Is the wave behind a duck a parabola or a hyperbola?

Question

I tried to solve this problem when I saw some ducks in a lake. Suppose a duck moves along a straight line with a constant speed. Is the wave behind it a parabola or half of a hyperbola. I checked the definitions of them but still have no clue about the problem. Should the roughly V shaped wave be a parabola or one branch of a hyperbola with the duck being the vertex? enter image description here

Answer 1

A lazy duck will generate a parabolic wake. But the duck in your picture is supersonic -- it is swimming faster than the speed of ripple propagation in the local medium -- so it generates a hyperbolic wake. (This is why we get sonic booms.)

Answer 2

The theoretical shape of the wake behind an object traveling in water at constant speed is called the Kelvin wake. It is a waveform that is trapped in a wedge that Kelvin showed has half-angular width $\arcsin(1/3)=19.472 ^\circ$.

However recent literature suggests that the actual observed angle has some dependence on speed.

"While the half-angle which encloses a Kelvin ship wave pattern is commonly accepted to be 19.47°, recent observations and calculations for sufficiently fast-moving ships suggest that the apparent wake angle decreases with ship speed. "

Pethiyagoda, Ravindra, Scott W. McCue, and Timothy J. Moroney. "What is the Apparent Angle of a Kelvin Ship Wave Pattern?" Journal of Fluid Mechanics, vol. 758, 2014, pp. 468-485.