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
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$\begingroup$ 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. $\endgroup$– PrincessEevCommented Mar 16, 2023 at 19:51
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$\begingroup$ 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. $\endgroup$– Community BotCommented Mar 16, 2023 at 20:00
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1$\begingroup$ 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. $\endgroup$– user170231Commented Mar 16, 2023 at 20:02
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5$\begingroup$ @PrincessEev I believe that the asker is talking about the wake pattern, and not the "waves". $\endgroup$– Xander Henderson ♦Commented Mar 16, 2023 at 20:05
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2 Answers
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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.
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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.)
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$\begingroup$ Thank you. But could you develop it further a little bit about the speed of a duck and the ripple propagation? $\endgroup$– J.LiuCommented Mar 17, 2023 at 4:52
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$\begingroup$ 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. $\endgroup$ Commented Mar 17, 2023 at 15:48
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$\begingroup$ @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. $\endgroup$– TonyKCommented Mar 17, 2023 at 19:38