= Density contrast of an entrained dust wave
{title2=$\delta_d=(ku/\Omega)\cos(kx-\Omega t)+C(x)$}
Let $\Sigma_{d0}$ be the uniform background dust <surface density of a disk> and $\delta_d=(\Sigma_d-\Sigma_{d0})/\Sigma_{d0}$ its fractional perturbation. Here $k$ is the radial <wavenumber>, $\Omega$ is the epicyclic frequency and $u$ is a real velocity amplitude. For that uniform background and late velocity $v_x=u\cos(kx-\Omega t)$, linear dust <continuity equation> gives $\partial_t\delta_d=-\partial_xv_x=ku\sin(kx-\Omega t)$. Integrating yields the displayed formula. The coherent travelling-wave fractional excess is $|ku|/\Omega$, with validity requiring this number much smaller than one. The static term $C(x)$ records initial density and transient displacement: drag damps velocity, but supplies no density diffusion. The long-wavelength gas wave has the same fractional density amplitude, so entrainment alone does not produce an additional dust-to-gas ratio enhancement at this order.
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