This article is from the Car Audio FAQ, by Ian D. Bjorhovde (firstname.lastname@example.org) with numerous contributions by others.
An "aperiodic membrane" is one part of a type of subwoofer enclosure.
It is an air-permeable sheet which has frequency-dependent acoustical
resistance properties. The original design goes back to Naim, for use
in home systems, but has been applied by several individuals and
companies in car audio.
The completed system will be aperiodic, which means it will prove to be
over-damped with a Q well below 0.7. In contrast, the most commonly
used sealed enclosures have Qtc's in the range of 0.8 to 1.1 which are
considered, by definition, to be underdamped. When improperly used, a
high-Q system may have poor transient response, nasty peaks in
frequency response, and high rates of roll-off. Aperiodic systems will
feature excellent Aperiodic systems are characterized by better
transient response, flatter frequency response and somewhat extended
low frequency response.
Another benefit of the system is that you can pretty much choose
whichever driver you'd like to use, as long as they are big. The
Thiele/Small parameters (which would normally determine what kind of
box would be used) are taken into consideration by the membrane
designers so that the response is extended and overdamped, regardless
of the characteristics of the driver.
Physically, the aperiodic membrane isn't for every car. It requires
sealing the trunk from the passenger compartment in an air-tight
manner, as well as sealing the trunk from the outside for best results.
The drivers are then mounted into the baffle between the passenger
compartment and the trunk, as would be standard in an
infinite-baffle/free-air set-up. The aperiodic membrane is then placed
either in front of the driver or behind the driver, depending on the
type. When mounting behind the driver, the membrane is used as the
rear-wall of a very small box which the driver sits in (as in Richard
Clark's infamous Buick Grand National). So, in short, it's not
suitable for trucks, jeeps, R/V's, or hatchbacks.
You should probably only get an aperiodic membrane if you've got money
to burn, lots of amplifier power, some big subs, a sedan, a desire for
trunk space, and no wish to boom. If your tastes lean towards
bass-heavy booming, as opposed to well-recorded acoustic instruments,
you're not going to be pleased with the result.