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Re: SNAP versus BOOM



Here's a riddle:

Suppose at time t=0 we release a smallish spherical region (radius r0) of
high-pressure air in an otherwise-ordinary atmosphere. This could be done
by setting off a firework or popping a balloon.

True fact: The air will rush outwards from the initial high-pressure region.

True fact: There will be an expanding spherical front outside of which the
air will be undisturbed. Behind the front there will be a pulse containing
high-pressure air.

From here on, let's consider just the far field, i.e. distances large
compared to r0.

We assume the amplitude is reasonably small, so we don't need to worry
about nonlinearities.

Supposition #1: By the law of conservation of air, you might suppose that
the height of the pulse decreases like 1/r^2, because
-- the area of the front is increasing like r^2, and
-- the amount of air in the world does not increase as the pulse propagates.

Supposition #2: By the law of conservation of energy, you might suppose
that the height of the pulse decreases like 1/r, because
-- the energy density of a wave goes like amplitude squared,
-- the area of the front is increasing like r^2, and
-- the amount of energy does not increase as the pulse propagates.

It appears that supposition #1 is inconsistent with supposition #2. So for
starters, the question is: Do you believe #1, or #2, or both, or neither?

Usual jsd rules of the road: This is not a word game. There is an
interesting physics-based answer. My statements are intended to be true
(i.e. very reasonable approximations) and non-misleading. OTOH I haven't
told you everything; for example I haven't told you the answer to the
riddle. Also, there are some widely-held misconceptions that I haven't
explicitly warned you about.