Wave Equation

Fourier Analysis · Axiom Academy

Exploring the mathematics of vibrating strings: from the fundamental wave equation through traveling waves to standing wave patterns and natural frequencies. For a vibrating string, the displacement u(x, t) at position x and time t satisfies the one-dimensional wave equation: This equation states that the acceleration of each point on the string (second derivative in time) is proportional to the curvature at that point (second derivative in space). The wave speed c emerges from the balance between two physical properties: where T is the tension in the string (force) and ρ is the linear mass density (mass per unit length). Tension T: Higher tension → faster wave propagation (tighter strings oscillate faster) Density ρ: Higher density → slower wave propagation (heavier strings oscillate slower) 3. D'Alembert's Solution: Traveling Waves D'Alembert discovered the general solution to the wave equation: where f and g are arbitrary functions. This represents two waves: f(x - ct): A wave traveling to the right at speed c g(x + ct): A wave traveling to the left at speed c 4. Standing Waves on a Fixed String For a string of length L fixed at both ends, we have boundary conditions u(0, t) = 0 and u(L, t) = 0 . This produces standing waves : These solutions oscillate in time but have fixed spatial patterns with nodes (zero displacement) and antinodes (maximum displacement). Each point oscillates with the same frequency but different amplitude

This is the written version of the interactive lesson above. See the full Fourier Analysis course.