heated rod
Heat the middle of a metal bar and watch the warmth spread and even out: hot spots cool, cold spots warm, and given enough time the whole bar settles toward a smooth, simple temperature profile. This was the very problem Joseph Fourier set out to solve, the one that forced him to invent his series. The key insight is that any starting temperature pattern can be written as a sum of sine-shaped pieces, and each piece simply fades away on its own clock.
The temperature u(x,t) in a rod of length L obeys the heat equation, u_t = k u_xx, where k is the thermal diffusivity. With the ends held at zero temperature, u(0,t) = u(L,t) = 0, separation of variables gives spatial modes sin(n pi x / L) — the same sines as the string — but now each mode decays exponentially in time rather than oscillating: u(x,t) = sum over n of B_n sin(n pi x / L) exp( -k (n pi / L)^2 t ). Expand the initial temperature as a half-range sine series to read off the B_n. The decisive feature is that exponent: the n-th mode dies at a rate proportional to n^2, so the wiggly high harmonics vanish almost instantly while the smooth fundamental lingers longest. That is exactly why heat smooths things out — fine detail is erased fastest.
This is the second canonical Fourier-to-PDE story, and the contrast with the vibrating string is illuminating: same sine modes, but the wave equation makes them oscillate (cos in time) while the heat equation makes them decay (exp in time). The n^2 in the decay rate is the mathematical signature of diffusion's relentless smoothing, and it explains why temperature, concentration, and other diffusing quantities forget their sharp initial features and relax toward equilibrium. Different boundary conditions — insulated ends, for instance — swap the sine series for a cosine series, but the decaying-mode structure is the same.
A rod starting at uniform temperature 100 with both ends suddenly clamped to 0 has u(x,t) = (400/pi) sum over odd n of (1/n) sin(n pi x/L) exp(-k (n pi/L)^2 t): after a short time only the n = 1 term matters, leaving a single smooth sine hump that decays away.
Diffusion erases the high harmonics first (rate ~ n^2), so a jagged start quickly becomes a single smooth hump.
The n^2 decay makes the heat equation a powerful smoother forward in time but disastrously ill-posed backward in time: reconstructing a past temperature amplifies the tiny high-harmonic remnants explosively, which is why you cannot reliably run heat conduction in reverse.