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

Introduce $t_0=t$, $T_1=ϵt$, $T={ϵ}^{2}t$ and write $x=x_0+ϵx_1+\cdots$. The method of multiple scales gives $x_0=A(T_1,T)e^{i{t}_0}+\overline{A}{e}^{-i{t}_0}$. At order $ϵ$, the resonant forcing is $-2{\partial }_{t_0}{\partial }_{T_1}{x}_0$, because multiplication by $\cos t_0$ generates only a constant and second harmonics. The solvability condition in the method of multiple scales therefore gives $A_{T_1}=0$. A nonresonant particular correction isAt order ${ϵ}^2$, the forcing is $-2{\partial }_{t_0}{\partial }_T{x}_0-f{x}_0-\cos t_0{x}_1$. Its $e^{i{t}_0}$ coefficient is $-2i{A}_T+(1/6-f)A+\overline{A}/4$. Removing this secular term givesPut $A=(a-ib)/2$, so $x_0=a(T)\cos t+b(T)\sin t$. The slow amplitudes for the second Mathieu instability tongue satisfyThe initial conditions here are at leading order; a homogeneous $O(ϵ)$ correction makes the original data exact without changing $x_0$. For example its initial cosine coefficient is $2/3$ in the chosen $x_1$ gauge. The slow squared growth rate is $(f+1/12)(5/12-f)/4$. Thus the robust leading stable ranges areSet $\nu =\frac{1}{2}\sqrt{(f+1/12)(f-5/12)}$ in those ranges. The required explicit leading solution isThe remainder is uniform for $0\le {ϵ}^{2}t\le C$, with fixed $C$ and fixed $f$ away from a changing instability edge. Inside $-1/12<f<5/12$, put $\sigma =\frac{1}{2}\sqrt{(f+1/12)(5/12-f)}$; replace the slow cosine and $\sin (\nu T)/\nu$ by $\cosh (\sigma T)$ and $\sinh (\sigma T)/\sigma$. The initial data excite this growing solution, so the leading solution is unstable.

At the leading endpoints the slow matrix is degenerate. Its continuous limits give $x_0=\cos t+(T/4)\sin t$ at $f=-1/12$, and $x_0=\cos t$ for these particular initial data at $f=5/12$; generic initial phases at the latter endpoint have slow algebraic growth. These are leading, not exact, instability edges. To specify the exact small-$ϵ$ endpoint status, put $t=2s$, so the Mathieu equation has characteristic parameter $a_M=4+4f{ϵ}^2$ and coupling $q_M=-2ϵ$. In the orthonormal cosine basis $1,\cos 2s,\cos 4s,\cos 6s$, the matrix has diagonal $0,4,16,36$ and adjacent couplings $\sqrt{2}{q}_M,q_M,q_M$; in the sine basis $\sin 2s,\sin 4s,\sin 6s$, the diagonal is $4,16,36$ and adjacent couplings are $q_M,q_M$. Expanding the roots near $4$ of their determinants yieldsThese truncations determine the displayed orders because reaching the next omitted mode and returning requires at least six coupling steps. The fourth-order edges of the second Mathieu instability tongue are thereforeFloquet theory gives the unstable gap $f_{-}<f<f_{+}$ near this resonance. In particular, both fixed values $-1/12$ and $5/12$ lie outside the exact gap for sufficiently small nonzero $ϵ$. Their fixed-$ϵ$ solutions are bounded, although the leading slow matrix alone does not show this and the amplification bound is not uniform as $ϵ\to 0$.

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.