= Initial conditions for an eternal monochromatic gravitational wave
{title2=$\xi=\tfrac12(h-h_0)X+(V-\tfrac12\dot h_0X)t$}
At first order, <geodesic deviation> gives $\ddot\xi_i=\ddot h_{ij}X^j/2$ for a small detector in a <plane gravitational wave in linearized gravity>. For physical initial displacement zero and initial <velocity> $V_i$, integration gives $\xi_i=[h_{ij}(t)-h_{ij}(0)]X^j/2+[V_i-\dot h_{ij}(0)X^j/2]t$. A bounded periodic response selects $V_i=\dot h_{ij}(0)X^j/2$. Prescribed positions alone do not select that response when the wave has no wave-free past.
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