Vibrating Membrane Examples

PDEs · Axiom Academy

EXAMPLE Vibrating Membrane Examples Step-by-step solutions for rectangular membrane vibrations using separation of variables Excellent work! You've completed this vibrating membrane example. Here's what we learned: Separation of Variables: The 2D wave equation can be separated into three independent ODEs (X, Y, and T), each with its own separation constant. Eigenvalue Structure: For a rectangular membrane with fixed edges, eigenvalues take the form λ mn = (mπ/a)² + (nπ/b)² , where m, n = 1, 2, 3, ... Normal Modes: Each mode u mn (x,y,t) oscillates at frequency ω mn = c√λ mn with spatial pattern sin(mπx/a)sin(nπy/b) . Initial Conditions: Coefficients for the general solution are determined by projecting initial displacement and velocity onto the orthogonal basis of eigenfunctions. Mode Frequencies: Higher mode numbers (larger m, n) correspond to higher frequencies and more complex spatial patterns on the membrane. This technique applies to many physical systems: drum heads, building floors, electromagnetic waveguides, and quantum particles in 2D boxes. Practice with different boundary shapes to deepen your understanding!

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