Vibroacoustics of a membrane-finite cavity resonator: Traditional and alternative approaches for analysis

Majdi Gzal, Alexander F. Vakakis, Oleg V. Gendelman

Research output: Contribution to journalArticlepeer-review


This paper investigates a vibroacoustic model consisting of a flexible rectangular membrane backed by an acoustic air cavity of large lateral extent. A traditional methodology (termed Approach I) for analyzing the interaction between the acoustic airfield in the cavity and the flexible membrane is initially revisited and employed. In this approach, the transversal deformation of the mid-membrane is represented as an infinite series of eigenfunctions (normal modes) that satisfy the clamped boundary conditions for the in-vacuo membrane, regardless of the presence of the cavity. However, numerical challenges and limitations are encountered in obtaining direct insights into system characteristics, as this method heavily relies on matrix inversions and computing the roots of determinants. These operations can be computationally expensive and prone to inaccuracies and numerical instabilities, particularly when dealing with high-dimensional, non-sparse (dense) and nearly singular matrices. To overcome these limitations, an alternative methodology (termed Approach II) is proposed, which enables a precise analysis of the acoustically forced membrane and provides explicit expressions for the steady-state response, mode shapes, governed by a single transcendental characteristic equation for determining the vibroacoustic natural frequencies. Through a comparative analysis, we demonstrate the effectiveness and reliability of the proposed approach, showcasing its potential advantages over the traditional method in addressing numerical challenges. The influence of cavity depth on the vibroacoustic natural frequencies is examined across various levels of membrane stiffness, including low, intermediate, and high values. The analysis reveals the presence of both softening and hardening effects with respect to the in-vacuo case, suggesting that the air pressure plays a dual role in the system, i.e., influencing both the overall stiffness and inertia. The vibroacoustic membrane mode shapes exhibit significant deviations for soft and intermediate-stiff membranes but resemble in-vacuo mode shapes for stiff membranes, with the resemblance dependent on the selected cavity depth. Additionally, two qualitatively distinct mode families for the acoustic pressure, namely, propagating, and localized modes, are identified. The propagating modes are spatially extended and are characterized by the deformation of the volume of the medium, rendering them volume modes. In contrast, the localized modes are distinguished by localized vibrations occurring in the near field close the membrane surface, without significant energy transport or sound wave propagation in the cavity; as a result, they can be regarded as surface modes. The proposed methodology reveals features common to more complex vibroacoustic systems and offers a comprehensive and efficient approach for predictively designing the dynamics of such systems.

Original languageEnglish (US)
Article number118214
JournalJournal of Sound and Vibration
StatePublished - Mar 17 2024
Externally publishedYes


  • Acoustic field
  • Air cavity
  • Elastic membrane
  • Fluid-structure interaction
  • Helmholtz equation
  • Inhomogeneous wave equation
  • Vibroacoustic resonator

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanics of Materials
  • Acoustics and Ultrasonics
  • Mechanical Engineering


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