= Solution
For a two-dimensional <internal gravity wave> with fixed horizontal <wave number> $k\ne0$, the vertical <velocity> amplitude satisfies
$$
\widehat w_{zz}+m(z)^2\widehat w=0,\qquad m(z)^2=k^2\left(\frac{N(z)^2}{\omega^2}-1\right).
$$
This follows from the <Linearized Boussinesq equations> just as in part (a), allowing the background <buoyancy frequency> to vary. The <WKB approximation> gives locally
$$
\widehat w\simeq\frac{A_\pm}{\sqrt{|m(z)|}}\exp\!\left(\pm i\int^z m(s)\,ds\right).
$$
It requires slowly varying stratification compared with the local vertical <wavelength>, in particular $|m_z|\ll m^2$, and a small-amplitude wave in the inviscid, nonrotating regime. Propagation requires $N>|\omega|$; where $N<|\omega|$, the local vertical <wave number> is imaginary and the solution is evanescent. Turning levels and abrupt changes need a separate connection calculation. Finite <viscosity>, background flow, rotation or nonlinear effects require modifications of this model.
The <ray tracing> equations use the local <dispersion relation> as a <Hamiltonian>:
$$
\dot x=\partial_k\omega,\qquad \dot z=\partial_m\omega,\qquad \dot k=0,\qquad \dot m=-\partial_z\omega.
$$
The stationary, horizontally uniform background conserves the <frequency> and horizontal <wave number>. In two dimensions the <group velocity> gives $dx/dz=-m/k$. For the specified <buoyancy frequency> and $\omega=1$, $m^2=256k^2z^2$. Hence for $z>0$ the two geometric <internal-wave rays> through the specified point are
$$
\boxed{x=8(z^2-1)\quad\hbox{or}\quad x=-8(z^2-1).}
$$
Each curve has both directed propagation senses, giving four local directed rays. Their formal intersections with $z=0$ are $x=-8$ and $x=8$.
At $z=0$, $N=\omega$ and $m=0$: this is a <double-zero internal-wave turning level>, since $m^2$ touches zero without changing sign. The ray becomes vertical geometrically, but its <group velocity> tends to zero. Taking $k>0$ without loss of generality, the downward branch has
$$
|c_{g,z}|=\frac{16z}{|k|(1+256z^2)},\qquad
\Delta t=\frac{|k|}{16}\log\frac{z_i}{z}+8|k|(z_i^2-z^2).
$$
The formal travel time diverges logarithmically as $z\to0$. Meanwhile $|m_z|/m^2=1/(16|k|z^2)$ grows, so the <WKB approximation> fails before that limit; its predicted $|m|^{-1/2}$ amplitude cannot be extrapolated to infinity. \b[The spatial ray reaches a finite limiting position, but finite-wavelength behaviour near the zero must be found beyond WKB.] There is no evanescent half-space supplied by this particular $N(z)$, so ordinary simple-turning-point reflection cannot be assumed solely from the local ray construction.
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