= Solution
Write $k=2\pi/\lambda$. For $T=Ae^{\alpha t}\cos(ky)$, direct <partial differentiation> gives
$$
T_t=\alpha Ae^{\alpha t}\cos(ky),\qquad
T_{yy}=-k^2Ae^{\alpha t}\cos(ky).
$$
Hence $T_t=\kappa T_{yy}$ exactly when \b[$\alpha=-\kappa k^2$], proving the claimed <diffusion equation> solution. At $t=0$, $e^0=1$, so \b[$T(0,y)=A\cos(2\pi y/\lambda)$]. The infinite spatial domain needs no end-boundary conditions. For positive <thermal diffusivity> the <amplitude> decreases, as described by the <thermal diffusion rate of a Fourier mode>.
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