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Re: but does the pilot hear the sonic boom?



On 12/01/2003 04:31 PM, Frohne, Vickie wrote:
>
> It is created because the duck swims faster
> than the speed of waves in the water.

The system is so dispersive that there is no well-defined
"speed of waves" on the water surface.

> A duck's wake has *exactly* the same origin as a sonic boom.

This brings new meaning to the word *exactly*.

I would have said that the physics is profoundly
different in the two cases (sonic boom versus wake
on water).

-- For starters, the Mach angle (the half-angle of the
Mach cone) in air depends on the speed of the emitter.
Specifically it equals arcsin(1/M) where M := v/c is
the Mach number. This is based on a non-dispersive
wave equation.

-- In contrast, the half-angle of the Kelvin wake on
water isindependent of the speed of the emitter.
Specifically it equals arcsin(1/3) = 19.47 degrees.
This is valid whether the emitter is a duck or a
battleship, if we neglect capillary effects, for which
it is sufficient to restrict attention to the case
where the emitter is large compared to the crossover
wavelength (1.7cm) between capillary waves and gravity
waves. The key idea here is that the wave equation is
dispersive.

For details see
"Governing equations for waves on the sea surface"
http://ocw.mit.edu/NR/rdonlyres/Civil-and-Environmental-Engineering/1-138JWave-PropagationFall2000/2DBF5F87-AA4E-42D1-AB68-75262A9C42A7/0/Wpchap4.pdf

Amusing pictures include:
http://www.stanford.edu/~jrdx/PICS/aircraft_kelvin_cropped.jpg
http://www.maths.ox.ac.uk/~mortimer/images/duck.gif
http://www.eng.vt.edu/fluids/msc/gallery/waves/duck1.htm