Derivation of Wave Equation

PDEs · Axiom Academy

LESSON Derivation of Wave Equation From a vibrating string to the fundamental wave equation in physics 1. Physical Setup: Vibrating String Consider a flexible string stretched along the x-axis under constant tension T . The string has linear mass density ρ (mass per unit length). When the string vibrates, we denote the vertical displacement at position x and time t as u(x,t) . Isolate a small element of the string between positions x and x + Δx . The tension T acts tangentially at both ends. The angles the string makes with the horizontal are θ 1 on the left and θ 2 on the right. For small oscillations, the angles θ are small. We can use the approximation: sin θ ≈ tan θ ≈ θ (in radians). Since tan θ equals the slope of the string, we have: 4. Applying Newton's Second Law The net vertical force on the element equals its mass times acceleration. The mass of the element is ρΔx, and the vertical acceleration is ∂²u/∂t². The net vertical force comes from the difference in tension components at the two ends: Divide both sides by Δx and take the limit as Δx approaches zero. The right-hand side becomes the definition of a partial derivative with respect to x : 6. The Wave Equation and Wave Speed Rearranging and defining the wave speed c = √(T/ρ), we obtain the one-dimensional wave equation. This elegant equation describes how waves propagate along the string at speed c , which depends on the tension and mass density.

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