Resistive wall impedance of an insert

The standard theoretical formulas for resistive wall impedance are usually derived in a model which assumes an infinitely long pipe. In practice, one often has to deal with resistive inserts with a conductivity different from the rest of the pipe. To address this case, we calculate the resistive wall impedance when the wall conductivity varies along the axis of the pipe. We show that at not very high frequencies the impedance of an insert per unit length is given by the same formulas as for an infinitely long pipe.