Concept

Wave resistance — where it appears

The drag a body pays for making waves on a free surface, equal to the energy the wave train carries away. It depends on the Froude number and rises and falls in humps and hollows as the waves from different parts of a hull reinforce or cancel.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

The free surface a submerged body leaves behind it, and the flat water in front. The linearised free-surface problem solved as a Fourier integral with a radiation condition. Behind the body a wave train of the wavelength that stands still relative to it, 2 pi U²/g; ahead of it, an amplitude a hundred and twenty times smaller. The asymmetry is the drag: an ideal fluid with a free surface can carry energy away.

The drag that is made of waves

D'Alembert's paradox says a body in a steady, irrotational, incompressible, inviscid flow feels no drag. Put a free surface above it and every one of those words still holds — and the drag is not zero. It is the energy walking away in the wave train behind.

inviscid · Dalembert

Hull speed is not the hump

The hull-speed rule says a displacement vessel meets a wall where its wave is as long as its hull, at a Froude number of 0.4. Michell's thin-ship integral, computed for a standard hull, puts that speed on the steep climb between the last hollow and the main hump, and the hump itself at 0.5. Past the hump the wave resistance grows more slowly than the speed. The hollows sit where the hull's own transverse wave switches off, and a bulb that cancels at one speed multiplies the resistance at another.

regimes · Froude

The hull Michell's integral prefers

Michell's integral gives a thin ship's wave resistance as a quadratic in the hull's offsets, so the hull of least resistance at a given speed and displacement is a quadratic minimisation. Allowed only to reshape its waterline, the integral rediscovers the naval architect's oldest rule: fine ends for a slow ship, full ends for a fast one. Allowed to reshape its depth too, it drains the waterline and piles volume at the keel until the hull is not a ship. And it cannot grow a bulb at the bow, because it cannot tell the bow from the stern.

regimes · Froude $With a wake, the bulb belongs at the bow. The Wigley hull's wave resistance with a spherical bulb, as a fraction of the bare hull's, against Froude number, with a wake fraction of 0.25: the bulb just ahead of the bow, and the same bulb just behind the stern. Without a wake the two curves are one. With it they separate: the stern's waves are made by slowed water and are weaker, so the bulb's cancelling wave has less to cancel there, and at the design speed of 0.30 the bow bulb leaves 0.443 of the bare resistance and the stern bulb 0.624.$

The stern's wake puts the bulb at the bow

Michell's thin-ship integral cannot tell a ship's bow from its stern: reverse any hull and its wave resistance is unchanged, so the least-resistance hull is symmetric and a bulb is worth as much at the stern as at the bow. Real ships are fuller aft and put their bulbs forward. Let the stern's waves be made by water the hull's own boundary layer has slowed, and both follow: the optimum hull leans aft, and a bulb at the bow cuts the waves by far more than the same bulb at the stern.

regimes · Froude $A wake held at the stern moves the volume aft a third as far. The least-resistance hull's centre of volume, in per cent of the half-length from midships, negative aft, against the design Froude number, holding the Wigley hull's length, draught, depth profile and displacement. With the linear wake it sits 2.8 per cent aft at Fr 0.30; with the shaped wake of the same propeller-disc fraction, 0.86 if uniform in depth, 1.5 if deepest at the waterline, and 0.11 if deepest at the keel, where the sources make the fewest waves.$

A wake held at the stern keeps the bulb and loses the lean

A wake that slows the water steadily from bow to stern makes Michell's least-resistance hull fuller aft and puts its bulb at the bow. A real ship's wake is nearly nothing along most of the hull and strong only in its last few metres, deepest at the keel. Held there, with the same wake at the propeller, it moves the hull's volume aft by a twentieth to a half as much, and it keeps a third to two-thirds of the bulb's preference for the bow. The lean was a property of the water along the run, and the bulb a property of the water at the stern.

regimes · Froude $The waves a wing makes on the sea cost it almost nothing. The wave drag of a wing skimming deep water, as a fraction of its lift, against speed, at heights of half a metre, one and two. Each curve peaks at the speed √(2gh) — 4.43 m/s at one metre, whatever the chord — and there the wave drag is 4.4·10⁻⁴ of the lift, the density ratio of air to water times Cₗ c/4eh. At a cruise of 50 m/s it is 9.31·10⁻⁶ of the lift.$

A wing over the sea barely touches it

A wing skimming the water presses its whole weight onto the surface beneath it, spread over a width about its own height. The sea is a deformable ground, and one might expect it to give way: a trough under the wing, waves behind, a drag to pay. It barely notices. Air is eight hundred times lighter than water, and that one ratio sets the dent at millimetres, the waves at a few millimetres high, and the wave drag at less than a two-thousandth of the lift even at the worst speed.

misconceptions · Ground cushion

Named alongside it

The objects these essays reach for when they reach for this one.

Froude number Model limit Linearisation Optimisation Boundary layer Dispersion Displacement hull Free surface Group velocity Hull speed Interference Source

All concepts