📢Constraint Dictates Drum Head Vibration

The boundary condition applied to a circular membrane is the decisive factor that determines the unique vibrational solution, including the mode shape and frequency. For example, the common physical setup of a drum, which has a rigid frame holding the membrane fixed at the edges (radius $R$), is mathematically described by the Dirichlet boundary condition where the transversal displacement ($u$) must be zero at the boundary: $u(\rho=R, \phi, t)=0$. This condition is sufficient, alongside initial conditions, to provide a unique solution to the wave equation. Critically, the sources illustrate that this specific Dirichlet condition produces the classic drum shape, featuring maximum amplitude at the center and resulting in a lower fundamental frequency. If the physical constraint were changed—for instance, to the Neumann condition (where the radial slope is zero at the boundary)—the resulting stable solution would be drastically altered, yielding a distinct mode shape and a significantly higher fundamental frequency, even though the spatial domain remains identical.

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Constraint Dictates Drum Head Vibration | Audiosviadean on Notion