The telegrapher's equations govern voltage/current pulses on coaxial cables, nerve fibers, and microwave guides. Click on the wire to inject a pulse and watch it propagate, reflect, and dissipate.
∂²u/∂t² + 2α·∂u/∂t = c²·∂²u/∂x² — interpolates between wave (α=0) and diffusion (α→∞)
Lord Kelvin (1855) derived the cable equation for the transatlantic telegraph; Heaviside (1880s) added inductance to obtain the full telegrapher's equations, revealing that distortionless propagation requires LC/RG = 1. Above the characteristic frequency ω_c = √(RG)/√(LC), waves propagate; below it they decay exponentially — the distinction between wave and diffusion regimes. Nerve impulses obey a nonlinear version (Hodgkin-Huxley, 1952).