Tubes and ducts are acoustic waveguides, and there are acoustic radiation losses when such a waveguide opens into a large domain. Idealized models for these losses have been implemented as boundary impedance models. Thus, instead of explicitly modeling the large domain, an appropriate impedance model Zend can be applied with Zi = Zend. These models all assume that the domain is infinitely big, that the propagation is in the direction of the waveguide axis, and that the propagating mode is a plane wave. As with all other impedance boundary models, only the boundary-normal velocity component is taken into account.

For a pipe of a user-specified radius a, the acoustic losses are given by (see Ref. 6)

where J1(x) is the Bessel function of the first kind of order 1, H1(x) is the Struve function and k is the wave number of the wave. This expression is also known as the impedance from a baffled piston.

For a rectangular duct of user-specified inner width wi and inner height hi, the acoustic losses are given by (see Ref. 5)

For an unflanged circular pipe of a user-specified radius a in the limit of small radius (low ka), the pipe end impedance is given by the classical expression (see Ref. 6)

For an unflanged pipe of any user-specified radius a relative to the wave number k, an approximate end impedance is given in Ref. 32. It is

where δ(ka) is a tabulated function reproducing the curve in Fig. 2 in Ref. 32 (where δ(ka) is referred to by l/a).