Past exam of the year 2 of the earth sciences (geology) course of the University of Oxford / 2024 / Paper 3 / 8 / c

For the sketches, take $A$ to multiply the annual component and $B$ the daily component. Their amplitudes are left as parameters, as in the model.

At the surface both attenuation factors are one and both lags vanish:The sketch is a rapid daily oscillation on a slowly varying annual background. The amplitudes cannot be inferred uniquely from the sampling intervals alone.

At $x=10{\delta }_D$, the daily component is multiplied by $e^{-10}\simeq 4.54\times 10^{-5}$ and has a lag of ten radians. The annual component instead has $x/{\delta }_Y=10/\sqrt{365}\simeq 0.5234$. HenceThe sketch is nearly a smooth annual sinusoid, with annual amplitude about $0.5925A$ and annual phase lag about $0.5234$ radians, corresponding to about $30.4$ days on the chosen year scale. The tiny daily component is retained in the formula rather than declared identically zero.

At $x=10{\delta }_Y$, the annual attenuation is $e^{-10}$ and the daily attenuation is $e^{-10\sqrt{365}}$, approximately $10^{-83}$. ThusOn the same vertical scale as the surface plot this is almost the constant ${\Theta }_0$. With a sufficiently enlarged scale, a residual annual oscillation of amplitude $4.54\times 10^{-5}A$ remains. Its phase lag is ten radians, interpreted modulo $2\pi$ when comparing periodic maxima.