Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 331 / 2

The linearized vorticity equation for a inviscid parallel shear flow isSubstituting $\psi =\varphi (y)e^{ik(x-ct)}$ gives Rayleigh's equation

No normal flow at a rigid wall gives $\boxed{\varphi (\pm 1)=0}$. At a free surface $y=1+\eta$, the linearized kinematic boundary condition isso $(U-c)\eta =-\varphi$. The linearized tangential momentum equation gives the pressure amplitude $p=(U-c){\varphi }^{\prime }-U^{\prime }\varphi$. Constant surface pressure therefore requires

For $U=y$, Rayleigh's equation reduces to ${\varphi }^{\prime \prime }-k^2\varphi =0$, so $\varphi =A\cosh ky+B\sinh ky$. Applying the free-surface condition at $y=\pm 1$ and setting the determinant to zero givesFor $k>0$, $k-\tanh k>0$, so instability occurs exactly when $c^2<0$, orThus $0<k<k_c$, where the unique positive cutoff satisfies $\boxed{k_c\tanh k_c=1}$.

Linearization about zero and expansion in the Fourier sine series give, for the $n$th mode,Every mode is oscillatory exactly when the smallest coefficient, at $n=1$, is nonnegative. Since $f^{\prime }(0)<0$,

Write $T=\epsilon t$ and expandAt order ${\epsilon }^2$, the assumptions $f^{\prime \prime }(0)=0$ and the orthogonality normalization leave a homogeneous equation for $u_2$ with no forcing, hence $\boxed{u_2=0}$.

At order ${\epsilon }^3$, project the equation onto the null mode $\sin z$. Sincethe Fredholm solvability condition obtained directly from the definitions printed in the question isThe paper asks for $A/f^{\prime }(0{)}^2$ in place of $f^{\prime }(0{)}^{2}A$, but that coefficient is incompatible with $R_c=-1/f^{\prime }(0)$ and ${\epsilon }^2=R-R_c$: differentiating $1/R$ at $R_c$ gives $-1/R_c^2=-f^{\prime }(0{)}^2$. Thus the displayed target appears to contain a reciprocal typo. It would agree with the expansion only under a correspondingly rescaled definition of the small parameter.

The imposed orthogonality of every $u_j$, $j\ge 2$, removes the freedom to transfer a multiple of $\sin z$ between $A$ and the higher-order terms.