Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 331 / 1 / b / ii

For an odd velocity potential, chooseAt $z=L$ the two kinematic conditions give $A=ic{B}_{+}$ and $C\cosh (kL)=i(V-c)B_{+}$. The dynamic condition is $-cA=(V-c)C\sinh (kL)$. Eliminating $A,C,B_{+}$ gives, with $s=\tanh (kL)$,The lower interface gives the same equation by odd parity of the velocity potential. The interior vertical velocity is even, so $B_{-}=B_{+}$: the displacement is a sinuous mode of a planar jet even though the potential is odd.

For $L>0$, $V\ne 0$ and $k>0$, $0<s<1$ and one root has $c_i>0$. Thus every nonzero wavenumber is unstable in this ideal top-hat planar jet. The phrase “all choices” is understood with these assumptions: $k=0$ has no exponential wave growth, and $V=0$ has no shear.