Hodograph transform

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

The Bernoulli integral and the 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. 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

[1]	S.A. Chaplygin, "On gas-like structures" , Moscow-Leningrad (1949) (In Russian)
[2]	N.E. Kochin, I.A. Kibel', N.V. Roze, "Theoretical hydrodynamics" , Interscience (1964) (Translated from Russian)

References

[a1]	N. Curle, H.J. Davies, "Modern fluid dynamics" , 1–2 , v. Nostrand-Reinhold (1971)