Hydraulic performance of permeable low-crested structures under asymmetric water levels: Extension of empirical reflection formulae

Existing formulae for low-crested structures (LCSs) neglect water-level differences. This study examined how asymmetric still-water levels affect the reflection, transmission and energy dissipation coefficients ( K R , K T , and K D ) of a permeable rubble-mound LCS in 216 wave-flume tests with independently controlled seaward and leeward levels, thereby imposing the water-level difference as a boundary condition distinct from wave-induced piling-up. Tests covered seaward and leeward depths ( h 1 and h 2 , respectively) of 17, 19 and 21 cm, yielding Δ h = h 1 − h 2 values from −4 to +4 cm in 2 cm increments. Measured significant wave heights ranged from 0.89 to 11.98 cm, and significant wave periods were 1.0, 1.3 and 1.6 s. For 72 conditions with h 1 = h 2 , existing reflection formulae yielded coefficients of determination (R 2 ) of 0.898 and 0.880, whereas an established transmission formula showed limited applicability. Mean K R increased from 0.377 at Δ h = + 4 cm to 0.539 at Δ h = − 4 cm , and mean K D decreased from 0.918 to 0.826; both trends persisted at fixed h 1 . The K T response reversed between overtopping and non-overtopping conditions. Run-up and run-down imagery was consistent with changes in effective infiltration associated with the hydraulic gradient and internal saturation state. Sensitivity analysis identified Δ h / h 1 as most strongly associated with the residuals of existing reflection formulae. Multiplicative corrections, which recover the baseline formulae exactly when h 1 = h 2 , reduced the root-mean-square error (RMSE) by 41.19% and 43.56% and increased R 2 from 0.756 to 0.916 and from 0.761 to 0.924, respectively, across all 216 conditions.