= Solution
Let $a(X)=k_0(X)^{-2}$, $F(y,X)=\cos(n\pi y/R(X))$, and $N(X)=\int_{-R}^{R}F^2dy$. At the next order of the <WKB approximation>, the transverse operator $L=a\partial_y^2+1-ak^2$ gives
$$
LP_1=i\bigl[(ak)_XAF+2ak(A_XF+AF_X)\bigr].
$$
The moving-wall <Neumann boundary condition> gives $P_{1,y}(R)=-ikR_XAF(R)$ and $P_{1,y}(-R)=+ikR_XAF(-R)$. Multiply by $F$ and integrate across the duct. Since $LF=0$ and $F_y=0$ at both walls, <integration by parts> gives
$$
\int_{-R}^{R}FLP_1dy=a[FP_{1,y}]_{-R}^{R}=-2iakR_XA.
$$
Also $N_X=2\int FF_Xdy+2R_X$, 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>:
$$
2akN A_X+(akN)_XA=0.
$$
Consequently $A\sqrt{akN}$ is a complex constant, including a constant phase. For $n\ge1$, $N=R$; for $n=0$, $N=2R$, with the factor two absorbed into the mode's arbitrary constant. Thus
$$
\boxed{A(X)=C\frac{k_0(X)}{\sqrt{k(X)R(X)}}.}
$$
Equivalently,
$$
\frac{A(X)}{A(X_*)}=\frac{k_0(X)}{k_0(X_*)}
\sqrt{\frac{k(X_*)R(X_*)}{k(X)R(X)}}.
$$
There is an independent flux check. The real-coefficient <Helmholtz equation> implies $\nabla\cdot\operatorname{Im}(p^*a\nabla p)=0$, and the wall condition gives zero normal flux. Hence the integrated longitudinal current is constant; at leading order it is $-ak|A|^2N$. 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.
Back to article page
