Chapter IV
Design of rubble mounds breakwater
57
shallow water conditions:
Table IV-4 Plunging and Surging Waves Van der Meer Formulae for shallow Water Conditions.
Van der Meer formulae
shallow water
Variables
For plunging waves (𝜉 𝑠−1,0 <𝜉 𝑐𝑟 ):
𝐻 𝑠
∆𝐷 𝑛50
= 𝑐 𝑝𝑙 𝑃
0.18 (
𝑆 𝑑
√𝑁
)
0.2 (
𝐻 𝑠
𝐻 2%
)ξ 𝑠−1,0
−0.5
and for surging waves (𝜉 𝑠−1,0 ≥ 𝜉 𝑐𝑟 ):
𝐻 𝑠
∆𝐷 𝑛50
=
𝑐 𝑝𝑙 𝑃
−0.13 (
𝑆 𝑑
√𝑁
)
0.2 (
𝐻 𝑠
𝐻 2%
)√𝑐𝑜𝑡𝛼ξ 𝑠−1,0
𝑝
where:
𝜉 𝑠−1,0 =
𝑡𝑎𝑛𝛼
√
2𝜋𝐻 𝑠
𝑔𝑇 𝑠−1,0
2
𝜉 𝑐𝑟 = (
𝑐 𝑝𝑙
𝑐 𝑠
𝑃
0.31 √𝑡𝑎𝑛𝛼)
1
𝑃+0.5
For 𝜉 𝑠−1,0 <𝜉 𝑐𝑟 waves are plunging
For 𝜉 𝑠−1,0 ≥ 𝜉 𝑐𝑟 waves are surging
𝑵: number of incident waves at the toe, which
depends on the duration of the wave conditions.
𝐇 𝐬 : significant wave height, H1/3 of the incident
waves at the toe of the structure (m).
𝑯 𝟐% : wave height exceeded by 2 per cent of the
incident waves at the toe (m).
𝛏 𝐦 : surf similarity parameter using the mean wave
period, Tm (s), from time domain analysis.
𝛏 𝐜𝐫 : critical value of the surf similarity parameter.
𝛼: slope angle (°).
𝛥: relative buoyant density.
𝑃: notional permeability of the structure.
𝑻 𝒔−𝟏,𝟎 : the (spectral) mean energy wave period (s).
𝐜 𝐩𝐥 : 8,4
𝐜 𝐬 : 1,3
Conditions: shallow water (water level at the toe, i.e., h < 3𝐻 s−toe ) at which 𝐻 s−toe < 70 % of
the deep-water wave height, 𝐻 so (offshore significant wave height).
NOTE: the locally determined values of H2% and 𝑇 m−1,0 should be used; a numerical wave
propagation model.
Geometric parameters of the armour stone
The geometric parameters of the armour stone in the design of coastal protection structures
include the volume of a block, nominal diameter of the block, armour stone thickness, and the
number of blocks per square meter. These parameters are determined based on the median mass
of the armour stone and the layer coefficient.
Table IV-5: The formulas and variables used to calculate geometric parameters of the armorstone.
Formula
variables
Volume of a block:
𝑉 =
𝑀 50
𝜌 𝑟
𝑴 𝟓𝟎 : the median mass of armour stone.
𝛒 𝒓 : the apparent rock density (kg/m³).
𝑛: number of layers.
𝒌 ∆ : layer coefficient.
Nominal diameter of bloc: 𝑫 𝒏𝟓𝟎 = √𝑽
𝟑
Armourstone thickness: 𝒆 = 𝒏𝒌 ∆ 𝑫 𝒏
Number of blocks per square meter:
𝑵 𝒔=
𝒏
𝑫 𝒏
𝟐
Design of under-layer
The design of the under-layer involves determining the size and thickness of the smaller rocks or
stones based on certain parameters. The formulas mentioned in the upcoming table are used to
calculate the dimensions of the under-layer blocks. These dimensions include the median mass,
volume, nominal diameter, and thickness.