Wave Equation

Mathematical Modeling · Axiom Academy

Modeling wave propagation: from vibrating strings to electromagnetic radiation A wave is a disturbance that propagates through space and time, typically transferring energy without permanently displacing the medium through which it travels. The key characteristic of wave motion is that the disturbance travels, while individual particles of the medium oscillate about their equilibrium positions. Amplitude (A): Maximum displacement from equilibrium Wavelength (lambda): Distance between successive crests or troughs Frequency (f): Number of oscillations per unit time Wave Speed (c): How fast the disturbance propagates 2. The One-Dimensional Wave Equation Consider a string under tension. When displaced slightly, the restoring force is proportional to the curvature (second spatial derivative), while the acceleration is the second time derivative. This leads to the classical wave equation : Here u(x,t) is the displacement at position x and time t, and c is the wave speed, determined by the medium's properties. Physical Derivation for a String For a string with tension T and linear mass density mu: Higher tension means faster propagation; higher mass density means slower propagation. The wave equation has a beautiful general solution discovered by Jean le Rond d'Alembert in 1746. It reveals that waves travel in both directions simultaneously: The solution is a superposition of two waves: f traveling right and g traveling left, both at speed c.

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