This is the 2D companion to the voxel-plate Rosenthal simulation: the same moving point-source heat conduction model, computed independently as a genuine transient thermal history instead of a single peak-temperature slice. In the frame moving with the arc, the temperature at a fixed plate point offset by ξ (along the weld direction, behind the current arc position) and r = √(y²+z²) (transverse + depth offset from the seam) is:
T(ξ,r) − T₀ = Q / (2π k R) · exp(−v(R+ξ) / (2α))
R = √(ξ² + r²), k = conductivity, α = diffusivity
The left panel is a true cross-section: a semicircular slice through the plate directly beneath the seam, banded by zone — not by evaluating T at ξ=0 (the instant the arc is directly overhead, the shortcut the 3D model uses) but by numerically maximizing T(ξ,r) over the entire weld pass at every r. Because the exponential's ξ term makes points slightly behind the arc retain heat longer than points at the same distance ahead of it, the true thermal peak at any offset actually lags a little behind the arc's closest approach — the right-hand readout above reports exactly how far.
The right panel plots that same T(ξ(t),r) formula as a genuine time series while a virtual arc sweeps past the probe offset you choose, tracing the real heating/cooling thermal cycle instead of just a static number. The shaded band marks the 800→500 °C window; its width is the t₈/₅ cooling time, the number welding codes use to judge hardening and hydrogen-cracking risk:
t₈/₅ = (Q/v) / (2πλ) · [ 1/(500−T₀) − 1/(800−T₀) ]
Zone banding follows the same peak-temperature thresholds as the 3D model: fusion (melted), coarse-grained HAZ (grain growth), fine-grained HAZ (refined, tough), intercritical HAZ (partial transformation), subcritical HAZ (tempered only). Stainless steel swaps the last two for a solution-annealed band and a chromium-carbide sensitization band instead. Higher heat input or preheat widens the bands and slows cooling; higher travel speed or a lower-conductivity material narrows them and speeds cooling up.