Natural Vibrations of a Gas in a Helmholtz Resonator with a Periodically Varying Cross-section

Within the framework of the long-wave approximation, the frequencies and shapes of gas natural oscillations in a Helmholtz resonator having the shape of a pipe of periodic cross-section have been studied. The problem is reduced to the Sturm–Liouville problem with boundary conditions of the first kind, the solution of which is carried out by the method of accelerated convergence. A detailed analysis of the dependences of eigenvalues and eigenfunctions on pipe parameters was carried out. A “self-similar” type of dependence of the natural frequency for various modes has been revealed. The values of the resonator periodicity parameters at which a sharp change in the natural frequency occurs are determined.