Re: shock wave as pressure builds in a vacuum?

["John S. Denker" <jsd@MONMOUTH.COM>]

On 05/28/2003 12:44 AM, Stefan Jeglinski wrote:
> I have a real-world situation that I am idealizing for the purposes
> of discussion. Consider an extremely thin membrane that is capable of
> withstanding an atmospheric pressure differential under steady-state
> conditions. IOW, one side of the membrane can be at high vacuum,
> while the other is at 1 atm. The membrane is likely flexed concave
> somewhat in steady state with a 1-atm differential (center deflection
> x toward the high vacuum, let's say), but it has a wire mesh on the
> high vacuum side to provide mechanical support.
>
> Now let's say that under normal operation, the high vacuum side is at
> 10^-9 Torr while the other side is at 10^-6. Consider what happens
> when the 10^-6 side undergoes "catastrophic venting."  That is, it is
> opened suddenly to atmospheric pressure. Air "rushes in," but it is
> my contention that there is no "front" associated with it that would
> cause a sudden overpressure at the membrane. It is tempting to make
> an analogy to water rushing into an empty vessel and the force it
> creates when it crashes against the inside walls, but I'm not sure
> the analogy is appropriate here. I know little about shock wave
> physics, don't know how that might enter in.
>
> It is experimentally observed that under these conditions
> ("catastrophic venting"), the membrane is sometimes broken. One
> colleague argues for some kind of shock wave, essentially an
> overpressure transient at the membrane as the air rushes in. I argue
> that there is little or no physics to support that, and that the
> membrane failure is due to an excessive deflection rate (dx/dt, as
> the pressure increases rapidly) that it cannot withstand from a
> mechanical standpoint.

a) Shocks are more-or-less by definition supersonic.
I don't see anything obviously supersonic here.
No supersonic pistons or suchlike.

b) Shocks happen when there is compression, not
rarefaction.

For these two reasons I am having a hard time
visualzing a shock wave hitting the membrane.
But peculiar things could happen if you have a
peculiar geometry.  (Focusing.  Bay of Fundy.)

But one thing's for sure, the membrane will see
a lot more that 1 atm pressure.  As an approximation,
analyze it in the linear regime.  Take 1 atm as
the "at rest" position.  Then the initial condition
is a very loud sound wave (negative 1 atm [gauge]
sound pressure).  This sound wave will reflect off
the membrane.  During the time of this reflection,
the incident wave and the reflected wave will
both be at the surface, so there will be a plus 1
atm [gauge] sound pressure, i.e. 2 atm [absolute].

That's just linear wave mechanics.  It's like asking
for the voltage at the end of an open-circuited coax.

So as a first approximation, the membrane must be
designed to withstand TWO atm.

The full nonlinear analysis might give a somewhat
different answer, but that would require actual
thinking and I haven't got time for that right now.

Options include:
 -- toughening up the membrane.
 -- softening the venting.  If the pressure-rise
waveform can be spread out even a little bit (a
smallish multiple of the round-trip time across
the cell) the peak pressure will be a lot less.