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

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

• Probably neither. In a perfect world, if the amplitude of the waves did not die down, you would get a sinusoidal pattern. A parabola would make a good approximation for part of a wavelength, though: consider the second-degree Taylor polynomial at the extrema of the sine wave. (Desmos visual,) Dampening the amplitude probably results in a function behaving more like $\sin(x)/x$, though. Commented Mar 16, 2023 at 19:51
• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
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Commented Mar 16, 2023 at 20:00
• This was apparently studied extensively(?) by Lord Kelvin. You might consider searching for papers on the topic to find more technical info on the phenomenon. Commented Mar 16, 2023 at 20:02
• @PrincessEev I believe that the asker is talking about the wake pattern, and not the "waves". Commented Mar 16, 2023 at 20:05

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.

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.)

• Thank you. But could you develop it further a little bit about the speed of a duck and the ripple propagation? Commented Mar 17, 2023 at 4:52
• The speed of ripple propagation in a liquid is not at all the same thing as the speed of sound in that liquid. The speed of sound in water is 1500m/s, the claim that the duck is supersonic is itself hyperbolic. A sonic boom has a hyperbolic wavefront when it intersects the ground at a different height, but at the same elevation of the plane (in this case, the surface of the water is analogous), it's just two straight lines. Commented Mar 17, 2023 at 15:48
• @NuclearHoagie: I agree with almost everything you say -- I was not claiming that the duck in the picture must be travelling at 3,350 mph. But note that (half of) a hyperbola looks like two straight lines from far enough away. If you include the bow wave of the duck (or the aircraft), it better approximates a hyperbola than two straight lines. Commented Mar 17, 2023 at 19:38