Wave + damping + dispersion — signal distortion on transmission lines
u_tt + 2α u_t + β²u = c² u_xx
0.40
0.00
1.50
The telegraph equation (Heaviside, ~1880) models electrical signals on transmission lines with resistance (damping α) and inductance effects (dispersion β).
When α=0 and β=0 we recover the pure wave equation — no distortion. As α increases, the pulse amplitude decays exponentially. Dispersion (β>0) spreads different frequency components at different speeds, smearing the pulse shape.
Heaviside showed that with proper loading coils, α=β makes the telegraph equation transform into a diffusion equation — crucial insight for transatlantic cables.