Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 76 / 4

Put $T=ϵt$ and use the method of multiple scales with $x=x_0(t,T)+ϵx_1+\cdots$. At leading order, $x_{0,tt}+x_0=k$, so write $x_0=k-r(T)\cos (t+\varphi (T))$ with $r(0)=k$, $\varphi (0)=0$. The order-$ϵ$ equation isUsing ${\cos }^{2}t\sin t=(\sin t+\sin 3t)/4$, its fundamental sine coefficient is $-2r_T-cr(1-k^2-r^2/4)$, and its fundamental cosine coefficient forces ${\varphi }_T=0$. Removing these resonant forcings is the solvability condition in the method of multiple scales. Thus the shifted Van der Pol amplitude evolution isWith $a=k^2-1$ and $R=r^2$, solve $R_T=caR+c{R}^2/4$ by putting $W=1/R$. This givesThe uniformly valid leading approximation is $x(t)=k-r(ϵt)\cos t+O(ϵ)$ on bounded slow-time intervals for which $r$ remains bounded. The exact initial derivative is recovered by the first-order correction; the displayed leading expression already satisfies the leading initial data.

There is an important limitation to the source's unqualified time-range wording. Starting from $r(0)=k$, the amplitude decays for $k^2<4/5$, is constant at leading order for $k^2=4/5$, and grows for $k^2>4/5$. In the latter case the denominator vanishes atA bounded-amplitude multiple-scale expansion is uniform for $0\le T\le C<T_b$, not through its predicted large-amplitude breakdown. For the decaying case every fixed bounded slow-time interval is available; the leading unstable equilibrium at $k^2=4/5$ also has bounded amplitude over such intervals. The envelope singularity is a breakdown warning, not a proof that the exact original solution has precisely this blow-up time. The coefficient in the PDF is $ϵc$, correcting the converted TeX's duplicated $ϵ$.

Again put $T=ϵt$. To leading order the oscillator has form $x_0=k-r(T)\cos (t+\varphi (T))$, while the second variable has a slow part $Y(T)$. Averaging its equation over the fast period givesThe oscillatory forcing of the second variable produces only an order-$ϵ$ correction: its leading fast derivative is $-r(T)\cos t$, whose bounded primitive is $-r(T)\sin t$. Thus it creates no secular term.

The oscillator solvability condition is the same as in part (i), with $c$ replaced by $Y(T)$:Introduce accumulated slow timeThe shifted Van der Pol amplitude evolution therefore gives, for $a=k^2-1$,ConsequentlyAs before, uniformity for $t=O({ϵ}^{-1})$ means a fixed bounded slow-time interval on which the envelope stays bounded. For $k^2>4/5$ it fails when $S(T)$ reaches $\log (5-4/k^2)/(k^2-1)$, with limiting value $4$ at $k=1$. There is no bounded leading approximation covering all such slow times for arbitrary positive $k$.

Use the Burgers reduction of weakly nonlinear rightgoing acoustics, with $\theta =t-x$, $X=Mx$, ${\partial }_t={\partial }_{\theta }$ and ${\partial }_x=-{\partial }_{\theta }+M{\partial }_X$. The equation of state expands asAt order one, mass conservation and momentum conservation give $({\rho }_0-u_0{)}_{\theta }=0$ and $(u_0-p_0{)}_{\theta }=0$. For the pure rightgoing acoustic wave with no independent leading mean density or velocity shift, ${\rho }_0=u_0=p_0$. This is the background choice implicit in the requested traveling-wave expansion; additional mean components would modify its phase convention. The source's duplicated phrase about expanding $p$ should refer to expanding $\rho$ as well as $u$.

At order $M$, the mass conservation equation becomeswhereas the momentum conservation equation becomesIn the latter equation, the two nonlinear products ${\rho }_0{u}_{0,\theta }$ and $-u_0{u}_{0,\theta }$ cancel. Substitute the equation of state relation into the former equation and add the two equations to eliminate the first corrections. The result isThe same calculation gives $u_1-p_1=-(\gamma +1)p_0^2/4+C(X)$, so the first corrections may be chosen without an explicit secular factor of $\theta$. This is why the retarded-time description can remain uniform for $|\theta |=O(M^{-1})$, provided the leading profile and its relevant derivatives remain bounded over that range and the slow propagation interval.

This Inviscid Burgers equation can develop a shock, so the source's uniformity assertion needs a pre-shock qualification. If $p_0(\theta ,0)=F(\theta )$ and $a=(\gamma +1)/2$, its method of characteristics givesThe smooth construction ends when this denominator vanishes; if $\max F^{\prime }>0$, the first such distance is $X_s=[a\max F^{\prime }{]}^{-1}$.

For viscosity write $\beta =M\nu$ with bounded positive $\nu$. Since $u_{0,xx}=p_{0,\theta \theta }+O(M)$, the order-$M$ momentum conservation equation instead reads $(u_1-p_1{)}_{\theta }+p_{0,X}=\nu p_{0,\theta \theta }$. Addition now yields the viscous Burgers equationIts positive diffusion resolves the inviscid steepening rather than supporting an unqualified smooth inviscid expansion through shock formation.