= Variable-sound-speed WKB duct amplitude
{title2=$A=Ck_0/\sqrt{kN},\quad N=\int_{-R}^{R}F^2dy$}
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 $p=A(X)F(y,X)\exp[-i\epsilon^{-1}\int^Xk(\xi)d\xi]$. 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 $n\ge1$ 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.
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