Directivity Analysis of A Vibrating Spherical Cap on A Rigid Sphere

This study investigates methods for modeling surface velocity on a vibrating spherical cap, offering valuable insights for loudspeaker design. While various methods exist, the spherical cap on a rigid surface stands out for its modeling accuracy and theoretical comprehensiveness. Despite extensive discussions on this model, there still exists a scarcity of in-depth theoretical analysis and performance comparisons. To address this gap, the paper first derives the theoretical directivity metric in the spherical harmonic (SH) domain for the spherical cap, enhancing understanding of its directional radiation characteristics. The maximum achievable bound of directivity factor (D-F) is then determined through a solution of a generalized Rayleigh quotient. Subsequently, a modified spherical cap model incorporating frequency-dependent and radial equalizing velocity is proposed, aiming to enhance directional radiation performance especially at lower frequencies. Finally, a comparison of different spherical cap surface velocity models is provided, revealing through simulation that the proposed model outperforms others in achieving better directivity within the frequency range from 0 to 3000 Hz.