About Wave Load Calculator (Morison Equation)
The wave load calculator estimates the maximum horizontal wave force on a slender vertical cylindrical pile, such as a jetty, jacket or monopile member, using the Morison equation. It first solves the linear dispersion relation for the wave number and wavelength, then evaluates the linear (Airy) water-particle velocity and acceleration and splits the load into an inertia component and a drag component.
Enter the wave height H, wave period T, water depth d, pile diameter D, the drag coefficient Cd (about 0.7 to 1.2) and the inertia coefficient Cm (about 1.5 to 2.0), in consistent SI units with seawater density 1025 kg/m^3. The tool reports the inertia force f_i = Cm rho (pi/4 D^2) a and drag force f_d = 0.5 Cd rho D u|u| per unit length, the depth-integrated total horizontal force on the pile and its overturning moment about the seabed.
How It Works
- Enter H (m), T (s), water depth d (m), pile diameter D (m), and the coefficients Cd and Cm.
- The angular frequency is omega = 2 pi / T; the wave number k is found from the dispersion relation omega^2 = g k tanh(kd) and the wavelength is L = 2 pi / k.
- The Airy kinematics give u_max = (pi H / T) cosh(k(z+d)) / sinh(kd) and a_max = (2 pi^2 H / T^2) cosh(k(z+d)) / sinh(kd).
- The Morison force per unit length is f = Cm rho (pi/4 D^2) a + 0.5 Cd rho D u |u|, separating inertia and drag.
- Integrating the peak envelopes over the depth gives the total inertia, drag and combined force, and integrating their moment about the bed gives the overturning moment.
Worked Example
A vertical pile of diameter D = 1 m stands in d = 10 m of seawater (rho = 1025 kg/m^3). The design wave has height H = 4 m and period T = 8 s, with Cd = 1.0 and Cm = 2.0. The angular frequency is omega = 2 pi / 8 = 0.7854 1/s; solving omega^2 = g k tanh(kd) gives k = 0.0886 1/m and a wavelength L = 70.9 m. At the still-water surface the peak velocity is u_max = (pi x 4 / 8) cosh(kd) / sinh(kd) = 2.214 m/s and the peak acceleration is a_max = (2 pi^2 x 4 / 64) cosh(kd) / sinh(kd) = 1.739 m/s^2. The per-unit-length inertia force is f_i = 2.0 x 1025 x (pi/4 x 1^2) x 1.739 = 2800 N/m and the drag force is f_d = 0.5 x 1.0 x 1025 x 1 x 2.214^2 = 2512 N/m. Integrating the peak envelopes over the 10 m depth gives a total inertia force of 22.4 kN, a total drag force of 16.3 kN, a combined maximum horizontal force of about 38.7 kN, and an overturning moment about the seabed of about 210 kN.m.
Formulas
- Dispersion relation
omega^2 = g * k * tanh(k * d) ; omega = 2*pi/T ; L = 2*pi/k- Airy water-particle velocity (amplitude)
u_max = (pi * H / T) * cosh(k(z+d)) / sinh(k*d)- Airy water-particle acceleration (amplitude)
a_max = (2 * pi^2 * H / T^2) * cosh(k(z+d)) / sinh(k*d)- Morison force per unit length
f = Cm * rho * (pi/4 * D^2) * a + 0.5 * Cd * rho * D * u * |u|- Depth-integrated peak forces
F_i = Cm rho (pi/4 D^2)(2 pi^2 H/T^2)/k ; F_d = 0.5 Cd rho D (pi H/T)^2/sinh^2(kd) * [d/2 + sinh(2kd)/(4k)]
Standards & References
- Morison equation (Morison et al., 1950)
- API RP 2A-WSD Fixed Offshore Platforms
- Water Wave Mechanics for Engineers and Scientists (Dean & Dalrymple)
- Shore Protection Manual / Coastal Engineering Manual
Frequently Asked Questions
When is the Morison equation valid for wave loading?
The Morison equation applies to slender members where the diameter is small relative to the wavelength, roughly D/L < 0.2, so the structure does not significantly scatter the incident wave. For large bodies diffraction becomes important and a diffraction (MacCamy-Fuchs) analysis is needed instead. This calculator assumes a slender pile and uses linear Airy wave kinematics.
Why are the inertia and drag forces added directly for the total?
The inertia force is maximum when the water-particle acceleration peaks (at the wave node) while the drag force is maximum when the velocity peaks (at the crest), so the two components are 90 degrees out of phase and do not actually peak together. Summing the two depth-integrated envelopes gives a conservative upper bound that is common in preliminary design; a phase-resolved time history gives the true maximum, which is smaller.
How are Cd and Cm chosen?
The drag coefficient Cd and inertia coefficient Cm depend on the Reynolds number, the Keulegan-Carpenter number and surface roughness. Typical design values are Cd around 0.7 to 1.2 and Cm around 1.5 to 2.0 (Cm = 2.0 is the theoretical value for a smooth cylinder in ideal flow). API RP 2A gives recommended values for smooth and marine-growth-roughened members.
What does the calculator integrate the force over?
It integrates the peak Morison force per unit length from the seabed (z = -d) up to the still water level (z = 0), which captures the bulk of the load. The contribution between the still water level and the wave crest is neglected here; for steep or shallow-water waves a stretched-kinematics or higher-order wave theory should be used to include that surface zone.