Methods of the Lagrangian and Hamiltonian Mechanics in Aeroacoustics Problems

It is well known that a sound source related to nonstationary motion of vortices at small Mach numbers can be obtained in the incompressible inviscid fluid approximation. In this study, it is proposed to describe the perturbation dynamics of an incompressible ideal fluid using the Lagrangian and Hamiltonian formalism with the displacement field and momentum density perturbation as canonical variables. Based on Noether’s theorem, the conditions of quadrupole moment conservation in the evolution of small perturbations of stationary flows have been formulated. It has been shown that these conditions are always satisfied for perturbations of uniform jet flows. The obtained results not only yield a solution to the general mechanics problem on motion integrals; they are also significant in aeroacoustics because the quadrupole moment of a vortex flow is the principal term of sound source expansion in the Mach number.