🎬the net forces and tension acting on a small element of a vibrating membrane

The demo clearly illustrates the key physical principle behind the 2D Wave Equation by precisely differentiating the vectors at the center of the membrane element. The green dashed arrow ( $n$ ) defines the local surface geometry (the direction perpendicular to the tangent plane), while the red solid arrow ( $dF _{ u , \text { net }}$ ) represents the net vertical restoring force. Critically, the red vector is constrained to be purely vertical and its magnitude tracks the local curvature (the Laplacian, $\nabla^2 u$ ). This separation demonstrates that while the local tension forces (orange and purple) are tangent to the surface, the net resultant force causing vertical acceleration is constrained to the vertical axis, thereby validating the fundamental force balance expressed in the wave equation ( $\rho \partial^2 u / \partial t^2=T \nabla^2 u$ ).

Brief audio

Condensed Notes

the net forces and tension acting on a small element of a vibrating membrane | Animated Resultsviadean on Notion