= Solution
The <diffusion equation> is linear, so <superposition> gives
$$
\boxed{T(t,y)=A_1e^{-\kappa(2\pi/\lambda_1)^2t}\sin\frac{2\pi y}{\lambda_1}
+A_2e^{-\kappa(2\pi/\lambda_2)^2t}\sin\frac{2\pi y}{\lambda_2}.}
$$
Each term separately satisfies the equation, and their sum recovers the stated initial data at $t=0$. Its e-folding times are $\tau_i=\lambda_i^2/(4\pi^2\kappa)$, so \b[$\lambda_1<\lambda_2$ implies $\tau_1<\tau_2$]. The shorter-wavelength oscillation loses a given fraction of its <amplitude> first. It approaches zero exponentially rather than disappearing at a finite instant. This preferential damping explains smoothing of fine-scale <temperature> structure.
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