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Coastal Hydraulics, Sea-Dike Overtopping & Wave Mechanics

Iribarren Surf Similarity Parameter & Wave Runup Formula

Calculate Deep-Water Wavelength L_0 (m), Iribarren Surf-Similarity Number xi_0, Smooth-Dike 2% Exceedance Wave Runup Height R_u2% (m), and Wave Breaker Regime.

Long-period swell surges higher up sea dikes than steep wind chop of the SAME wave height—governed by Iribarren xi_0!

Two coastal storms with identical offshore significant wave height H_0 = 2.5 m produce vastly different sea-dike overtopping depending on wave period T (s) and foreshore/dike slope tan_beta! Deep-water dispersion fixes Offshore Wavelength L_0 = (g * (T^2)) / (2 * PI()) = 1.5613 * T^2 (m), defining Battjes/Iribarren Surf-Similarity Parameter xi_0 = tan_beta / SQRT(H_0 / L_0)! Gentle beaches/short chop (xi_0 < 0.5) break gradually as "SPILLING" foam; intermediate slopes (0.5 <= xi_0 < 3.3) curl violently as "PLUNGING" breakers; steep seawalls (xi_0 >= 3.3) surge upward as "COLLAPSING-SURGING"! Under EurOtop / Hunt’s Smooth-Slope Runup Law, 2%-Exceedance Vertical Wave Runup is R_u2% = MIN(3.2 * H_0, 1.6 * H_0 * xi_0)—rising linearly with both slope AND wave period!

Verify on a 1V:10 Smooth Sea-Dike Slope—Dike_Slope tan_beta = 0.10 (B2), Deep_Water_Wave_Height H_0 = 2.50 m (B3), and Swell_Period T = 9.0 s (B4): Deep-Water Wavelength L_0 hits (9.81 * 81) / (2 * PI()) = 126.47 m (E2), Iribarren Parameter xi_0 equals 0.10 / SQRT(2.50 / 126.47) = 0.711 (E3), 2% Wave Runup Elevation R_u2% reaches 1.6 * 2.50 * 0.711 = 2.84 m above still-water storm surge (E4), and Breaker Type classifies as "PLUNGING" (E5)!

Iribarren Dike Runup KPI (tan_beta = 0.10, H0 = 2.5 m, Period T = 9.0 s)FormulaVerified Output
Deep-Water Wavelength L0 (m)(9.81 * (Period_T^2)) / (2 * PI())126.47 m
Iribarren Number xi_0Slope_tan_beta / SQRT(H0 / L0)0.711 (1.778 @ 1V:4 slope 0.25)
Smooth 2% Wave Runup Ru2% (m)MIN(3.2 * H0, 1.6 * H0 * xi_0)2.84 m (7.11 m @ 1V:4 slope!)
Surf-Zone Breaker Regime<0.5 Spilling; <3.3 Plunging; >=3.3 SurgingPLUNGING

EurOtop / Hunt Iribarren 2% Exceedance Wave Runup Formula

Validate Slope tan_beta in (0.005, 1.0], Wave_Height H_0 > 0, Wave_Period T > 1.0 s, and enforce deep-water physical wave steepness H_0 / L_0 <= 0.142 (Miche wave-breaking limit in open water)!

=LET(tb_s, B2, h0_m, B3, tp_s, B4, IF(OR(tb_s <= 0.005, tb_s > 1, h0_m <= 0, tp_s <= 1), "Invalid Wave Input", LET(l0_m, (9.81 * (tp_s ^ 2)) / (2 * PI()), IF((h0_m / l0_m) > 0.142, "WAVE EXCEEDS MICHE LIMIT", LET(xi_0, tb_s / SQRT(h0_m / l0_m), ROUND(MIN(3.2 * h0_m, 1.6 * h0_m * xi_0), 2))))))

Steeper 1V:4 Dike Crest Overtopping Surge Check (Set Slope tan_beta = 0.25)

Steepen Dike_Slope B2 from 0.10 (1V:10 gentle dike) to 0.25 (1V:4 steep revetment): Iribarren Number E3 jumps 2.5x to 1.778 and Vertical 2% Wave Runup E4 surges from 2.84 m up to 7.11 m—demanding a much higher dike crest or rough rock berm!

=IF(ISTEXT(E4), E4, LET(l0_m, (9.81 * (B4 ^ 2)) / (2 * PI()), xi_0, B2 / SQRT(B3 / l0_m), IF(xi_0 < 0.5, "SPILLING", IF(xi_0 < 3.3, "PLUNGING", "COLLAPSING-SURGING"))))

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