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When $k_y=0$, $k_x$ is constant and the system becomesThusFor time dependence $e^{-i\omega t}$, the dispersion relation isExplicitly,and $\tilde{\delta }=2i{k}_x{\tilde{u}}_y/\Omega$. This is an axisymmetric inertial-acoustic wave: pressure supplies the $c_s^2{k}_x^2$ restoring term and epicyclic motion supplies the ${\Omega }^2$ term.

The radial buoyancy frequency describes restoration of a radially displaced parcel in a stratified astrophysical disk. With a dimensionless specific entropy $S$, its square is proportional to $-{\rho }^{-1}(dP/dr)(dS/dr)$. Negative $N_r^2$ means an adverse entropy stratification, but epicyclic motion can still stabilize the parcel. For smooth thin disk profiles, $|N_r^2|$ is usually of order $(H/r{)}^2{\Omega }^2$.

If the prescribed gas velocity is itself a free epicyclic motion solution, subtract its two momentum equations from the forced dust epicycle with aerodynamic drag equations. The velocity difference obeys a damped epicycle and decays as $e^{-ϵ\Omega t}$. Therefore dust approaches the gas velocity with identical phase and amplitude for every positive drag rate. At zero drag the difference does not decay, so the late-time limit is nonuniform as drag vanishes. A finite-wavelength inertial-acoustic wave is detuned from the free epicycle; matching its phase and amplitude in the weak-drag limit additionally requires $|\omega -\Omega |\ll ϵ\Omega$. For a gas density wave of frequency $\omega$, the exact radial transfer amplitude isobtained by solving the two dust equations and using $U_y=-i\Omega U_x/(2\omega )$. At $\omega =\Omega$ it is exactly one.

A small planar velocity kick on a stellar circular orbit produces a local epicyclic motion in the source's rotating reference frame. With outward $x$, forward $y$, and initial displacements zero, Hill equations give $x/(\gamma a)=2\cos \theta (1-\cos nt)+\sin \theta \sin nt$ and $y/(\gamma a)=\cos \theta (4\sin nt-3nt)+2\sin \theta (\cos nt-1)$. Tangential kicks combine an epicycle with Keplerian shear; a radial kick gives a closed leading-order epicyclic ellipse.