236
DOC. 39 THEORY OF WATER WAVES
flow itself. But if
gravity
acts
in the direction of the
arrows
S,
it will
generate
additional
pressure
forces in the
liquid
and
they
will increase toward the bottom. If
gravity
alone would
act,
there would be lesser
pressure
at
points
B
than
at
points T.
Consequently,
flow and
gravity generate pressure
differences
opposite
in
sign
between
B
and
T,
and
it is obvious that the
velocity
of the fluid flow
can
be chosen
such that the
pressure
differences
between
B
and
T,
due
to
the
two
causes,
vanish.
In
that
case we can remove
the wall
W
without
disturbing
the motion of the
liquid.
We
then have
a
fluid flow with wave-like curved surface such
as we
often observe behind
an
obstacle in
a
stream.
We notice this when
we
stand
at
a bridge-pier, looking
downstream and
watching
the
water
behind the
pier.
Imagine
that the entire
process
is described
by an
observer who
moves
with the
internal
velocity
of the
liquid
stream to the
right;
then
we
have the
ordinary case
of
water
waves
before
us.
For
the
observer
deep
down,
the
liquid
is
at rest
and the
mountains B and
valleys
T
propagate
with constant
velocity
toward
the left.
The
possibility
of
the
wave-process
is
therefore based
upon
the
fact that both the
statistically
and
dynamically produced
pressure
differences at various
heights
just
balance
one
another.
The
carrying capacity
of
a
wing
has
a
very
similar
explanation.
Let
an
air
stream
or
fluid flow
tangential
to
a
solid,
cylin-
drical
wall
W
(Fig.
5),
the
latter
perpen-
dicular to the
plane
of
the
paper,
and with
Fig
5.
an
upward-directed bulge.
If
this
bulge
were missing
there would be
no
force
on
the surface
apart
from the action
of
the
unavoidable friction. The
bulge,
however,
will affect the fluid flow
above
and below
the
wall, whereby pressures
are
generated.
For the lower
flow
the
bulge represents a
local
increase
in
cross
section,
which
means a
slowdown in the flow
and, thus,
higher
pressure
at
U.
On
top, however,
the
bulge
contracts the
cross
section,
increases the flow
velocity locally,
and
causes
diminished
pressure
at
O.
The
dynamic pressure
forces caused
by
the
flow
generate
an upward-directed
force
upon
the wall. In order to
generate
this
force,
one
obviously
needs to realize
only
a
large enough portion
of the wall
as
is
necessary
to
cause an
effective
bulge
in the fluid flow.
We
then have
the
supporting wing
of
a flying
machine
or
of
a
bird
soaring
without
moving
its
wings.
One
sees
from this
simple analysis
that
flying requires
the exertion
of
work
only
insofar
as
unavoidable obstacles
of
friction must
be
overcome.
Without
friction,
a
bird could
fly any
horizontal
paths
without
doing any
work.
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