Compact rotating two-blade loading source (${\rho }^{\prime }\propto R^{-1}{\sin }^2\theta \sin (2\Omega {\tau }_0)$)

Two diametrically opposite rotating blades with equal tangential loads have zero total rotating force. Their leading compact acoustic dipoles cancel, but their retarded positions and moving-surface retarded Jacobians leave an acoustic quadrupole. For line load $2Fr/a^2$ and angular speed $\Omega$, the first radiating density is $2F{\Omega }^{2}a{\sin }^2\theta \sin (2\Omega {\tau }_0)/(3\pi c_0^{4}R)$. The twofold geometry gives frequency $2\Omega$, while projection into the rotor plane gives the amplitude directivity ${\sin }^2\theta$.