= Solution
Write $\delta_j=\sqrt{2K/\omega_j}$ for each component's <thermal penetration depth>. Its contribution at depth is
$$
C_j e^{-x/\delta_j}\cos(\omega_jt-x/\delta_j).
$$
Thus its <amplitude> is multiplied by $e^{-x/\delta_j}$ and its <wave phase> lags the surface by $x/\delta_j$ radians. The corresponding unwrapped time lag is $x/(\delta_j\omega_j)$; a phase measured from a single periodic signal is observable only modulo $2\pi$.
Higher-frequency variations have smaller penetration depths. In particular,
$$
\boxed{\frac{\delta_Y}{\delta_D}=\sqrt{\frac{\omega_D}{\omega_Y}}\simeq\sqrt{365}\simeq19.1.}
$$
The daily oscillation is therefore strongly attenuated at depths where a substantial annual oscillation survives. This is a <thermal diffusion wave>, whose phase lag is not the arrival time of a sharp thermal front. The displayed solution is the established periodic response; an additional transient would depend on initial subsurface data not needed for this interpretation.
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