🎬Boundary conditions define the allowed solutions (eigenmodes) and the natural frequencies

Boundary conditions define the allowed solutions (eigenmodes) and the natural frequencies (eigenvalues)

The comparison illustrates how boundary conditions fundamentally determine the vibrational behavior of a circular membrane governed by the wave equation. The Dirichlet (Fixed Edge) condition, where the displacement $u$ is zero at the boundary $R(u(R, t)=0)$, produces the classic drum shape with maximum amplitude at the center and a lower fundamental frequency ( $\omega \approx 2.40 ps$ ). Conversely, the Neumann (Free Edge) condition, where the radial slope is zero at the boundary ( $\partial_\rho u(R, t)=0$ ), yields a distinct mode that still maximizes at the center but features a zero slope at the edge; this constraint results in a significantly higher fundamental frequency ( $\omega \approx 3.83 ps$ ). In essence, the analysis demonstrates that while the spatial domain remains the same, changing the physical constraint at the boundary drastically alters both the shape (eigenmode) and the rate (eigenfrequency) of the stable solution.

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Boundary conditions define the allowed solutions (eigenmodes) and the natural frequencies (eigenvalues) | Animated Resultsviadean on Notion