<DOC>
<DOCNO>
157
</DOCNO>
<TITLE>
the hodographic transformation in transonic flow .
</TITLE>
<AUTHOR>
lighthill,m.j.
</AUTHOR>
<BIBLIO>
proc. roy.s. a, 191, 1947, 323.
</BIBLIO>
<TEXT>
  the author studies the problem of finding the shape of a
symmetrical nozzle with the velocity along the axis (x-axis)
specified .  the velocity along each streamline is assumed to
increase steadily .  the singularity at the sonic velocity and
to the axis of the nozzle) is first studied in the physical
plane by using a power series in .  in the hodograph plane,
the two characteristics of the hodograph differential equation
passing through the sonic point and are lines
of branch points .  the region between these lines is a
region of triple-valuedness for the stream function .  outside
this region is single-valued .  there are also singularities
at the sonic point and the point corresponding to
the specified condition at the exit of the nozzle .  the author
then proposes to construct in the hodograph plane by
at the exit velocity and (3) a finite sum regular throughout ..
sin, where r is the square of the velocity
and the are hypergeometric functions .  the a's are
fixed by the required approximation to the specified velocity
distribution along the axis .  this solution is single-valued,
convergent and represents except a region near the sonic
point in the nozzle .  for this excluded region, the author
inverts the solution to obtain a power series in for 0 .  this
is shown to be convergent for the region of interest .  the
type of solution considered by the author gives a nozzle
having an infinitely long supersonic part .
</TEXT>
</DOC>
