Axisymmetric acoustic oscillations of a rigid spherical shell with two symmetric circular apertures

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The plane wave scattering problem for a rigid spherical shell with two coaxial circular holes has been solved rigorously by the Method of Analytical Regularization. An original approach to solving the triple series equations with Legendre polynomials as kernels that arise led to its solution, which is uniformly valid with respect to aperture size. The solution presents two independent infinite systems of linear algebraic equations of the second kind for the even and odd Fourier coefficients respectively. The fast convergence of the systems when truncated is the key to accurate and comprehensive studies of scattering characteristics (total and sonar cross-sections, mechanical force on the walls of doubly-connected sphere) in a wide frequency range. The solution is valid for investigation of the Rayleigh scattering region (a≪λ), the resonance region (a~λ), and also the deep quasi-optics region (a≫λ), where radius a of the sphere may attain thousands of wavelengths λ (a⁄(λ>1000)). The unique properties of the obtained solution have been used for calculation of the first 16 complex eigenvalues (in the rigid doubly-connected sphere) with a guaranteed accuracy of 8 significant decimal digits. The new phenomenon of a huge drop in the sonar cross-section near the Helmholtz mode (σ_B⁄(πa^2~10^(-10) )) was discovered and is studied both analytically and numerically. The resonance behaviour of the reflectivity and mechanical force factor is studied in a wide frequency range.
Original languageEnglish
Article numberJSV-D-20-01735
Number of pages34
JournalJournal of Sound and Vibration
Publication statusSubmitted - 15 Aug 2020


  • acoustic resonance scattering
  • acoustic resonator
  • rigid open sphere
  • complex eigenvalues
  • sonar cross-section
  • mechanical force

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