the telegrapher's equation
When a signal travels down a long cable — an old telegraph line, a coaxial cable, a nerve fibre — it does two things at once: it propagates like a wave and it leaks and damps like diffusion. The telegrapher's equation is the model that captures both at the same time. It sits exactly between the wave equation (pure propagation) and the heat equation (pure spreading), and which behaviour dominates depends on the cable's electrical properties.
The equation reads u_tt + (a + b) u_t + a b u = c^2 u_xx (one common form), where u is the voltage or current along the line, c is a propagation speed, and the first-derivative terms u_t come from resistance and leakage — they are friction. Strip out the u_t and u terms and you are left with the wave equation u_tt = c^2 u_xx; keep only a single dominant u_t and drop u_tt and you recover a diffusion equation. The damping term u_t is the crucial new ingredient: it makes signals decay and, for a dispersive line, spread out and distort as they travel, which is precisely the problem telegraph engineers had to fight.
It is a showcase for transform methods. Laplace-transform in time and the equation becomes a boundary-value ODE in x whose solution carries a factor like e^(-sqrt(...) x); inverting through the Bromwich integral reveals both a sharp wavefront travelling at speed c and a diffusive tail trailing behind it. The transform also hands you the dispersion relation directly, telling you how each frequency travels and decays — and the historical punchline, due to Heaviside, is that with the right balance of the cable's parameters (the 'distortionless condition') all frequencies travel at the same speed, so a signal arrives attenuated but undistorted. That insight made long-distance telegraphy and telephony possible.
Send a sharp pulse down a leaky line. Without the damping terms it would be a clean wave arriving undistorted at speed c. With resistance and leakage present, transform analysis shows the pulse arrives smaller and smeared into a tail — unless Heaviside's distortionless balance holds, in which case every frequency still moves at speed c and the shape survives, merely shrunk.
A wave equation with friction: propagation plus diffusion, the model of a real cable.
Do not picture it as 'either wave or diffusion' — its signature is that both coexist, a wavefront at finite speed followed by a diffusive wake. Drop the u_tt term and you wrongly collapse it to pure diffusion with infinite speed.