= Solution
Write the density as a hydrostatic reference profile $\widehat\rho(z)$ plus a small perturbation $\rho'$, and define buoyancy and <buoyancy frequency> by
$$
b=-\frac g{\rho_0}\rho',
\qquad
N^2=-\frac g{\rho_0}\frac{d\widehat\rho}{dz}>0.
$$
The <Boussinesq approximation> to the <Navier-Stokes equation>, together with <mass conservation>, is
$$
\nabla\mathbin\cdot\mathbf u=0,
$$
$$
\frac{D\mathbf u}{Dt}
=-\nabla\pi+b\mathbf e_z+\nu\nabla^2\mathbf u,
\qquad
\frac{Db}{Dt}+N^2w=\kappa\nabla^2b,
$$
where $D/Dt$ is the <material derivative>, $\nu$ is <kinematic viscosity>, and $\kappa$ is <mass diffusivity>. Linearizing about rest and eliminating pressure and buoyancy gives
$$
\left[(\partial_t-\kappa\nabla^2)(\partial_t-\nu\nabla^2)\nabla^2
+N^2\partial_x^2\right]w=0.
$$
For a <plane wave> proportional to $e^{i(kx+mz-\omega t)}$, with $K^2=k^2+m^2$, the viscous-diffusive <dispersion relation> is
$$
(-i\omega+\nu K^2)(-i\omega+\kappa K^2)K^2+N^2k^2=0.
$$
In the inviscid limit this becomes
$$
\omega_0^2=\frac{N^2k^2}{k^2+m^2}.
$$
For weak diffusion the two oscillatory roots are
$$
\omega=\pm\omega_0-\frac i2(\nu+\kappa)K^2
+O\bigl((\nu-\kappa)^2K^4/\omega_0\bigr),
$$
so a freely evolving Fourier mode decays.
A single plane wave is also an exact solution of the nonlinear equations. Every field depends only on its phase $\Theta=\mathbf k\mathbin\cdot\mathbf x-\omega t$, while <incompressible flow> gives $\mathbf u\mathbin\cdot\mathbf k=0$. Therefore $\mathbf u\mathbin\cdot\nabla$ annihilates both $\mathbf u(\Theta)$ and $b(\Theta)$, and all nonlinear advection terms vanish.
For a boundary-forced wave with real $\omega$, nonzero $\nu$ or $\kappa$ instead makes the bulk vertical wavenumber complex, attenuating the propagating beam. Because diffusion raises the spatial order of the equations, additional short vertical-wavenumber roots form viscous and scalar <boundary layers>; they allow a no-slip velocity condition and a scalar no-flux condition to accompany impermeability. These layers and bulk attenuation become essential near <critical internal-wave reflection>, where the inviscid reflected wavelength collapses.
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