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TOWER WITH CONSTANT PRESSURE, OR FUNNEL SURFACE

Cylindrical equation:  (tower) or , i.e.  (funnel).
Cartesian parametrization:  (tower).
Cartesian parametrization the coordinate lines of which are the asymptotic lines:  (view opposite).

The tower with constant pressure is the surface of revolution obtained by rotating a logarithmic curve around its asymptote.
Its name comes from the fact that, if this surface is filled with a homogeneous material, then the pressure applied on any horizontal section by the upper part is constant.
 
Derivation of the equation:
The pressure at the altitude z is equal to ; assuming P constant and differentiating, we get the differential equation:
which immediately gives the result, with .
Remark: if we take the variation of g due to the altitude into account, , we get the surface , with , represented opposite.

Much as the first one has a straight line as asymptote, this one has an asymptote cylinder. Note that this tower, extended to infinity, would have an infinite mass, but finite weight...

See also Gabriel's horn.
 
 
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© Robert FERRÉOL  2017