If α is the whole angle of the pattern left by a swimming little duck, then
sin(α/2) = 1/3
α/2 ≈ 19∘
Some years ago, while being part – President 😉 – of the organizing committee of a BIOMAT congress in Chile, I got acquainted with this fact that really surprised not only me! So here are some – more or less – clear statements of this math-in-nature fact! Just N – Joy!:
This pattern consists of two wake lines that form the arms of a “V” with the source of the wake at the vertex of the V. For sufficiently slow motion, each wake line is offset from the path of the wake source by around arcsin(1/3) = 19.47°.
The inside of the V (of total opening 39° as indicated above) is filled with transverse curved waves, each of which is an arc of a circle centered at a point lying on the path at a distance twice that of the arc to the wake source. This pattern is independent of the speed and size of the wake source over a significant range of values.
However, the pattern changes at high speeds (only), viz., above a hull Froude number of approximately 0.5. Then, as the source’s speed increases, the transverse waves diminish and the points of maximum amplitude on the wavelets form a second V within the wake pattern, which grows narrower with the increased speed of the source.
The angles in this pattern are not intrinsic properties of merely water: Any isentropic and incompressible liquid with low viscosity will exhibit the same phenomenon. Furthermore, this phenomenon has nothing to do with turbulence. Everything discussed here is based on the linear theory of an ideal fluid, cf. Airy wave theory.