Hodograph transformation

Remarks

The inverse function theorem gives $u_x = J y_v$, $u_y = -J x_v$, $v_x = -J y_u$ and $v_y = J x_u$. Substituting and dividing by $J$, the equations become $$(c^2 - u^2)\, y_v + uv\, (x_v + y_u) + (c^2 - v^2)\, x_u = 0, \qquad x_v = y_u.$$ These are linear in $(x, y)$, since $c$ depends only on $(u, v)$. With the velocity potential $\varphi$, the Legendre transformation $\Phi = xu + yv - \varphi$ satisfies $x = \Phi_u$ and $y = \Phi_v$, and the first equation becomes the linear equation $(c^2 - v^2)\Phi_{uu} + 2uv\, \Phi_{uv} + (c^2 - u^2) \Phi_{vv} = 0$. In polar coordinates of the hodograph plane this is Chaplygin's equation.

Example: in the incompressible limit $c \to \infty$ the equations become $x_u + y_v = 0$ and $x_v = y_u$, so $x - iy$ is a holomorphic function of $u + iv$ wherever it is defined.

Non-example: a uniform flow, or a simple wave such as the Prandtl–Meyer expansion fan, has $J = 0$ everywhere. Such flows have no hodograph image of positive area and are not obtained by the transformation.