= Solution
Choose $t_1=1/(\kappa k^2)$ to make the sketch explicit. The three profiles are
$$
T(0,y)=A\cos ky,\qquad T(t_1,y)=Ae^{-1}\cos ky,\qquad
T(2t_1,y)=Ae^{-2}\cos ky.
$$
They have \b[the same <wavelength>, zero crossings and phase], with <amplitudes> $A$, $0.3679A$ and $0.1353A$. <Diffusion> reduces the <temperature gradient> and <amplitude>; it does not advect or change this individual mode’s <wavelength>.
\Image[/past-exam-of-the-year-2-of-the-earth-sciences-geology-course-of-the-university-of-oxford/2022/paper-1-diffusive-smoothing.png]
{title=Original diffusion solutions: one Fourier mode loses amplitude, and a two-mode profile loses its short-wavelength structure faster}
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