Past exam of the year 2 of the earth sciences (geology) course of the University of Oxford / 2023 / Paper 1 / 10 / c / Solution

The reference to part “(c)(i)” in the source is circular; the relevant modes are those verified in part (b)(i). Linearity of the heat equation and its homogeneous Dirichlet boundary conditions allows superposition of those modes:Match the initial uniform interior temperature $T_0$ using the orthogonality of the Fourier sine series:Thus $b_n=0$ for even integers $n$ and $b_n=4T_0/(n\pi )$ for odd integers $n$, givingFor every $t>0$, the exponential factor makes the series and the derivatives needed in the heat equation converge uniformly on later time intervals. At $t=0$, the Fourier sine series equals $T_0$ at interior points but zero at the endpoints: the instantaneous cooling boundary condition is incompatible with the pre-immersion endpoint values. This corner mismatch is physically the idealization of an instantaneous bath contact and does not invalidate the solution for $t>0$.