Rubble-mound and dike design

How much rock you need per statute mile of waterfront is set by wave height to the cube, slope, density, how open the mound is, and the trapezoid you actually build. This page uses published design formulae. It is not a bid. It is the state of the art the grown wall is priced against; the sandbar page runs both at the same sea.

peer-reviewed formula from tests   manual CEM / Rock Manual / SPM   claim rule of thumb   input your number

What geometry does

A conventional mound is a trapezoid: seaward slope, crest, lee slope, core, filter, armour, toe. The Rock Manual (CIRIA C683, 2007) is the geometry source used here.

• Slope — generally not steeper than 1:1.5 (natural dumped repose can be ~1:1.2). Steeper saves volume but Hudson stone mass scales as 1/cot α, so the armour units get heavier.
• Crest width B — at least three armour stones (three to four typical). If dump trucks build the core, B is often set by plant, not by hydraulics.
• Freeboard R_c — crest minus still-water level. Positive = emerged; negative = submerged. This is the first-order control on transmission and overtopping. Tide and surge move R_c.
• Armour thickness — randomly placed stone needs a double layer: t_a = n k_t D_n50 with n = 2. Rock Manual Table 3.9, double standard, blocky rock, reference survey: k_t = 0.91, n_v = 32%.
• Toe — at least three stones wide. On sand, scour protection should reach at least 6 m or H_s from the toe if scour is expected.

A dike/revetment is the same seaward face on an impermeable core (clay/sand). The core is not armourstone. Van der Meer’s notional permeability for that case is P = 0.1, not the volumetric porosity n_v of the armour.

Porosity vs permeability

Two different numbers. Do not swap them.

Stone size

manual Hudson (SPM 1984 / EM 1110-2-1614 Eq. 2-15): M₅₀ = ρ_r H³ / [K_D Δ³ cot α], with Δ = ρ_r/ρ_w − 1 and D_n50 = (M₅₀/ρ_r)^1/3.

SPM Table 7-8 / EM Table 2-3, rough angular quarry stone, two layers, random, trunk, breaking waves: K_D = 2.0 (smooth rounded: 1.2). Non-breaking trunk, rough angular: K_D = 4.0. Roundhead K_D is lower (more mass). SPM “no damage” is still up to ~5% of units moved. SPM 1984 wants H = H_1/10 in Hudson, not H_s (H_1/10 ≈ 1.27 H_s for Rayleigh). The 1973 SPM used K_D = 3.5 for breaking; 1984 cut it to 2.0 and switched to H_1/10 — about 3.5× the stone mass if you apply both changes (van der Meer 1998).

peer-reviewed van der Meer (1988; restated 2021): H_s/(Δ D_n50) = 6.2 P^0.18 (S/√N)^0.2 ξ_m^−0.5 (plunging); = 1.0 P^−0.13 (S/√N)^0.2 √(cot α) ξ_m^P (surging). ξ_m = tan α / √s_m, s_m = 2π H_s/(g T_m²).

Use van der Meer for rock: it includes period, storm length, P, and a defined damage S. Hudson is the simple check and the only K_D table for many concrete units. Range (van der Meer 1998 / CEDD): cot α = 1.5–6; N = 1000–7500 (cap at 7500); s_m = 0.005–0.06. For cot α > 3, the plunging/surging switch uses the cot α = 3 value of ξ_mc.

Erosion — three different things

1. Armour damage S = A_e / D_n50². A_e is the eroded area of the slope profile. S = 2–3 is start of damage, the same “Hudson 5%” idea. Failure = underlayer visible: S = 8 on 1:1.5 and 1:2; S = 12 on 1:3; S = 17 on 1:4 and 1:6 (CEDD Table C1 / van der Meer Table 1). Actual rocks displaced in a D_n50-wide strip ≈ 0.7–1.0 × S.
2. Toe / seabed scour — standing-wave streaming in front of the structure. Sumer & Fredsøe (2000): rubble-mound scour is smaller than at a vertical wall; it decreases as the slope gets gentler; irregular waves scour less than regular (often ~half). Stones sitting on the bed can slump into the hole (~30% less toe scour). There is no single S/H formula that is safe to use without the original diagrams — this page does not invent one.
3. Beach in the lee of a low-crested or gapped breakwater can accrete or retreat. Transmission over the crest plus rip currents through gaps can export sand. Dean et al. (1997) is the Palm Beach submerged-breakwater field case where the net sand balance landward of the structure went negative. K_t from a 2-D formula does not give that 3-D cell.

Wave transmission K_t = H_t/H_i

peer-reviewed d’Angremond, van der Meer & de Jong (1997): K_t = −0.4 (R_c/H_i) + c₂ (B/H_i)^−0.31 [1 − exp(−0.5 ξ)], with c₂ = 0.80 impermeable and 0.64 permeable (porosity ~0.5). Clip to 0.075–0.80. If B/H_i > 10, Briganti et al. (2004) replace the coefficients (−0.35, 0.51, −0.65, −0.41) because the original overestimates wide crests.

Ferrario et al. 2014 (~97% dissipation) is natural coral reefs, not a rubble mound and not Biorock. Do not use it as K_t = 0.03 for a mesh cage.

How each knob moves the answer

Material for a mile of waterfront

Empty cells are omitted, not zero. Defaults that are filled are the published design values named in the label. Type H_s before anything sizes. Van der Meer and K_t also need a period. Volume needs depth and freeboard. Trunk section only — roundheads need more armour. Optional length in feet scales the mile bill down or up.

Sources

• Hudson, R.Y. 1959/1961 — WES rubble-mound tests. Formula as in SPM 1984 and EM 1110-2-1614 Eq. 2-15
• USACE Shore Protection Manual, 4th ed. 1984 — Table 7-8 K_D; H_1/10; underlayer mass 1/10–1/15 of armour (cited in HR Wallingford SR 631)
• USACE EM 1110-2-1614 (1995) — Design of Coastal Revetments, Seawalls, and Bulkheads. Table 2-3 K_D; layer thickness r = n k_Δ (W/γ_r)^1/3
• Van der Meer, J.W. 1988 PhD / ICCE 21; 1998 in Herbich ch. 11; 2021 J. Coastal & Hydraulic Structures doi:10.48438/jchs.2021.0008
• CEDD Port Works Design Manual App. C — Hudson + van der Meer as in BS 6349-7:1991; Table C1 damage S; P = 0.3 if no data
• CIRIA/CUR/CETMEF C683 Rock Manual (2007) — Ch. 3 Table 3.9 k_t, n_v; Eq. 3.23–3.28; Ch. 6 crest ≥ 3 stones, double-layer t_a = 2 k_t D_n50, toe ≥ 3 stones
• d’Angremond, K., van der Meer, J.W., de Jong, R.J. 1997. Wave transmission at low-crested structures. Coastal Engineering 1996, ASCE, 2418–2427
• Briganti, R. et al. 2004. Wave transmission behind low-crested structures. Coastal Structures 2003
• Sumer, B.M. & Fredsøe, J. 2000. 2D scour at a rubble-mound breakwater. Coastal Engineering 40: 59–87. doi:10.1016/S0378-3839(00)00006-5
• HR Wallingford / Defra SR 631 — armour porosity 25–40%; SPM underlayer 1/10–1/15
• Ferrario et al. 2014 doi:10.1038/ncomms4794 — reef dissipation; not a rubble-mound K_t
• Huang et al. 2024. Wave transformation across impermeable and porous artificial reefs. Coastal Engineering 189, 104488 (n = 0.68 reefs)
• Dean, R.G., Chen, R., Browder, A.E. 1997. Palm Beach submerged breakwater monitoring. Coastal Engineering 29: 291–315