Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 82 / 3 / b

Use $X=ϵx$ and the WKB approximationAt leading order, the divergence-form acoustic Helmholtz equation becomes $P_{0,yy}+[k_0(X{)}^2-k(X{)}^2]P_0=0$. The sloping-wall Neumann boundary condition becomes $P_{0,y}=0$ at $y=\pm R(X)$. In the even transverse family this has normal modes $P_0=A(X)\cos (n\pi y/R(X))$, where $n=0,1,\dots$, andFor the right-going propagating branch take real positive $k$. The displayed local expression $e^{-ikx}$ in the question must be read as a frozen-coefficient shorthand: the actual slowly varying phase is $\exp [-i{\int }^{x}k(ϵs)ds]$. Differentiating $e^{-ik(X)x}$ would instead give the erroneous local wavenumber $k+X{k}_X$.

The symmetric duct also has odd transverse normal modes $\sin [(n+1/2)\pi y/R]$. The integer-cosine form selects the even family requested here, rather than representing every duct mode. The approximation requires smooth slow variation and a positive $k$ separated from cutoff; it fails in a cutoff transition.

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.