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

[jsd at AV8N.COM (John Denker)]

John SOHL wrote:

> Let me try to express my "new" understanding. Especially, let me try to
> express it at the 6th grade level for the science teacher that asked me
> about it.
>
> 1. Shock is a term to avoid because it has a special use/definition.
> Fair enough.
>
> 2. The sudden release of air from the popped bag or balloon causes a
> disturbance or wave in the surrounding air (in simple terms it "shakes"
> the surrounding air for a moment). This sudden disturbance (release of
> energy) to the surrounding air then propagates outward as an ordinary
> sound (i.e., pressure) wave. This is what you hear.
> (I have absolutely zero problems with this pop being heard a long
> distance away from the source, it is simply a sound wave.)

That all seems fine.

> Now, my take of the two "explanations" on the net that I found (phrased
> in very non-physics terms):
>
> They are saying that the "extra molecules" that pressurize the air
> inside the balloon are what you "hear" when the air expands outward
> rapidly with the release of the air. (PV=NRT, increase N and you
> increase P and slightly increase V.) I.e., > this pressure moves outward
> and your ears respond by giving the sensation of a "bang."

There are two parts there.  Taking them in reverse order:

> this pressure moves outward
> and your ears respond by giving the sensation of a "bang."

Yes, that's spot-on correct.

> They are saying that the "extra molecules" that pressurize the air
> inside the balloon are what you "hear"

In the far field, it is safe to say that we hear the pressure, not
the "extra molecules".  See below for comments about the near field.

The wave has a reality that transcends the medium in which it travels.
   http://www.av8n.com/physics/reality-reductionism.htm
For most purposes, especially in the far field, it makes more sense
to focus attention on the wave, and not so much on the medium.

> This is a
> conservation of air molecules argument that implies that the extra air
> molecules that are released have to go somewhere and as they escape they
> cause a local increase in air pressure that your ears hear since the air
> pressure momentarily increases at your ears as the molecules in the
> surrounding air adapt to the extra molecules you just released.
> This argument makes no sense to me and without actually calculating the
> increase in the air pressure 100 meters away, I just don't see that it
> would be a very loud pressure sensation to your ear.

That argument makes a limited amount of sense in the near field.

As I loudly hinted in my previous note, to really understand what
happens when a balloon is popped requires _two_ arguments based
on _two_ scaling laws derived from _two_ conservation laws.

Conservation of air molecules tells you that if you look at the area
under the pulse (with due regard to signs), the area falls off like
1/r^2.  This is the "excess air" scaling law.  This describes a
wind-like permanent net displacement of the air.

Conservation of energy tells you that if you look at the RMS (i.e.
root-mean-square) of the pulse (which is insensitive to signs), it
falls off like 1/r, i.e. the mean square is falling off like 1/r^2.
This is the "energy" scaling law.  The sound wiggles the air, but
after the sound has passed, there is very little net displacement.

So in the far field, the plain old sound dominates.  That is, 1/r
dominates 1/r^2.

   !!  I reeeally recommend working through these   !!
   !!  conservation+scaling arguments.              !!

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.  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.