 Variable-sound-speed WKB duct amplitude

For $\nabla \cdot (k_0(X{)}^{-2}\nabla p)+p=0$ in a slowly varying rigid duct, a real local transverse normal mode $F$ with squared norm $N(X)$ has longitudinal wavenumber $k(X)$. The WKB approximation is $p=A(X)F(y,X)\exp [-i{ϵ}^{-1}{\int }^{X}k(\xi )d\xi ]$. The moving-wall Neumann boundary condition and next-order solvability give $2kN{k}_0^{-2}{A}_X+(kN{k}_0^{-2}{)}_{X}A=0$, hence the displayed amplitude. For $F=\cos (n\pi y/R)$, $N=R$ for $n\ge 1$ and $N=2R$ for $n=0$. The factor $k_0$ is essential when the coefficient in the divergence-form equation varies. This approximation requires a mode separated from cutoff and from other modes.