Ocean Engineering

Every wave problem on this paper starts with the same question: how does the seabed feel to this particular wave? A 12 second swell in 200 m of water behaves nothing like the same-period wave running into a 15 m approach channel, and the dispersion relation is what connects wave period, wavelength and depth so you can tell the two situations apart before reaching for a formula.

Physically, tanh(kd) is a bottom-feeling term: it climbs toward 1 once kd is large (deep water — the bed is too far away to matter) and collapses toward kd itself once kd is small (shallow water — the whole water column moves together). Between those limits, roughly 0.05 < d/λ < 0.5, neither shortcut is valid and the full transcendental relation has to be solved, normally by iterating on k with a calculator or a short spreadsheet loop.

The deep-water shortcut is the one that gets misapplied most often, because it is the formula everyone memorises first. It assumes the orbital motion has died away completely before it reaches the seabed, which only holds once the wave "feels" more than about half its own wavelength of water beneath it. Applying λ = 1.56T² to a wave that is actually in intermediate depth doesn't just introduce a small arithmetic error — it quietly shifts every downstream number: wavenumber, phase speed, and especially group velocity.

A single sinusoidal wave train is a convenient fiction — real energy travels in groups, and the envelope of a wave group moves at the group velocity c_g, not at the phase speed c of the individual crests inside it. Watch a group of waves arrive at a breakwater and you'll see crests appear at the back of the group, run forward through it, and vanish at the front: the crests move faster than the parcel of energy carrying them.

In deep water the bracket collapses to ½, so c_g = c/2 exactly — energy takes twice as long to cross a stretch of open ocean as the individual wave crests do. In shallow water the bracket climbs to 1 and c_g → c: once the whole water column moves together, there is no longer a distinction between how fast the shape moves and how fast the energy moves. This matters directly for wave power, because the energy flux past a line is proportional to c_g, not c.

Taking c_g as c/2 regardless of depth is the classic slip: in intermediate or shoaling water the bracket term climbs well above ½, so the deep-water shortcut understates both group velocity and wave power — sometimes by a large margin, as Example 1 shows. Group velocity is also what governs shoaling: as a wave group moves into shallower water c_g falls, and because the energy flux has to be conserved, the wave height rises to compensate.

A real sea surface is not one wave but a superposition of many, and the honest way to describe it is statistically, through a wave spectrum S(ω) that shows how much energy sits at each frequency. The spectral moments compress that spectrum into single numbers a designer can actually use.

H_s is not the height of any one particular wave — it is defined as the mean of the highest third of all waves in the record, and it turns out to work out to almost exactly four times the square root of the area under the spectrum. For a narrow-banded sea, individual wave heights follow a Rayleigh distribution, which is what lets you go one step further than H_s and estimate the largest wave you're actually likely to meet in a given storm duration, rather than quoting H_s itself as though it were a design maximum.

The zero-crossing period T_z, read straight off a wave trace as the mean time between successive upward crossings of the mean water level, is what lets you convert a storm duration into a wave count, and from there into an extreme-value estimate using the Rayleigh distribution — worked through in full in Example 2.

A ship or floating structure has six rigid-body degrees of freedom — surge, sway, heave, roll, pitch and yaw — and in a seaway each one responds to the waves passing beneath it. The response amplitude operator, or RAO, is simply the ratio of the motion amplitude to the wave amplitude that caused it, plotted against frequency.

The frequency that matters to the vessel is not the wave frequency ω an observer on the shore would measure, but the encounter frequency ω_e — the rate at which the moving vessel actually meets successive crests, which depends on heading as well as speed.

In head seas the vessel meets crests faster than a stationary observer would, so ω_e is raised above ω; running with the sea in following conditions lowers it, and at the right speed and heading ω_e can pass through zero — the vessel then sits almost stationary relative to the wave pattern, which is exactly the condition to watch for near a roll or pitch resonance. Every RAO curve peaks close to the structure's own natural frequency in that mode, and the practical design task is keeping that peak away from the encounter frequencies the vessel will actually see in service, particularly for roll, where damping is naturally low and resonant amplification can be severe.

A moored structure is, in effect, a mass-spring system with the mooring lines supplying most of the spring. In a catenary mooring, the restoring force comes from lifting line weight clear of the seabed as the structure is pushed off station — soft at small offsets and stiffening as more chain lifts. A taut mooring instead relies on the elastic stretch of the line itself, giving a stiffer, more linear response and a smaller seabed footprint, at the cost of higher peak line tension.

Either way, the mooring system has its own natural period in surge, sway and yaw, set by the platform's mass (plus added mass) and the system stiffness. The design aim is the same one that runs through this whole topic: keep that natural period well clear of the energetic band of the sea states the structure will experience, typically by pushing it out beyond the wave-period range entirely, into the tens-of-seconds region where wave-frequency energy is negligible.

Slow-drift forces — second-order, arising from the wave groups rather than the individual waves — are what excite that long-period resonance in practice, even though the first-order wave loads themselves are too fast to do it directly. A mooring check that only compares first-order loads against a stiff spring model, and never asks where the system's own natural period sits relative to the group (envelope) period of the sea state, misses the mechanism that most often governs mooring-line fatigue and peak tension.

A slender vertical member in waves — a pile, a jacket leg, a riser — sees a horizontal force built from two physically distinct mechanisms, and Morison's equation keeps them as two separate terms rather than folding them into one.

The drag term is the familiar velocity-squared resistance force, in phase with the water particle velocity u; the inertia term comes from the fluid having to be accelerated out of the way of the member, and is in phase with the particle acceleration u̇ instead. Because velocity and acceleration in a wave are 90° apart in phase, so are the two force terms — they do not peak at the same instant, and simply adding their individual maxima overstates the true peak load, sometimes substantially.

Which term matters is governed by the Keulegan–Carpenter number, KC = u_max T/D, which compares how far a fluid particle travels in one wave period against the member's diameter. At low KC the particle barely moves relative to the member and the flow never really separates, so inertia dominates almost completely; at high KC the flow separates and sheds vortices every half-cycle, and drag takes over. Most real design cases fall somewhere between the two limits, which is exactly where both terms have to be carried through the cycle together, as in Example 3.

Three worked problems below cover the three places this topic most often costs marks: picking the right depth regime and following its consequences through to group velocity, turning a sea state's statistics into a design wave, and combining drag and inertia correctly on a loaded member.