Wave Equation in 2D

PDEs · Axiom Academy

Understanding vibrating membranes, separation of variables, and normal modes in two dimensions 1. The Two-Dimensional Wave Equation The wave equation in two spatial dimensions describes the displacement u(x, y, t) of a membrane at position (x, y) and time t : where c is the wave speed and the Laplacian operator ∇² represents the sum of second derivatives in both spatial directions. This PDE is: Second-order in time (acceleration) Second-order in space (curvature in both x and y) 2. Vibrating Rectangular Membrane Consider a rectangular membrane with dimensions a × b , fixed at all edges (like a drumhead stretched over a rectangular frame). The boundary conditions are: We apply separation of variables by assuming a solution of the form: This separates the PDE into three ODEs, one for each variable. Each spatial function satisfies a 1D wave equation with fixed endpoints. 3. Normal Modes for Rectangular Membranes Solving the separated equations with boundary conditions gives the normal modes : where m and n are positive integers (1, 2, 3, ...). Each mode has a characteristic frequency: The integers m and n count the number of half-wavelengths in the x and y directions. The mode (1,1) is the fundamental, (1,2) has one nodal line parallel to x-axis, (2,1) has one nodal line parallel to y-axis, and so on. Each normal mode has characteristic nodal lines where the displacement is always zero. These are curves where:

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