DOC. 39 THEORY OF WATER WAVES
235
p
=
const
-
1/2pq2,
where
p
is
the
density
of
the
liquid.
Next,
we
consider several
generally
known
examples
of
this theorem. The
outflow of
a
liquid
that is contained under
pressure
in
a
vessel
(Toricelli).
The
pressure
at
J
(Fig. 2)
is
higher
and the
velocity
is lower than
at
A,
such that
p
+
1/2pq2 Fig.
2.
is constant
during
the outflow
process.
The atomizer
(Fig.
3)
can
serve as a
second
example.
The
air
stream
fed
through L
widens
to
all sides
after
its
exit
into free air while its
velocity
is
decreasing.
Therefore,
P
is
at
a
lower
pressure
than
G
and also
at
lower
pressure
than
the
surrounding
air,
which
is
at rest.
Due
to the lower
pressure
at P,
liquid
is sucked
up
through
feed
pipe S
from
the
vessel G
and
swept
away by
the
air stream
in
the
form
of little
droplets.
(The
fact that
it is
an
air
stream and not
a
stream
of
an
incompressible
liquid
does not
essentially change our
considerations.)
After these
preparations,
we
turn
to
a
Fig.
3.
Fig.
4.
consideration of
water
waves.
Let
W
(Fig.
4)
be
a
cylindrical
and
wave-shaped rigid
wall,
perpendicular
to
the
plane
of the
paper;
on
one
side it forms the
boundary
of
a
fluid that flows from left to
right.
We
ask
for the
pressure
forces exerted
by
the
fluid toward
the
wall.
Obviously,
the
cross
section offered
to
the
streaming
fluid
is
larger
at the
B
positions
than
at
the
positions
T.
The
liquid
at B
will therefore flow
slower,
and
at
T
faster,
than
at
any
locations
deeper
inside the
fluid, i.e.,
farther removed from the wall
W.
The
streaming
fluid
will therefore
generate
a
higher pressure
at B
and
a
lower
one
at T.
Consequently,
the
fluid will
press against
the wall
such
as
to
enlarge
the
already existing
outward-
pointing bulges
of
the
wall.
Therefore,
the flow could not maintain itself under
a
free
surface
of the
fluid,
that
is,
if the wall would be
infinitely
moldable and stretchable.1
However,
all these considerations
are,
like
our previous ones,
based
on
the
assumption
that
there
are
no
other
causes
to
generate pressure
in the fluid
except
the
1It
is
well known that
fluttering flags can
be understood based
on
these considerations.