*3.2.2 Acoustic power balance between vortex layers and acoustic vortices*

Acoustic power generation by the unbalance between the upper and lower vortex layers (cf. Section 2.4) will be balanced with acoustic power dissipations by the wall boundary effects (cf. Section 3.1.2), sound radiation (cf. Section 3.1.3), and acoustic vortices in the sense of time average:

$$
\langle \Pi\_{\mathbf{G}}(\mathbf{t}) \rangle = \langle \Pi\_{\mathbf{a}}(\mathbf{t}) \rangle + \langle \Pi\_{\mathbf{D}}(\mathbf{t}) \rangle,\tag{41}
$$

where *Π*Gð Þ*t* is given by Eq. (14) with *ω* ¼ *ω*vl concerning the vortex layer, *Π*αð Þ*t* is the power lost from the bore that is given by the total attenuation constant of Eq. (40), and *Π*Dð Þ*t* is given by Eq. (13) with *ω* ¼ *ω*av concerning the acoustic vortex. A more exact description of *Π*Gð Þ*t* derived from the unbalance between the upper and lower vortex layers will be given in Section 4.1.
