Examples covering the application of Hudson's Formula for rubble mound breakwater armor unit design.

A rubble mound breakwater is being designed to protect a harbor entrance. The design significant wave height ($H$) is 4.5 meters. The breakwater will be armored with rough, angular quarrystone placed randomly in two layers. The specific weight of the stone ($\gamma_r$) is 26.0 kN/m³, and the specific gravity relative to seawater ($S_r$) is 2.65. The stability coefficient ($K_D$) for this stone type and placement is 4.0. The seaward slope of the breakwater is 1V:2H ($\cot \theta = 2.0$). Determine the required minimum weight ($W$) of an individual armor stone using Hudson's Formula.

Due to a lack of local quarrystone large enough to meet the 65.9 kN requirement from the previous example, engineers propose using interlocking concrete Tetrapods. The design wave height ($H = 4.5 \text{ m}$) and slope ($\cot \theta = 2.0$) remain identical. The specific weight of the concrete ($\gamma_r$) is 23.5 kN/m³ ($S_r = 2.40$). The stability coefficient for Tetrapods ($K_D$) is significantly higher at 8.0. Determine the required weight of an individual Tetrapod.

Using the rough angular quarrystone from the first example ($\gamma_r = 26.0 \text{ kN/m}^3$, $S_r = 2.65$, $K_D = 4.0$), the design wave height suddenly increases to an extreme $H = 6.0 \text{ m}$ due to updated climate models. The heaviest stones the local quarry can physically produce weigh 100 kN. The original slope was 1V:2H ($\cot \theta = 2.0$). To make the 100 kN stones stable under the new 6.0 m wave, determine the required slope ($\cot \theta$) the breakwater must be built at.