The telegraph equation interpolates between wave (hyperbolic) and diffusion (parabolic) regimes. Click to excite the field. Tune damping from pure wave to diffusion-dominated.
Telegraph equation: u_tt + 2α u_t = c² u_xx. With α=0: pure wave equation (non-dispersive). With α→∞: diffusion equation (parabolic). For α < c·k: propagating damped waves; for α > c·k: overdamped diffusion. Describes signal propagation in lossy transmission lines (Heaviside, 1880s).