Sound radiation of a vibrating circular plate set in a hemispherical enclosure

Wojciech P. Rdzanek (Corresponding author), Augustus J.E.M. Janssen, Krzysztof Szemela, Marek Pawelczyk

Research output: Contribution to journalArticleAcademicpeer-review

1 Citation (Scopus)


This study rigorously addresses the Neumann–Robin problem of sound radiation from a vibrating flexible disk embedded into a hemispherical enclosure. The enclosure is acoustically soft on its inside surface and hard on its outside surface. The method of superimposed solutions, expressed in terms of Dini series and spherical harmonics, has been applied for this purpose. The presented solution is applicable to any vibration velocity profile on the radiator, with an elastically supported thin plate excited asymmetrically used as an example. The results presented here can be valuable for modeling vibroacoustic responses of various speakers, sensors, and microphones with edges either clamped to the enclosure or nearly free. Achieving this involves modifying the values of the boundary stiffness for resisting transverse displacement and the rotation of the plate's edge. Additionally, when applying these results to model the responses of alarm or signaling transducers, high-amplitude responses are expected for selected resonant frequencies. In such cases, the boundary stiffness values can be set to ensure multiple strong and wide resonant maxima in the frequency characteristics. In addition, the results presented in this study include solving the dynamic equation of motion for the plate or rigid piston with external excitation. This equation is coupled with the wave equation inside and outside the hemispherical enclosure, allowing the inclusion of the enclosure's effect on the plate-enclosure system's responses. This approach can be useful for modeling and improving the acoustic responses of spherical speakers. In such cases, the boundary stiffness values should be relatively small to emulate a soft suspension of the plate. The impedance of the enclosure's internal surface can be arbitrarily selected, ranging from acoustically hard to acoustically soft. In this study, the impedance of mineral wool has been applied. Numerical analysis is presented for frequencies below 4 kHz, with errors smaller than 0.2%. The results have been validated using the finite element method.

Original languageEnglish
Article number118136
Number of pages41
JournalJournal of Sound and Vibration
Publication statusPublished - 3 Feb 2024


The research presented in this paper was partially supported by The Centre for Innovation and Transfer of Natural Sciences and Engineering Knowledge Project at The University of Rzeszow, Poland. The research reported in this paper has been supported by the European Union within Horizon Europe MSCA grant IN-NOVA, no. 101073037 .

FundersFunder number
European Commission
Uniwersytet Rzeszowski


    • Acoustic field
    • Acoustic impedance
    • Circular plate
    • Fluid–structure interactions
    • Hemispherical enclosure
    • Robin boundary value problem
    • Zernike circle polynomial


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