Kelvin Wake: Why Every Boat Leaves the Same 39° V

A Hull Is an Antenna for Gravity Waves

Start by forgetting the boat and keeping only what the water feels: a travelling dent in the pressure at the surface, moving at speed V. At every instant that dent launches an expanding ring of waves — a broadband pulse containing every wavelength at once. The wake is not carved by the bow shoving water sideways. It is the interference pattern of an unbroken chain of such rings, each launched a moment later and a little further along the track.

What sorts them is dispersion. Deep-water gravity waves obey

ω² = gk, so c = ω/k = √(g/k) = √(gλ/2π).

Long waves are fast, short waves are slow: a 100 m swell runs at 12.5 m/s, a 1 m ripple at 1.25 m/s. Depth only has to exceed about λ/2 for this to hold, which is why a pond a metre deep is already deep water for a duck making 10 cm waves.

A wave train can only stay locked to the boat if its crests keep pace with the boat's motion measured along the wave's own direction of travel. If the wave normal makes an angle θ with the track, that condition is c = V cos θ. Every θ from 0° to 90° therefore selects its own wavelength, λ(θ) = 2πV² cos²θ / g. The wake is the superposition of that entire one-parameter family, and almost everywhere the family cancels itself out.

The Hidden Half: Energy Travels at Half the Crest Speed

Cancellation is not the whole story, because crests are not what carries the wave. Energy travels at the group velocity, c_g = dω/dk. Differentiate ω = √(gk):

c_g = dω/dk = ½√(g/k) = c/2.

Deep-water gravity waves are the textbook case in which the factor is exactly one half, and it is directly observable: in a packet from a dropped stone, crests appear at the rear, sprint forward through the group, and dissolve at the front. Thomas Havelock made this quantitative in his 1908 Royal Society paper on groups of waves in dispersive media.

This factor of two is the entire reason a boat wake is a narrow V rather than a broad circular disturbance. Wavelets satisfying c = V cos θ stay phase-locked to the hull, but their energy falls behind at half that rate — and where all that lagging energy piles up is fixed forever by the number ½.

Kelvin got there by inventing the tool the calculation needed. In two 1887 papers he introduced what is now called the method of stationary phase: in an integral over all wave directions, contributions cancel except where the phase is stationary, dφ/dθ = 0. George Gabriel Stokes had used the same idea for the Airy integral in 1850, but Kelvin turned it into a general asymptotic method — arguably a bigger legacy than the wake itself.

The Circle Construction: Why the Answer Is arcsin(1/3)

Here is the derivation with no integrals at all. Take a packet emitted when the boat was a distance D = V t astern of where it is now. That packet travels in direction θ at c_g = c/2 = (V cos θ)/2, so after time t it sits at distance

r = c_g t = (D/2) cos θ

from its emission point. In polar coordinates, r = a cos θ is the equation of a circle — one of diameter D/2, so radius D/4, centred D/4 from the emission point along the track, in the boat's direction of travel. Every packet born at that instant now lies somewhere on that single circle.

The boat is a distance D from the emission point, hence 3D/4 from the circle's centre. The largest angle at which the boat can see any point of the circle is the tangent line, and for a circle of radius R seen from distance d the half-angle obeys sin α = R/d:

sin α = (D/4)/(3D/4) = 1/3 → α = 19.4712°, a full V of 38.94°.

Now look at what dropped out. D cancelled, so it does not matter how long ago the packet was launched, and V and g never appeared at all: the angle is pure geometry plus the number ½ from the dispersion relation. A duck paddling at 0.4 m/s (transverse wavelength 2πV²/g = 10 cm) and a tanker at 10 m/s (λ = 64 m) draw the same V, differing only in scale — a 640-fold difference in wavelength, identical angle. On Titan, where g = 1.35 m/s², a boat would trail the same 39° with waves 7.3 times longer.

Inside the V: Transverse Crests, Diverging Crests, and a Cusp Caustic

The envelope is only the boundary. Within it, two wave systems coexist, and both come from the fact that for any point at angle α < 19.47° behind the boat there are two stationary-phase solutions θ.

• Transverse waves (small θ): crests lying almost perpendicular to the track, marching along behind the stern. Their wavelength is λ_T = 2πV²/g — the number you can invert to read a ship's speed straight off a photograph, V = √(gλ/2π).
• Diverging waves (large θ): the feathered oblique crests that sweep out to the arms and run up a beach long after the boat has gone.

The two solutions merge exactly on the arms. Setting d(tan α)/dθ = 0 with tan α = cos θ sin θ/(1 + sin²θ) gives sin θ = 1/√3, i.e. θ = 35.26° for the wave normal at the cusp, and back-substituting returns tan α = 1/(2√2), sin α = 1/3. The arms are therefore caustics — cusped ones, in the catastrophe-theory sense. Fritz Ursell's 1960 Journal of Fluid Mechanics analysis showed the consequence: along the cusp line the amplitude decays only as r^−1/3 (an Airy function), versus r^−1/2 elsewhere. That is why the arms are the bright, sharp, persistent, shoreline-eroding part of the wake.

The pattern also sets the price of going fast. λ_T equals the waterline length L when 2πV²/g = L, i.e. Froude number Fr = V/√(gL) = 1/√(2π) = 0.399 — the classic hull speed, 1.34√L knots for L in feet, where a displacement hull sits in the trough of its own transverse wave and wave-making resistance climbs steeply.

How the Wake Is Actually Measured

Towing tanks. William Froude built the first ship model basin at Torquay in 1872 with Admiralty funding, precisely to separate wave-making from frictional resistance using his 1868 law of comparison — match Fr, not speed.

Optical laboratory methods. Free-surface synthetic Schlieren (Moisy, Rabaud & Salsac, Experiments in Fluids, 2009) recovers the full surface topography by watching a printed dot pattern refract through the water, resolving surface deformations of a few micrometres across an entire wake at once.

Aerial and satellite imagery. Marc Rabaud and Frédéric Moisy's much-cited 2013 study measured dozens of real wakes from Google Earth airborne images. Synthetic-aperture radar sees them even better, because Bragg-scattering modulation makes the arms bright. ERS-1/2 (both long since retired), RADARSAT-2, TerraSAR-X (launched 2007) and Sentinel-1A (launched 3 April 2014) have all been used for maritime surveillance, with a Radon or Hough transform extracting the wake lines to recover heading and — from λ_T — speed. A characteristic giveaway is SAR azimuth displacement: a ship with a few m/s of radial (line-of-sight) velocity is imaged displaced along-track, roughly half a kilometre off the apex of its own wake.

Coastal wave gauges. Tarmo Soomere's monitoring of the Tallinn–Helsinki fast ferries showed vessel wakes can supply a large share of the wave energy reaching the shore in a nearly tideless sea.

Where the 39° Breaks Down

The angle is universal only under the assumptions that produced it: deep water, pure gravity waves, a point-like disturbance, still uniform fluid. Remove any one and it moves.

Shallow water. When kh ≪ 1 the dispersion relation flattens to c = √(gh) and dispersion vanishes, so c_g = c and the half-factor is gone. As the depth Froude number F_h = V/√(gh) rises toward 1, the wake widens from 19.47° toward 90°; above F_h = 1 the transverse system disappears and what is left is a genuine Mach cone, sin μ = √(gh)/V. This is why a fast ferry in a shallow channel can throw a single destructive long wave at the bank.

Floating ice. Replace the free surface with an ice plate and the dispersion gains a flexural term, with a critical speed near √(gh) at which a vehicle resonates with its own wave and the deflection peaks — the reason ice roads carry speed limits (Vernon Squire, Moving Loads on Ice Plates, 1996).

Capillarity. Below the minimum phase speed of clean water, c_min = (4gσ/ρ)^1/4 = 23.1 cm/s at λ = 1.73 cm, no steady wave pattern can be radiated at all — a loophole open to small surface insects and larvae that skim slowly enough, although striders and whirligig beetles routinely exceed c_min and do radiate ripples. Just above it, short capillary ripples run ahead of the object, because for capillary waves c_g = 3c/2.

Stratification. A ship can pour its energy into internal waves on a pycnocline instead of the surface. Fridtjof Nansen's Fram was nearly halted by this dead water in August 1893; V. Walfrid Ekman explained it in 1904.

Look-alikes, and What Is Still Argued About

The Kelvin wake's most persistent impostor is the sonic boom. Both are V-shaped, both come from a source outrunning something — and they run on opposite physics. A Mach cone lives in a non-dispersive medium where one signal speed c exists, so sin μ = c/V and the cone tightens without limit as speed rises. Water offers a wave of every speed, so there is nothing to outrun — the boat always finds a wave that matches, and the angle never budges. Cherenkov radiation is the electromagnetic cousin, cos θ = c/(nv), with mild dispersion entering through n(ω). The dark stripe down the middle of a radar wake image is not Kelvin's pattern either but the turbulent propeller wake, damping the short Bragg ripples the radar needs; and the white lines ships leave in cloud images are ship tracks, aerosol-seeded cloud, not waves.

Open questions are mostly about real hulls rather than point sources. Rabaud and Moisy (Physical Review Letters 110, 214503, 2013) reported that above Fr = V/√(gL) ≈ 0.5 the brightest waves lie at an angle shrinking roughly as 1/Fr, arguing that a hull of length L simply cannot radiate waves much longer than L. Darmon, Benzaquen and Raphaël (JFM 738, R3, 2014) reproduced the scaling analytically; Noblesse and colleagues (2014) attributed it instead to interference between the bow and stern wave systems. All agree the cusp line itself is still at 19.47°: what narrows is where the eye, or the radar, sees the most energy.

Kelvin wakes versus the non-dispersive cones they are mistaken for

Phenomenon	Medium	Half-angle law	Behaviour as speed rises
Kelvin ship wake (deep water)	Dispersive: ω² = gk, c = √(g/k)	sin α = 1/3 → 19.47°	Angle fixed; wavelength grows as V², amplitude peak may narrow at high Froude number
Sonic boom / Mach cone	Non-dispersive air, c ≈ 343 m/s	sin μ = c/V	Cone narrows steadily; μ = 30° at Mach 2, 11.5° at Mach 5
Cherenkov radiation	Weakly dispersive dielectric, c/n	cos θ = c/(nv)	Cone widens toward a ceiling set by the refractive index
Shallow-water ship wake	Non-dispersive if kh ≪ 1, c = √(gh)	sin μ = √(gh)/V for V > √(gh)	Widens toward 90° at the critical speed, then narrows like a true Mach cone
Bow capillary ripples	Capillary-dominated, c ∝ √(σk/ρ)	No steady wake below 23.1 cm/s	Ripples run ahead of the object because c_g > c