The Kelvin ship wake is the V-shaped wave pattern visible behind any vessel moving through deep water. Lord Kelvin proved in 1887 that the half-angle of this wedge is always 19.47° (= arcsin 1/3), regardless of the ship's speed — a surprising result that arises purely from the deep-water dispersion relation ω² = gk. The wake is the superposition of two wave families: transverse waves with crests nearly perpendicular to the track, and diverging waves whose crests fan outward at steeper angles.
Adjust Ship speed to see wavelength scale with V² while the envelope angle stays fixed at 19.47° (deep water). Increase Wave sources for a denser, more realistic superposition. Switch to Shallow water to activate the tanh dispersion relation — at low Froude depth numbers the angle widens, and above the critical Froude number Fr = 1 the pattern collapses into a narrower Mach-cone-like wedge. The yellow dashed lines always mark the Kelvin angle boundary.
In 2014 satellite measurements showed that the 19.47° rule breaks down for very large or very fast ships in practice — when the ship's hull length is comparable to the dominant wavelength, the single-point-source model fails and the visible angle narrows. This sparked renewed interest in the century-old theory and led to refined predictions for modern supertankers and container ships.
The Kelvin wake is one of the most elegant results in classical fluid mechanics. Lord Kelvin showed in 1887 that the half-angle of the V-shaped wake is always 19.47° in deep water, arising from the unique dispersion relation of surface gravity waves. This simulation superposes contributions from multiple point sources distributed along the ship's trail, each emitting circular waves with wavelength λ = 2πV²/g, recovering the familiar transverse and diverging wave arms of the Kelvin pattern.
Deep-water surface gravity waves obey ω² = gk, giving phase speed c = √(g/k) and group speed c₁ = c/2. The Kelvin angle is determined by the condition that wave crests from all past ship positions constructively interfere along a line making arcsin(1/3) with the track. This 19.47° result is speed-independent.
Use Ship speed to scale wavelengths (λ ∝ V²); increase Wave sources for richer interference; toggle Shallow to switch to ω² = gk·tanh(kh) dispersion. The yellow dashed guide lines show the 19.47° Kelvin boundary. Pause/Play stops the animation; Reset clears the trail.
The 19.47° angle is the same as the latitude of the Great Red Spot on Jupiter and of Olympus Mons on Mars — a numerical coincidence, but a useful mnemonic. It equals arcsin(1/3), which is why it appears in the geometry of the Kelvin construction.
The angle follows from deep-water wave dispersion: ω² = gk gives group speed c₁ = c/2. The wave packet emitted when the ship passed a point travels at c₁, while the ship travels at V. Stationary phase analysis shows constructive interference along a line at arcsin(1/3) ≈ 19.47° regardless of V.
Transverse waves have crests nearly perpendicular to the ship's track and are responsible for the characteristic "ripples" in the middle of the wake. Diverging waves have crests at about 35° to the track and form the outer arms of the V. Both are produced by the same mechanism and together make up the Kelvin pattern.
The ship's past positions are recorded at regular intervals as point sources. Each source emits a circular wave with wavelength λ = 2πV²/g and amplitude decaying as 1/√r. The surface elevation at each pixel is the sum over all sources, evaluated with the correct phase for the time elapsed since emission. The resulting interference pattern recovers the Kelvin arms.
In deep water, speed only changes the wavelength (λ ∝ V²) — faster ships produce longer waves — but the 19.47° envelope angle is strictly independent of speed. In shallow water the picture changes: if the ship's speed exceeds the maximum shallow-water wave speed √(gh), the pattern narrows like a Mach cone.
When water depth h is finite, the dispersion relation becomes ω² = gk·tanh(kh). The wave speed is limited to √(gh). Below the critical Froude depth number Fr = V/√(gh) = 1, the angle widens; above it (transcritical flow) waves pile up into a narrower, Mach-cone-like wedge and the Kelvin angle is no longer 19.47°.
For a clean pattern, 12–20 sources per characteristic wavelength work well. Too few sources produce sparse, spotty interference; too many become computationally expensive in a per-pixel sum. The slider lets you explore this trade-off. Real ocean wakes involve a continuous distribution of sources that integrates to the idealized Kelvin result.
Yes. Synthetic aperture radar (SAR) satellites routinely image Kelvin wakes at sea. The dark V-shaped region behind ships is clearly visible because the wave pattern modifies the radar backscatter from the sea surface. Ship detection via wake signatures is a practical application in maritime surveillance.
In 2014, Rabaud and Moisy published satellite measurements showing that large, fast vessels (hull Froude number Fr₁ = V/√(gL) close to or above 0.5) produce visible wake angles narrower than 19.47°. When the dominant wavelength λ = 2πV²/g approaches the ship's hull length L, the ship cannot be treated as a point source, and the interference shifts to shorter, more oblique waves, narrowing the apparent angle.
The simulation captures the essential physics — dispersion, superposition, the 19.47° envelope, shallow-water effects — but is a pedagogical tool rather than an engineering model. Real ship wake prediction requires panel methods or boundary element codes that account for hull geometry, forward-speed corrections, finite draft, and nonlinear free-surface effects.
In shallow-water mode the panel shows the depth Froude number Fr = V/√(gh), where h is a reference depth scaled to the current wavelength. Fr < 1 is subcritical (standard wide Kelvin wake), Fr = 1 is critical (wave system breaks down), and Fr > 1 is supercritical (Mach-cone-like narrow wake). In deep-water mode it reads "deep" since depth effects are negligible.