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[Phys-L] Re: Acoustics question about popped balloons - try 2



In the context of:

... let me try to
express it at the 6th grade level for the science teacher that asked me
about it.

If we select the good part of what I wrote, we get:

To me, scaling laws and conservation laws are the heart and soul
of physics. With a little practice, scaling and symmetry/conservation
arguments become easy to make. They are very powerful. I'm not
saying they solve all the world's problems, but there are quite
a few problems (including the balloon-pop problem) where they
really help. .... The scaling argument gives you an almost-quantitative
picture of what happens ....

... which is fine. Alas, what I really wrote ended this way:

Sure, you could write down the fluid equation of
motion in spherical polar coordinates and solve it ... but why
bother? The scaling argument gives you an almost-quantitative
picture of what happens, with incomparably less work.

I really shouldn't have said that. There is nothing wrong with the
equations-of-motion approach. It is in relative terms more work
than the scaling-law + conservation-law approach, but in absolute
terms it isn't very much work. It is worth bothering with because
it provides details that the other approach.

If you do it right, the equations-of-motion approach works just fine
when the customer is a sixth-grade teacher, or even a sixth grader.
The trick is that the customer never needs to see the equations. In
particular, one possible way to proceed would be to
-- solve the equations
-- make a movie of the solution
-- show the movie to the customer.

I'm thinking in analogy to my movie of waves on a rope with
non-uniform tension (i.e. dangling with one free end):
http://www.av8n.com/physics/rope/movie1.gif
which was prepared using the program
http://www.av8n.com/physics/rope/rope

Modifying the program to produce a movie of the sound of popping a
balloon should be a piece of cake. The balloon version is in fact
slightly simpler than the rope version, because the spatial eigenfunctions
for Laplace's equation with spherical symmetry are just
|k> == f_k(r) = exp(i k r) / r [1]
which are familiar and easy to deal with. (This is in contrast to
the rope version, which involves Bessel functions, which are not
quite as easy to deal with, because their zeros are unevenly spaced.)
The time-dependence is just exp(i w t).

You have seen this method of solution a jillion times before. It is
just expansion in orthogonal functions, i.e. a generalized Fourier
expansion.

Let the initial condition F(r,0) be a square bump, i.e. high pressure
inside the balloon, and ambient pressure everywhere else:

______
|
|_____________________
0 r -->


You also need to impose the condition that the velocity is zero at t=0,
in accordance with the statement of the problem.

Decompose F(r,0) by multiplying by unity in the form of Sum_k |k><k|
to obtain
F(r,0) = (Sum_k |k> <k|) F(r,0) [2]

= Sum_k |k> (<k| F(r,0)) [3]

= Sum_k a_k |k> [4]

which tells you the value of the scalar coefficients a_k.

Then evolve each |k> forward in time, to find

F(r,t) = Sum_k a_k exp(i w t) |k> [5]

which is easy to evaluate numerically. Then it is just a
matter of drawing the graph for each t-value, and assembling
the graphs into a movie using ImageMagick.

Note that in equation [3], the contraction (<k| F) may be most
easily done by imagining the balloon is in the middle of a large
spherical room of size R, and later (if desired) letting R become
very large. This is typically easier than starting out with an
infinite-sized room.

====================

One of the major milestones of an undergraduate education in physics
is mastering the technique of expansion in orthogonal functions. This
technique has been the stock-in-trade of physicists for almost 200
years (Théorie Analytique de la Chaleur, 1822). It works for heat,
sound, electrostatics, electrodynamics, QM, etc. etc. etc.

I'm not sure ImageMagick should be a mandatory part of the physics
curriculum, but it should be strongly recommended. Compared to the
math/physics stuff, it's easy, and it's fun, because it makes the
equations come to life.

====================

Having said all that, recognizing that orthogonal functions are the
stock-in-trade of physics, I still believe that conservation laws
and scaling laws are the heart and soul of physics.

Without making a movie, without a computer, and indeed without more
than a pencil and a tiny scrap of paper, I can explain to a sixth-grader
why in the balloon-popping scenario, the near field is very different
from the far field. That's kinda neat.

Also BTW there is a nice hands-on demonstration you can do. When it's
my birthday, at the end of the song, I like to hold a balloon over the
cake. The candles pop the balloon, whereupon the balloon blows out all
the candles, all at once.

The physics lesson here is that the ability to blow out candles falls
off rather rapidly as a function of distance (near field), where as
the ability to hear the popping noise falls of much more slowly (far
field).

Scaling laws have been part of physics for a very long time (Discourse
on Two New Sciences, 1638). One of the "new sciences" was the laws of
motion ... the other was scaling laws!

I was given a really good education ... far beyond what most people can
even imagine. Just now, I spent almost an hour trying to recall what
I was taught about scaling, symmetry, and conservation. It's strange.
I have vivid memories from when I was a freshman, seeing half the class
turn in a big fat ten-page homework assignment ... while the other half
turned in half a page, having found a way to trivialize the problem by
an appeal to symmetry. At first I thought the half-page solution was
almost "cheating", in the sense that it defeated the purpose of the
assignment. And the TA initially thought it was "cheating", too.
But an upperclassman explained it this way: "If they wanted you to
do it the ugly way, they would've assigned a less-symmetric problem."
And the professor explained it to the TA this way: "The half-page
solution is the way Feynman would have done it. We don't necessarily
expect the frosh to do it that way, but the more we see of that, the
happier we are." And indeed they fed us a steady diet of problems
that could be solved two ways (the grungy way and the wizardly way).

But let me emphasize that scaling and symmetry/conservation ideas
are not just for wizards. These are ideas you can teach to a
bright 12-year-old. These are ideas you can teach in a "Physics
for Poets" course, to students who have never heard of Laplace or
Fourier, and never will.

Scaling and symmetry/conservation arguments are simple, elegant,
and powerful. Tremendous bang for the buck. I think they're just
plain fun. IMHO they should be emphasized at every stage in the
curriculum, from grade school on up ... emphasized far more than
they typically are at present.