  We show that two families of equations, the generalized inviscid
Proudman-Johnson equation, and the $r$-Hunter-Saxton equation (recently
introduced by Cotter et al.) coincide for a certain range of parameters. This
gives a new geometric interpretation of these Proudman-Johnson equations as
geodesic equations of right invariant homogeneous $W^{1,r}$-Finsler metrics on
the diffeomorphism group. Generalizing a construction of Lenells for the
Hunter-Saxton equation, we analyze these equations using an isometry from the
diffeomorphism group to an appropriate subset of real-valued functions. Thereby
we show that the periodic case is equivalent to the geodesic equations on the
$L^r$-sphere in the space of functions, and the non-periodic case is equivalent
to a geodesic flow on a flat space. This allows us to give explicit solutions
to these equations in the non-periodic case, and answer several questions of
Cotter et al. regarding their limiting behavior.
