 Solution

ID: 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 π / λ

. For

T = A e^{α t} cos (k y)

T_{t} = α A e^{α t} cos (k y), T_{yy} = - k^{2} A e^{α t} cos (k y) .

Hence

T_{t} = κ T_{yy}

exactly when $α = - κ k^{2}$ , proving the claimed diffusion equation solution. At

t = 0

e^{0} = 1

, so $T (0, y) = A cos (2 π y / λ)$ . 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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