the analytical solution for a specific mode of the Helmholtz equation using Bessel functions

The analysis demonstrates that the standing wave patterns (vibration modes) in a circular domain, governed by the Helmholtz equation, are fundamentally defined by the zeros of the Bessel function of the first kind, $J_m$. Specifically, for the boundary condition where the amplitude must be zero at the radius $R$, the possible wavenumbers ( $k$ ) are constrained such that $J_m(k R)=0$. This means the integer mode indices $m$ and $n$ act as "quantum numbers": the angular index $m$ determines the number of straight nodal diameters, while the radial index $n$ determines the number of nodal circles (including the boundary), ultimately dictating the precise geometric shape and standing wave pattern observed in the 2D solution.

🎬Narrated Video

https://youtu.be/w0dch32zJqY

🏗️Structural clarification of Condensed Notes

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🗒️Downloadable Files - Recursive updates

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