The analytic properties of adjoint solutions are examined for the quasi-one-dimensional
Euler equations. For shocked flow, the derivation of the adjoint problem
reveals that the adjoint variables are continuous with zero gradient at the shock,
and that an internal adjoint boundary condition is required at the shock. A Green's
function approach is used to derive the analytic adjoint solutions corresponding to supersonic,
subsonic, isentropic and shocked transonic flows in a converging–diverging
duct of arbitrary shape. This analysis reveals a logarithmic singularity at the sonic
throat and confirms the expected properties at the shock.