= Seasonal source depth for a reversible heat pump
{title2=$z_{\rm winter}=3\pi\delta/4$}
For a sinusoidal ground-temperature model whose coldest surface day is $t=0$, source <temperature> on that day is $T_0-T_Ae^{-x}\cos x$, where $x=z/\delta$. The first and largest thermal maximum occurs at $x=3\pi/4$, not at infinite depth. This maximizes the winter <coefficient of performance> for a reversible pump with a fixed indoor <temperature>. With a heating load even about that day, the same optimum minimizes annually integrated reversible heating <work>. Excavation cost, finite <heat> extraction and real heat-pump behaviour can change the optimum.
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