Initial conditions for an eternal monochromatic gravitational wave ($\xi =\frac{1}{2}(h-h_0)X+(V-\frac{1}{2}{\dot{h}}_{0}X)t$)

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.