particularly severe threats, and Hong Kong, with its densely developed shoreline, frequent south-easterly wave exposure, and a population exceeding seven million, is especially vulnerable. Recent decades have brought recurring impacts, including Typhoon Hato in 2017 and Typhoon Mangkhut in 2018, both of which caused severe coastal damage in Hong Kong and nearby cities. These events have sharpened interest in protection strategies that reduce wave impacts while minimising ecological disturbance and the environmental footprint of construction, precisely the niche that conventional hard defences such as seawalls and breakwaters struggle to fill.

Nature-based and bioinspired solutions have emerged as promising candidates for sustainable coastal protection in urbanised settings. Oyster reef restoration and 3D-printed artificial reef structures can provide simultaneous wave attenuation and ecological co-benefits, such as habitat provision and biodiversity enhancement, that hard defences rarely deliver. In Hong Kong, this prospect is particularly relevant because oyster farming has a long history, and restoration initiatives have recently been undertaken at Lau Fau Shan, Tolo Harbour, and the waterfront of Hong Kong International Airport. When such reef formations are arranged in spatially periodic arrays, they function as submerged bar arrays capable of attenuating wave energy through Bragg resonance, a mechanism in which surface waves are constructively reflected by periodic bottom topography when the bar spacing is approximately half the incident wavelength. Because the reflecting elements remain fully submerged and attenuate waves through interference rather than blocking, such arrays offer a complement to conventional defences with lower ecological and visual impact.

The physics of Bragg reflection on seabed topography has been established across theory, laboratory confirmation, engineering application, and nonlinear simulation. Early theoretical work by C. C. Mei in 1985 laid the foundation, while Davies and Heathershaw provided laboratory confirmation in 1984 that reflection is strongest when bar spacing is approximately half the incident wavelength and increases with the number of bars. Subsequent studies extended the framework to oblique incidence, higher-order resonance conditions arising from bottom and free-surface nonlinearity, and practical applications such as mitigating harbour oscillations induced by long waves and short wave groups. More recent analytical work has examined graded trapezoidal arrays that exploit a rainbow-reflection mechanism to produce broad-band attenuation, in which the spacing increment can be adjusted to broaden the effective frequency bandwidth of reflection. However, most of these studies considered simplified bottom geometries or idealised periodic bar arrays under unidirectional waves over prescribed horizontal, sloping, or reef-type bed profiles.

The key practical question addressed by the new research is whether engineered Bragg arrays remain effective when deployed over natural bathymetry under broadband tropical-cyclone forcing. In a realistic coastal setting, natural bathymetry continuously modifies local wavelength, wave direction, and group velocity through shoaling and refraction, so a spacing strategy optimised on an idealised bed may become detuned. Tropical-cyclone forcing further complicates performance assessment because the incident sea state is energetic, broadband, and directionally spread, requiring attenuation to remain effective beyond a single narrow resonance peak. For typhoon-exposed coastal cities such as Hong Kong, Macao, and Shenzhen, where conventional seawalls and breakwaters are constrained by ecological regulation, visual impact, and high construction cost in deep water, submerged alternatives that can be integrated into existing coastal management frameworks represent an important emerging option.

To answer this question, the team employed a fully nonlinear potential flow model, the Enhanced Spectral Boundary Integral (ESBI) solver, which combines efficient spectral computation with variable-depth capability and is applicable to large coastal domains. The model was first validated against three benchmark problems covering the three capabilities required for the case study: Bragg reflection by a graded trapezoidal bar array under regular waves, Bragg reflection of normally incident irregular waves over sinusoidal bars, and Bragg reflection under oblique wave incidence over sinusoidal ripples. In the first case, the model reproduced both the overall envelope and the local features of the reflection curve for an eight-bar graded array, achieving a root-mean-square error of 0.0255 against the analytical solution. In the second, comparisons against numerical results and analytical estimates for two test configurations yielded RMSE values between 0.0157 and 0.0484. In the third, the model reproduced the angular dependence of oblique Bragg reflection across incident angles from 0 to 45 degrees, with an RMSE of just 0.0084 against perturbation theory.

With the model validated, the researchers applied it to the Junk Bay region of Hong Kong, a semi-enclosed embayment on the southeastern coast of the New Territories whose mouth opens toward the South China Sea. The densely developed Tseung Kwan O district occupies the inner shoreline and is exposed to tropical-cyclone waves. The computational domain was a 19.6 kilometre by 19.6 kilometre square region discretised on a 4096 by 4096 grid with a spacing of approximately 4.8 metres, rotated so that the primary model axis aligned with the dominant incident wave direction. High-resolution bathymetry data were interpolated onto the rotated grid, allowing the natural seabed to be represented directly. The incident wave conditions came from a hindcast of Typhoon Mangkhut at the instant of maximum significant wave height, synthesised at the wave-making boundary as a linear superposition of 36 frequency bins and 24 directional bins, corresponding to a significant wave height of 9.92 metres, a peak period of 15.55 seconds, and a mean wave direction of 129.7 degrees.

The reference bar array consisted of eight trapezoidal bars, each 4 metres high with a 12-metre top width and 20-metre base width, extending approximately 850 metres transversely to cover the principal wave exposure zone of the bay. Bar heights at the transverse edges were tapered smoothly to zero using a Gaussian weighting function to avoid abrupt lateral depth transitions. The target mean spacing was selected from the local wave conditions: a baseline simulation over the natural bathymetry estimated a mean local wavelength of approximately 110 metres in the candidate deployment area, where the mean water depth is 10.5 metres, giving a target spacing of approximately 55 metres for the primary Bragg resonance condition. Performance was quantified by comparing simulations with and without the array under identical forcing, using the wave energy flux ratio, which accounts for the frequency- and depth-dependent propagation speed of wave energy, together with the protected-area fraction, which measures the share of the region of interest achieving at least a 40 percent reduction in wave energy flux.

The results revealed a pronounced attenuation zone shoreward of the bar array, where the energy flux ratio dropped to values between 0.4 and 0.8 over a broad area of the inner bay. Seaward of the array, a striped pattern of alternating enhancement and near-unity flux extended southward, consistent with a partial standing-wave field formed by the superposition of incident and reflected components. The graded array, with a grading ratio of 1.10, substantially outperformed the uniform array of the same mean spacing and bar count: the protected-area fraction increased from 0.291 to 0.655, more than doubling the areal coverage of cells experiencing at least 40 percent reduction, while the mean flux ratio decreased from 0.693 to 0.634. Frequency-resolved spectral analysis along a shore-normal transect showed that the uniform array concentrated attenuation within approximately 0.06 to 0.08 hertz, whereas the graded array extended this band to approximately 0.05 to 0.09 hertz, yielding a wider simulated reflection bandwidth that agrees qualitatively with the rainbow-reflection behaviour reported in earlier analytical work.

Sensitivity analyses revealed important design trade-offs. Increasing the number of bars from 6 to 12 produced a non-monotonic response: the protected-area fraction rose sharply from 0.386 to 0.655 between 6 and 8 bars, changed marginally to 0.661 at 10 bars, and then decreased to 0.590 at 12 bars. Two site-specific effects may explain this reversal. As the array extends seaward, the outermost bars sit in progressively deeper water, where the reduced relative bar height weakens the wave-bottom interaction and diminishes their Bragg resonance contribution. The longer array may also introduce more complex wave interactions at the lateral edges that redistribute energy within the protected region. Bar height, by contrast, showed a monotonic improvement: raising the height from 3.2 to 4.8 metres decreased the mean flux ratio from 0.752 to 0.531 and increased the protected-area fraction from 0.046 to 0.788, with the largest single gain occurring between 3.6 and 4.0 metres. Taller bars reduce crest submergence and strengthen the wave-bottom interaction but require more construction material and provide less navigational clearance.

The authors caution that the physical scope of the model follows from its fully nonlinear potential-flow formulation, which assumes inviscid and irrotational flow, so turbulence and viscous effects are not explicitly represented, and wave breaking is handled through a high-wavenumber filter and dissipation treatment rather than resolved explicitly. Coastlines were represented as wave-absorbing boundaries, so shoreline run-up and overtopping were not simulated, meaning the reported reductions in wave energy flux should not be interpreted as direct predictions of overtopping reduction. Uncertainty from random-phase realisations was assessed with three additional realisations, showing that the mean attenuation was relatively insensitive to phase realisation, whereas the protected-area fraction showed larger variation and should be interpreted with caution. No site-specific wave measurements were available for direct field verification. Nevertheless, for the simulated case, approximately 60 percent of the protected region was predicted to experience at least a 40 percent reduction in wave energy flux under the selected Mangkhut sea state. Because the structures remain fully submerged, they impose no visual barrier above the waterline, and when constructed from biocompatible substrate, such arrays could support natural oyster colonisation over time, potentially combining the simulated wave-attenuation function with reef habitat provision. The findings suggest that coastal managers in Hong Kong and similar South China Sea cities may consider graded submerged bar arrays as a complementary element within integrated coastal defence strategies, particularly for exposed embayments where hard-structure alternatives face regulatory, ecological, or aesthetic constraints, and that effective deployment requires site-specific hydrodynamic design matched to the local wavelength distribution rather than simply maximising array length or height.

Subject of Research: Site-scale simulation of graded submerged bar arrays for Bragg-resonant wave attenuation over natural bathymetry under tropical-cyclone wave conditions in Junk Bay, Hong Kong.

Article Title: Graded submerged bar arrays for nature-based coastal wave attenuation under tropical-cyclone conditions

Article References: Zhang, J., Gao, C., Wang, J., Stocchino, A., Duan, H., & Ning, D. (2026). Graded submerged bar arrays for nature-based coastal wave attenuation under tropical-cyclone conditions. Results in Engineering, 32, Article 113456. https://doi.org/10.1016/j.rineng.2026.113456

Image Credits: AI Generated

DOI: 10.1016/j.rineng.2026.113456

Keywords: Bragg resonance, coastal protection, nature-based solutions, submerged bar arrays, wave attenuation, tropical cyclones, Typhoon Mangkhut, Junk Bay, oyster reefs, rainbow reflection, Hong Kong, coastal resilience