Normal Modes in External Acoustics. Part II: Eigenvalues and Eigenvectors in 2D

Part II of this paper deals with normal modes for two-dimensional problems in external acoustics. We will present a formulation that is based on the so-called conjugated infinite Astley-Leis elements to evaluate these eigenvalues and eigenvectors in a state–space formulation as they are solutions of a quadratic eigenvalue problem. In our specific case, a circle is used for the circumferenting geometric structure that couples regions of finite and infinite elements. In that case, the mass matrix possesses a rank deficiency and, thus, the eigenvalue solution requires special treatment. We will present two computational examples, a radiating ellipse and an open box structure. For these examples, specified modes of the external problems are found and discussed. Among other modes, solution clearly reveals the well-known multipole shapes e.g. monopole, dipole, quadrupole etc. For the open box, it is possible to identify most of the low-frequency normal modes of the corresponding closed cavity. One remarkable observation is that the frequency of vibration increases for the open cavity compared to the closed one. Additionally, distribution of eigenvalues and their convergence are discussed.