Energy conservation, time-reversal invariance and reciprocity in ducts with flow

A sound wave propagating in an inhomogeneous duct consisting of two semi-infinite uniform ducts with a smooth transition region in between and which carries a steady flow is considered. The duct walls may be rigid or compliant. For an irrotational sound wave it is shown that the three properties of the title are closely related, such that the validity of any two implies the validity of the third. Furthermore it is shown that the three properties are fulfilled for lossless locally reacting duct walls provided the impedance varies at most continuously. For piecewise-continuous wall properties edge conditions are essential. By an analytic continuation argument it is shown that reciprocity remains true for walls with loss. For rotational flow, energy conservation theorems have been derived only with the help of additional potential-like variables. The inter-relation between the three properties remains valid if one considers these additional variables to be known. If only the basic gasdynamic variables in both half-ducts are known, one cannot formulate an energy conservation equation; however, reciprocity is fulfilled.