Wave Equation Summary

PDEs · Axiom Academy

Key concepts, formulas, and solution techniques for the wave equation PDE Classification: Hyperbolic partial differential equation describing wave propagation in space and time Physical Interpretation: Models vibrating strings, sound waves, electromagnetic waves, and seismic waves Wave Speed: Parameter c determines how fast disturbances propagate through the medium Key Property: Solutions propagate without changing shape when traveling in opposite directions Starting Point: Newton's second law applied to an infinitesimal element of a vibrating string Force Balance: Tension forces and mass-times-acceleration yield the PDE Assumptions: Small displacement, uniform tension, negligible gravity compared to tension Result: Second-order in both time and space with wave speed c = where T is tension and is linear density General Form: u(x,t) = f(x-ct) + g(x+ct) representing left and right traveling waves Traveling Waves: Functions of form f(x-ct) move right at speed c , while g(x+ct) move left Initial Conditions: Position u(x,0) and velocity u_t(x,0) determine the functions f and g Physical Insight: Any initial disturbance splits into two waves traveling in opposite directions Ansatz: Assume solution u(x,t) = X(x)T(t) to separate spatial and temporal parts Eigenvalue Problem: Leads to X'' + X = 0 and T'' + c^2 T = 0 Boundary Conditions: Determine allowed values of (eigenvalues) and corresponding spatial modes General Solution: Superposition of infinitely many normal mode solutions

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