View source for Hodograph transform

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A mapping realizing a transformation of certain differential equations of mathematical physics to their linear form.

The [[Bernoulli integral|Bernoulli integral]] and the [[Continuity equation|continuity equation]] of a plane-parallel potential stationary motion of a barotropic gas  $  ( \rho = F( p)) $,

$$ 
\rho  =  \rho _ {0} \left (
1 - 
\frac{u  ^ {2} + v  ^ {2} }{2 \alpha }

\right )  ^  \beta  ,\ \ 

\frac{\partial  \rho u }{\partial  x }
 +

\frac{\partial  \rho v }{\partial  y }
  =  0,
$$

where

$$ 
\alpha  = \ 

\frac{c  ^ {2} }{\gamma - 1 }
 ,\ \ 
\beta  = \ 

\frac{1}{\gamma - 1 }
 \ \ 
( c  \textrm{ is }  \textrm{ the }  \textrm{ velocity }  \textrm{ of } \ 
\textrm{ sound }  \textrm{ for }  \rho = \rho _ {0} ),
$$

lead to the equation

$$ 

\frac \partial {\partial  x }
 \left [ \left (
1 - 
\frac{v  ^ {2} }{2 \alpha }

\right )  ^  \beta  u \right ] +

\frac \partial {\partial  y }
 \left [ \left (
1 - 
\frac{v  ^ {2} }{2 \alpha }

\right )  ^  \beta  v \right ]  =  0,
$$

which is used for determining the velocity potential

$$ 
u  =  
\frac{\partial  \phi }{\partial  x }
 ,\ \ 
v  =  
\frac{\partial  \phi }{\partial  y }
 ,
$$

where  $  u $
and  $  v $
are the velocity components. By introducing new independent variables  $  \tau = v  ^ {2} / 2 \alpha $
and  $  \theta $
equal to the slope of the angle made by the velocity vector with the  $  x $-
axis, equation

is reduced to linear form:

$$ 

\frac \partial {\partial  \tau }
 \left [

\frac{2 \tau ( 1 - \tau ) ^ {\beta + 1 } }{1 - ( 2 \beta + 1) \tau }

\frac{\partial  \phi }{\partial  \tau }

\right ] +

\frac{( 1 - \tau )  ^  \beta  }{2 \tau }

\frac{\partial   ^ {2} \phi }{\partial  \theta  ^ {2} }
  =  0.
$$

This is the first hodograph transformation, or the Chaplygin transformation. The second Chaplygin transformation is obtained by applying the tangential [[Legendre transform|Legendre transform]]. The function

$$ 
\Phi  =  x

\frac{\partial  \phi }{\partial  x }
 +
y 
\frac{\partial  \phi }{\partial  y }
 - \phi
$$

is selected as the new unknown; it is expressed in terms of new independent variables  $  u $
and  $  v $, 
which replace  $  x $
and  $  y $
by the formulas

$$ 
u  =  
\frac{\partial  \phi }{\partial  x }
 ,\ \ 
v  =  
\frac{\partial  \phi }{\partial  y }
 .
$$

The equation

assumes a linear form:

$$ 
\left [ 1 -

\frac{v  ^ {2} }{2 \alpha }
 -

\frac \beta  \alpha 
 v  ^ {2}
\right ]

\frac{\partial   ^ {2} \Phi }{\partial  u  ^ {2} }
 +

\frac{2 \beta } \alpha 
 u v

\frac{\partial   ^ {2} \Phi }{\partial  u \partial  v }
 +
$$

$$ 
+ 
\left [ 1 - 
\frac{v  ^ {2} }{2 \alpha }
 - {
\frac \beta  \alpha 
 }
u  ^ {2} \right ] 
\frac{\partial   ^ {2} \Phi }{\partial  v  ^ {2} }
  =  0.
$$

Hodograph transforms are employed in solving problems in the theory of flow and of streams of gases flowing around curvilinear contours.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.A. Chaplygin,   "On gas-like structures" , Moscow-Leningrad  (1949)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.E. Kochin,   I.A. Kibel',   N.V. Roze,   "Theoretical hydrodynamics" , Interscience  (1964)  (Translated from Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Curle,   H.J. Davies,   "Modern fluid dynamics" , '''1–2''' , v. Nostrand-Reinhold  (1971)</TD></TR></table>