Q: What is the horizontal velocity formula in Airy wave theory?

The horizontal particle velocity from linear (Airy) wave theory is u(z,t) = (πH/T)×cosh(k(z+d))/sinh(kd)×cos(kx−ωt), where H is wave height, T is period, k is wave number (k=2π/L), d is water depth, z is depth below the free surface (negative), and ω is angular frequency (ω=2π/T). The horizontal acceleration is du/dt = (2π²H/T²)×cosh(k(z+d))/sinh(kd)×sin(kx−ωt).

Q: How is the Morison equation applied to jacket structures?

For jacket structures, each member is divided into segments and the Morison equation wave force is integrated over the full depth. Total force F_total = Σ F(z_i)×Δz and overturning moment M = Σ F(z_i)×(z_i+d)×Δz. A four-leg jacket is typically evaluated as four times the single-pile result (accurate directional combination required in practice). Detailed design to DNV-OS-J101 or ISO 19902 combines this with nonlinear wave theory (Stokes 5th order).

Question 1

What is the Morison equation?

Accepted Answer

The Morison equation is a semi-empirical formula for calculating wave forces on offshore structural members. Written as F = ρ×Cm×V×(du/dt) + ½ρ×Cd×A×u|u|, the first term is the inertia force (due to fluid acceleration) and the second is the drag force (proportional to the square of fluid velocity). Cm is the inertia coefficient (typically 1.5–2.0) and Cd is the drag coefficient (typically 0.6–1.2), both varying with the Keulegan-Carpenter (KC) number.

Question 2

What is the KC number?

Accepted Answer

The Keulegan-Carpenter number KC = U_max×T/D, where U_max is the maximum horizontal velocity, T the wave period, and D the cylinder diameter. For KC < 5, inertia forces dominate; for KC > 20, drag forces dominate; and for 5 < KC < 20, both are important. Slender jacket legs (D ≈ 1 m) often have large KC values and are drag-dominated.

Question 3

What is the horizontal velocity formula in Airy wave theory?

Accepted Answer

The horizontal particle velocity from linear (Airy) wave theory is u(z,t) = (πH/T)×cosh(k(z+d))/sinh(kd)×cos(kx−ωt), where H is wave height, T is period, k is wave number (k=2π/L), d is water depth, z is depth below the free surface (negative), and ω is angular frequency (ω=2π/T). The horizontal acceleration is du/dt = (2π²H/T²)×cosh(k(z+d))/sinh(kd)×sin(kx−ωt).

Question 4

How is the Morison equation applied to jacket structures?

Accepted Answer

For jacket structures, each member is divided into segments and the Morison equation wave force is integrated over the full depth. Total force F_total = Σ F(z_i)×Δz and overturning moment M = Σ F(z_i)×(z_i+d)×Δz. A four-leg jacket is typically evaluated as four times the single-pile result (accurate directional combination required in practice). Detailed design to DNV-OS-J101 or ISO 19902 combines this with nonlinear wave theory (Stokes 5th order).

Offshore Structure Wave & Current Force

Morison Equation