Kelvin Deep-Water Wake Cusp

The wave pattern generated by a ship moving steadily across deep water forms a distinctive V-shape. By analyzing the linear gravity waves stationary relative to the moving source, we can determine the maximum angle of this envelope, known as the Kelvin wake angle.

worked-example

Let the source move at speed U in the +x direction. In the frame of the moving source, stationary waves must have a phase speed equal to the component of the source's speed in the direction of wave propagation. With normalized gravity g=1 and deep water dispersion omega = sqrt(k), we can calculate the properties for a specific wave normal angle where q = cos^2(theta) = 2/3 and U = 1.

The resonance condition is k = 1 / (U^2 cos^2(theta)) = 1 / (U^2 q). For U=1, q=2/3, we find wavenumber k = 1.5.
Wavelength lambda = 2*pi / k = 2*pi / 1.5 approx 4.18879.
Group velocity relative to the source is given by components (v_x, v_y) = (U(q/2 - 1), U sqrt(q(1-q)) / 2). For q=2/3, (v_x, v_y) = (1(1/3 - 1), sqrt(2/3(1/3)) / 2) = (-2/3, sqrt(2) / 6) approx (-0.6666, 0.2357).
The wake ray angle is alpha = arctan(v_y / -v_x) = arctan((sqrt(2)/6) / (2/3)) = arctan(sqrt(2) / 4) approx 19.47°.

experiment

Using the module controls, we can explore how the ray angle changes with the squared normal cosine, q.

Observe that changing the "Source speed" scales the wavenumber and wavelength, but the Ray angle and Cusp angle remain unchanged.
Select "Squared normal cosine" to "1". The wave propagates exactly in the direction of the source, and the relative group velocity is directly backward with a ray angle of 0°.
Select "Squared normal cosine" to "2/3". This is the critical angle where the ray angle reaches its maximum of approx 19.47°, the Kelvin cusp angle.
Select "Squared normal cosine" to "0". The resonance condition cannot be met for finite waves, showing "No finite resonant wave".

limitations

This derivation assumes a linear gravity wave pattern from an idealized moving point source in inviscid, infinite-depth water with no surface tension. In reality, the apparent wake angle from a physical boat can differ. The cusp envelope derived here is an asymptotic idealization, not a claim that every finite-hull, finite-depth, or nonlinear boat photograph has this exact apparent wake angle.

references

For more details on the stationary wave patterns and derivation of the Kelvin wedge, refer to:

Fitzpatrick, R. (2013). "Ship Waves", The University of Texas at Austin.

Kelvin Deep-Water Wake Cusp