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duck-pond puzzle



Hi --

At 06:13 AM 6/26/00 -0400, David Bowman asked for more puzzles that
illustrate physics principles. The recent "wave" and "hypercube" threads
make the following puzzle somewhat timely.

Compared to some recent puzzles, this one is very easy.

Suppose we have a large quiet pond. Near the middle is a duck. The duck
wiggles up and down sinusoidally at a constant frequency. This causes a
wave to be radiated in all directions. We are concerned with the surface
ripples, not sound waves in the bulk of the water or air. There are no
reflections from the edge of the pond, and no other wave sources. Make
whatever additional simplifying assumptions you like.

Your mission, should you decide to accept it, is to give a scaling law that
describes how the _amplitude_ of this wave falls off as a function of
distance. That is, if you are given the amplitude at one distance, what is
the amplitude at other distances? Is it an exponential law? Is it a power
law?

Standard hint: Everything I've said is intended to be true and
non-misleading. This is not a trick question. It is not a word game. It
illustrates a principle of physics. If you understand the physics, the
answer is clear.

Note: We've all had students who love to make wild guesses. Guessing,
without thinking about the physics, has a near-zero chance of success in
this case.

Big hint: If you find yourself writing down the wave equation you've
missed the point of the puzzle.