Why doesn’t the angle change when I move the speed slider?

In this **idealized** deep-water sketch featuring the dominant V arms, the classical cusp angle is fixed at **arcsin(1/3)** and does not depend on source speed. The slider only changes **animation** speed. Real finite-depth, nonlinear, and multi-family interference effects modify the picture.

Is this the same as a Mach cone?

No. A **Mach cone** arises for supersonic motion with sin μ = 1/M in a nondispersive acoustic sketch. **Kelvin’s wake** follows from **water-wave** dispersion ω(k); the **19.47°** result is specific to deep-water gravity waves.

Are transverse waves between the arms shown?

No. Only the **outer cusp lines** are drawn for clarity; the full pattern includes feather-like interference from many components. The simulator teaches the **key angle**, not a complete CFD field.

How does this relate to tsunami shallow-water sims?

Tsunamis are often modeled in **shallow water** with c ≈ √(gH), where depth variation dominates coastal amplification. **Kelvin wakes** concern **deep** water and ship scales; both involve gravity waves, but dominant physics and scales differ.

Kelvin Ship Wake (Cusp Angle)

The visualization highlights the geometry of Kelvin’s ship wake—the distinctive V-shaped wave pattern behind a vessel (or any body) moving at constant speed through deep water. In the linear theory of gravity waves with dispersion ω² ≈ g|k|, wave energy from a steady disturbance concentrates along two arms that make an angle arcsin(1/3) ≈ 19.47° with each side of the track (full opening ≈ 38.94°). This angle is independent of ship speed in the ideal deep-water limit—unlike many everyday intuitions. Real wakes are richer: they include transverse and divergent wave families, finite-depth effects, viscosity, and nonlinearity; here only the classic cusp lines are emphasized as a signature of dispersion.

Who it's for: Students studying water waves, hydrodynamics, and dispersion; anyone linking the observed “Kelvin wedge” to a simple angle formula.

Key terms

Kelvin wake
Wave dispersion
Gravity waves
Deep water
Kelvin wedge angle
Wavenumber
Group velocity

How it works

A ship steadily disturbing the water radiates wave energy into a characteristic V; in the ideal Kelvin pattern the arms make a fixed angle set by dispersion, not by hull length.

Frequently asked questions

Why doesn’t the angle change when I move the speed slider?: In this idealized deep-water sketch featuring the dominant V arms, the classical cusp angle is fixed at arcsin(1/3) and does not depend on source speed. The slider only changes animation speed. Real finite-depth, nonlinear, and multi-family interference effects modify the picture.
Is this the same as a Mach cone?: No. A Mach cone arises for supersonic motion with sin μ = 1/M in a nondispersive acoustic sketch. Kelvin’s wake follows from water-wave dispersion ω(k); the 19.47° result is specific to deep-water gravity waves.
Are transverse waves between the arms shown?: No. Only the outer cusp lines are drawn for clarity; the full pattern includes feather-like interference from many components. The simulator teaches the key angle, not a complete CFD field.
How does this relate to tsunami shallow-water sims?: Tsunamis are often modeled in shallow water with c ≈ √(gH), where depth variation dominates coastal amplification. Kelvin wakes concern deep water and ship scales; both involve gravity waves, but dominant physics and scales differ.

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