In chapter 2 a method was presented for calculating the impedance at any point along an acoustic horn from the impedance at any other part of the horn. For this to be useful we must get a starting point. In practice we are dealing with horns with a narrow input end and a wide flaring bell which radiates some of the sound into free space. While the input impedance will depend on the whole of the horn we are dealing with, the radiation impedance only depends on the geometry of the opening into free space and serves as our boundary condition.
Various methods of analysing the behaviour of the open end of a duct are discussed. As mentioned previously, in a pipe of constant cross-section, the higher order modes propagate independently, while at a change of cross-section there exists mode coupling. If the rate of flare of an acoustic horn is large, the coupling of modes has a significant effect on the musical performance of the horn. An expression for the radiation impedance at an open end should therefore include mode coupling. In order to achieve a multimodal expression for the radiation impedance, it is currently necessary to assume the duct is terminated in an infinite baffle. Obviously, practical examples will not feature an infinite baffle at the opening and a small error in the input impedance calculation results. Numerical evaluation of the multimodal radiation impedance is performed by reference to Zorumski  for a cylindrical duct and to the current author, Kemp et al. , for a rectangular duct.