= Solution
The time-dependent components of the <second mass moment tensor> contain $\sin(2\Omega t)$ and $\cos(2\Omega t)$. Taking two time derivatives for the <retarded quadrupole field> leaves that frequency unchanged. Hence the <gravitational-wave frequency> is
$$
\boxed{\omega_{\mathrm{GW}}=2|\Omega|,\qquad
f_{\mathrm{GW}}=\frac{|\Omega|}{\pi}.}
$$
Here $\omega_{\mathrm{GW}}$ is angular frequency and $f_{\mathrm{GW}}$ counts cycles per unit time. If $I_1=I_2$, the <mass quadrupole moment> is time independent, so the leading <gravitational wave> amplitude vanishes and there is no emitted quadrupole frequency to measure.
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