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Let $a(X)=k_0(X{)}^{-2}$, $F(y,X)=\cos (n\pi y/R(X))$, and $N(X)={\int }_{-R}^R{F}^{2}dy$. At the next order of the WKB approximation, the transverse operator $L=a{\partial }_y^2+1-a{k}^2$ givesThe moving-wall Neumann boundary condition gives $P_{1,y}(R)=-ik{R}_{X}AF(R)$ and $P_{1,y}(-R)=+ik{R}_{X}AF(-R)$. Multiply by $F$ and integrate across the duct. Since $LF=0$ and $F_y=0$ at both walls, integration by parts givesAlso $N_X=2\int F{F}_{X}dy+2R_X$, because $F^2=1$ at both walls. Substituting these identities in the next-order equation produces the transport equation for the variable-sound-speed WKB duct amplitude:Consequently $A\sqrt{akN}$ is a complex constant, including a constant phase. For $n\ge 1$, $N=R$; for $n=0$, $N=2R$, with the factor two absorbed into the mode's arbitrary constant. ThusEquivalently,There is an independent flux check. The real-coefficient Helmholtz equation implies $\nabla \cdot \mathrm{Im}(p^{\ast }a\nabla p)=0$, and the wall condition gives zero normal flux. Hence the integrated longitudinal current is constant; at leading order it is $-ak|A{|}^{2}N$. This agrees with the transport equation. The factor $k_0$ must be retained: using only $(kR{)}^{-1/2}$ would ignore the variable coefficient in the given divergence-form equation.