= Solution
A <heat pump> delivers $Q_H=Q_S+W$: electrical <work> moves <heat> from a source as well as contributing <energy> itself. Its heating <coefficient of performance> can therefore exceed one, whereas direct resistance heating supplies at most its electrical input as useful room <heat>. For an ideal reversible pump,
$$
\mathrm{COP}_H=\frac{T_H}{T_H-T_S},\qquad W=Q_H\frac{T_H-T_S}{T_H},
$$
with absolute <temperatures> and $T_S<T_H$. This is an ideal limit; actual costs also depend on compressor performance and tariffs.
For the <annual ground-temperature wave>, model the soil as a homogeneous semi-infinite conductor, with depth positive downward and a prescribed surface <temperature>. Here $\kappa$ is <thermal conductivity> and $C$ is <heat capacity> per unit volume, so the <thermal diffusivity> is $\kappa/C$. Conservation of <energy> gives $C\partial_tT=\kappa\partial_z^2T$. After transients decay, put $T-T_0=\operatorname{Re}[A e^{i\omega t-qz}]$. Substitution gives $q^2=i\omega C/\kappa$; the solution bounded at depth uses $q=(1+i)/\delta$. Matching the surface then gives
$$
\boxed{\delta=\sqrt{\frac{2\kappa}{\omega C}},\quad
T_S=T_0-T_Ae^{-z/\delta}\cos(\omega t-z/\delta).}
$$
This approximation identifies the ground surface with air <temperature> and neglects the pump's perturbation of the ground, as required by the model. The attenuation and phase delay arise from <heat> <diffusion>, not propagation at one fixed wave <speed>.
The graph below covers a full annual cycle. At $z/\delta=\pi/2$, the amplitude is $20e^{-\pi/2}=4.16$ K and the minimum lags January by a quarter-year. At $z/\delta=\pi$, the amplitude is only $0.864$ K and the minimum lags by half a year: the ground is slightly warmer than its annual mean in January.
\Image[/past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2013/paper-2-b2-seasonal-source.png]
{title=Annual air and ground temperatures, with attenuation and depth-dependent phase delay}
The <seasonal source depth for a reversible heat pump> has a subtle limit: a deep source approaches $T_0$ and avoids severe winter cooling, but infinite depth is not the thermal optimum of this single-frequency model. In January, with $x=z/\delta$, maximize $T_S=T_0-T_Ae^{-x}\cos x$. The first and largest maximum is at $x=3\pi/4$, where $\cos x+\sin x=0$, giving \b[$z=3\pi\delta/4$ and $T_S\simeq16.34^\circ\mathrm C$]. It is also the optimum of the ideal annually integrated heating <work>: the heating load is even about January, so its weighted sine term integrates to zero and the depth dependence is again proportional to $e^{-x}\cos x$. Installation cost, finite tube spacing, <heat> extraction, weather and real pump performance alter this result. The supplied model therefore supports a finite seasonal thermal optimum, while not fixing a practical burial depth in metres.
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