Past exam of the year 2 of the earth sciences (geology) course of the University of Oxford / 2022 / Paper 1 / 10 / b / i / Solution

Write $k=2\pi /\lambda$. For $T=A{e}^{\alpha t}\cos (ky)$, direct partial differentiation givesHence $T_t=\kappa T_{yy}$ exactly when $\alpha =-\kappa k^2$, proving the claimed diffusion equation solution. At $t=0$, $e^0=1$, so $T(0,y)=A\cos (2\pi y/\lambda )$. 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.