Wave Spectra, Hs and Tp for Offshore Lifting

Describing a sea state

• Significant wave height, H_s. Originally the mean height of the highest third of the waves. In spectral work it is computed from the area under the wave spectrum, its zeroth moment m_0: H_s = 4\sqrt{m_0}.
• Peak period, T_p. The period at which the spectrum peaks, where the wave energy is most concentrated.
• Zero-up-crossing period, T_z. The average time between upward crossings of the still-water level, T_z = 2\pi\sqrt{m_0/m_2}. It is shorter than T_p: T_p \approx 1.40\,T_z for Pierson-Moskowitz and 1.29\,T_z for JONSWAP with \gamma = 3.3.
• Direction and spreading. The mean direction the waves come from, relative to the vessel’s heading, and how widely their energy is spread around it. The vessel’s response, and so the crane-tip motion, changes strongly with heading, so a lift is analysed over a range of headings around the planned one.

A sea state is usually treated as stationary for anything from 20 minutes to 3 to 6 hours. Within it, individual wave heights follow approximately a Rayleigh distribution, and the most probable largest of N waves is

so in a three-hour sea, which holds 10{,}800\ \text{s}/T_z waves, about 1,000 to 2,000, the most probable largest wave is about 1.9 times H_s (Figure 1).

Pierson-Moskowitz and JONSWAP

The wave spectrum S(\omega) says how the energy of the sea is spread over the wave frequency \omega. DNV-RP-N103 gives the two standard forms. The Pierson-Moskowitz (PM) spectrum describes a fully developed sea, where the wind has blown long enough over a long enough fetch for the waves to stop growing:

The JONSWAP spectrum, from the Joint North Sea Wave Project, describes a sea still growing under the wind. It multiplies the PM peak by a factor \gamma and scales the whole spectrum by A_\gamma so that H_s stays the same; for \gamma = 3.3 the peak ends up about 2.2 times the PM peak:

with \sigma = 0.07 below the peak and 0.09 above it. The average \gamma from the JONSWAP measurements is 3.3, and \gamma = 1 gives back the PM spectrum. Where no \gamma is given, DNV-RP-N103 takes it from the steepness of the sea: \gamma = 5 for T_p/\sqrt{H_s} \le 3.6, \gamma = 1 for T_p/\sqrt{H_s} \ge 5, and \gamma = \exp(5.75 - 1.15\, T_p/\sqrt{H_s}) between, with T_p in seconds and H_s in metres. JONSWAP is a reasonable model for 3.6 < T_p/\sqrt{H_s} < 5, and should be used with caution outside it.

Where wind sea and swell arrive together, the spectrum has two peaks, and a two-peak form such as Ochi-Hubble or Torsethaugen describes it better. Swell can arrive from a different direction than the wind sea, and its long period sits closest to the natural period of a passive compensator. The metocean basis of the project says which spectrum and which periods to use.

From waves to crane-tip motion

The vessel turns the waves into motion through its response amplitude operators (RAOs), one for each motion and wave direction. At the crane tip, heave, roll and pitch combine with their phases: roll acts through the tip’s sideways offset and pitch through its fore-and-aft offset, both measured from the point the vessel’s RAOs refer to. The crane-tip motion spectrum is the wave spectrum times the square of that combined RAO. Four times the square root of its area is the significant (double-amplitude) motion, as H_s = 4\sqrt{m_0} is for the waves, and the most probable largest motion follows from it for the duration of the operation: for a pass through the wave zone of up to 30 minutes, DNV-RP-N103 takes 1.80 times the significant response. Our vessel motions and RAOs article works through it.

The shape of the spectrum matters most where its peak sits near a natural period: the vessel’s own roll or heave, or the lift’s. A narrow JONSWAP peak there drives a larger response than a broad PM spectrum with the same H_s; a little away from it the narrow peak does less, and far from it the two give about the same response. See resonance avoidance.

What the waves mean for a compensator

The compensator’s stroke comes from the motion between the crane tip and the load in the design sea states: the crane-tip motion from the RAOs and the spectrum, applied in a lift model with the wire, the rigging, the compensator and the payload’s hydrodynamics. Multiplying H_s by a heave RAO leaves out roll, pitch, phase, the spread of the wave energy over period and the statistics of the largest motion, so it does not give a design stroke. The stroke also has to cover the shift in the compensator’s mid-position as the load goes from air into water.

The peak period matters as much as the height. A passive compensator isolates the load only from waves shorter than about its natural period divided by √2, with the compensator in line, and it amplifies motion near that period, so the lift model checks the wave periods against it (see resonance avoidance). For the splash zone, where the wave period is taken into account, the simplified method of DNV-RP-N103 covers zero-up-crossing periods from 8.9\sqrt{H_s/g} up to 13 s for each design H_s: for H_s 4.0 m, T_z from 5.7 to 13 s, which is T_p of about 7.3 to 16.7 s for JONSWAP with \gamma 3.3.

Sea states, operating limits and weather windows

A lift’s limit is not one number but an envelope over H_s and T_p: at each peak period and heading, the largest H_s at which every load, motion and stroke check still passes. Operability screening sets that envelope against a site’s wave statistics, and a weather window needs the forecast to stay inside it for the whole reference period, the planned operation plus contingency, with the forecast’s uncertainty covered by the alpha factor of DNV-ST-N001.