= Solution
The linearized horizontal and vertical <momentum>, density, and incompressibility equations are
$$
\rho_0u_t=-p_x,
\qquad \rho_0w_t=-p_z-g\rho',
\qquad \rho'_t+w\rho_0'=0,
\qquad u_x+w_z=0.
$$
Differentiate the <momentum> equations in time, take $\partial_x$ of the vertical equation minus $\partial_z$ of the horizontal one, and use incompressibility to eliminate $u,p,$ and $\rho'$. Neglecting the stated slow-background term gives
$$
\boxed{\nabla^2w_{tt}+N^2(z)w_{xx}=0,
\qquad
N^2(z)=-\frac g{\rho_0}\frac{d\rho_0}{dz}.}
$$
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