= Solution
Use the <time-harmonic wave> convention $\operatorname{Re}(\psi e^{-i\omega t})$, with a positive background <wavenumber> $k$. The scalar <Helmholtz equation> for a constant-density acoustic model is
$$
\psi_{xx}+\psi_{zz}+k^2n^2\psi=0.
$$
Writing $\psi=e^{ikx}E$ gives the exact envelope equation
$$
E_{xx}+2ikE_x+E_{zz}+k^2(n^2-1)E=0.
$$
The <paraxial approximation> neglects $E_{xx}$ relative to $2ikE_x$, giving the <parabolic wave equation>
$$
\boxed{2ikE_x+E_{zz}+k^2(n^2-1)E=0,\qquad
E_x=\frac{i}{2k}E_{zz}+\frac{ik}{2}(n^2-1)E.}
$$
The choice of carrier and the direction of propagation matter: this is a forward, slowly varying envelope approximation. Sufficient scale conditions are $|E_{xx}|\ll2k|E_x|$, transverse spectral components $|p|\ll k$, and medium/envelope variation on longitudinal scales large compared with $1/k$. With the carrier fixed at the background $k$, small $|n-1|$ makes the refractive phase vary slowly too. Large-angle propagation, appreciable backscattering, or rapid longitudinal variation violates the approximation. Acoustic models with variable <mass density> can have additional gradient terms, so the scalar <Helmholtz equation> itself is a model assumption. For the <Gaussian beam> in part (b), useful initial conditions are $kD\gg1$ and $D/|F|\ll1$, with the second condition absent for an initially uncurved beam. A narrow angular spectrum, rather than merely the label “Gaussian”, justifies the <paraxial approximation>.
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