Modeling Steady Acoustic Fields Bounded in Cavities with Geometrical Imperfections

A mathematical method is derived within the framework of classical Lagrangian field theory, which is suitable for the determination of the eigenstates of acoustic resonators of nearly spherical shape. The method is based on the expansion of the Helmholtz differential operator and the boundary condition in a power series of a small geometrical perturbation parameter e{epsilon} . The method extends to orders higher than e²{epsilon^2} the calculation of the perturbed acoustic eigenvalues, which was previously limited by the use of variational formalism and the methods of Morse and Ingard. A specific example is worked out for radial modes of a prolate spheroid, with the frequency perturbation calculated to order e³{epsilon^3} . A possible strategy to tackle the problem of calculating the acoustic eigenvalues for cavities presenting non-smooth geometrical imperfections is also described.