= Solution
For the supplied <plus polarization>, the perturbation is traceless, so $h_{\mu\nu}=\phi_{\mu\nu}$. At a fixed longitudinal coordinate $z_0$, put $H(t)=h_\oplus\cos[\omega(t-z_0)]$. With the $+---$ convention chosen above, the transverse spatial metric measuring positive squared lengths is
$$
d\ell^2=(1-H)dx^2+(1+H)dy^2.
$$
The initial coordinate separation is $(\Delta\cos\theta,\Delta\sin\theta)$ and remains constant in <transverse-traceless gauge>: $h_{00}=h_{0i}=0$ implies $\Gamma^i{}_{00}=0$, so initially stationary particles stay at fixed spatial coordinates. Their measured <proper length> is instead
$$
\boxed{\ell(t)=\Delta\sqrt{1-h_\oplus\cos(2\theta)\cos[\omega(t-z_0)]}
=\Delta\left[1-\frac{h_\oplus}{2}\cos(2\theta)\cos[\omega(t-z_0)]\right]+O(h_\oplus^2).}
$$
The first expression is understood within the linearized metric; only its first-order expansion is controlled. Separation along the two transverse axes varies oppositely; at $45$ degrees there is no first-order length modulation. This is the <angular response of a gravitational-wave separation>, despite constant coordinate separation.
If the background is instead chosen as $-+++$ while keeping the same numerical spatial amplitude, the displayed length modulation has the opposite sign. The question does not state a signature here; the convention above agrees with its earlier explicitly specified <Minkowski metric>. The physical alternating stretch-compress pattern is unchanged by consistently redefining the signed amplitude.
\Image[/past-exam-of-the-relativity-course-of-the-university-of-cambridge/2021-q5-wave-separations.png]
{title=Plus and cross gravitational-wave distortions of a freely falling particle ring}
{description=Original first-order proper-frame displacement calculation for an illustrative amplitude 0.2. The undeformed coordinate ring is shown faintly. The columns show opposite extrema and a zero of the wave phase; the cross pattern is rotated by 45 degrees relative to the plus pattern. Signs follow the $+---$ metric convention used in the solution.}
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