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

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:

F_{n} (x) = sin \frac{nπ x}{L}, λ_{n} = κ \frac{n ^{2} π ^{2}}{L ^{2}}, T (x, t) = \sum_{n = 1}^{\infty} b_{n} F_{n} (x) e^{- λ_{n} t} .

(1)

Match the initial uniform interior temperature

T_{0}

using the orthogonality of the Fourier sine series:

b_{n} = \frac{2}{L} \int_{0}^{L} T_{0} sin \frac{nπ x}{L} d x = \frac{2 T _{0}}{nπ} {1 - (- 1)^{n}} .

(2)

Thus $b_{n} = 0$ for even integers $n$ and $b_{n} = 4 T_{0} / (nπ)$ for odd integers $n$ , giving

T (x, t) = \frac{4 T _{0}}{π} \sum_{j = 0}^{\infty} \frac{s i n (( 2 j + 1 ) π x / L )}{2 j + 1} exp [- \frac{( 2 j + 1 ) ^{2} π ^{2} κ t}{L ^{2}}] .

(3)

For 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