= Solution
At $x=10\delta_Y$, the annual attenuation is $e^{-10}$ and the daily attenuation is $e^{-10\sqrt{365}}$, approximately $10^{-83}$. Thus
$$
\boxed{\begin{aligned}
\Theta-\Theta_0={}&A e^{-10}\cos(\omega_Yt-10)\\
&+B e^{-10\sqrt{365}}\cos(\omega_Dt-10\sqrt{365}).
\end{aligned}}
$$
On the same vertical scale as the surface plot this is \b[almost the constant $\Theta_0$]. With a sufficiently enlarged scale, a residual annual oscillation of <amplitude> $4.54\times10^{-5}A$ remains. Its phase lag is ten radians, interpreted modulo $2\pi$ when comparing periodic maxima.
\Image[/past-exam-of-the-year-2-of-the-earth-sciences-geology-course-of-the-university-of-oxford/2024/paper-3-q8-thermal-depth-response.png]
{title=Daily and annual thermal oscillations at the surface and at ten daily or annual penetration depths}
{description=Original calculation of the established periodic solution. The illustrative amplitudes are chosen as $A=1$ and $B=0.3$ in normalized temperature units; they are not fitted to the examination plots. The deepest response is enlarged by a factor of $10^5$ to make its tiny residual annual oscillation visible.}
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