For the <normal mode>
$$
w'=\widehat w(z)e^{ikx-i\omega t},
\qquad
c=\frac\omega k,
$$
the linear material derivative becomes
$$
D_t\longmapsto ik(\overline u-c).
$$
Substitution into the equation from part b and division by $-k^2(\overline u-c)^2$ gives the <Taylor–Goldstein equation>
$$
\boxed{
\widehat w_{zz}+m^2(z)\widehat w=0},
$$
where
$$
\boxed{m^2(z)=l^2(z)-k^2},
$$
and the <Scorer parameter> is
$$
\boxed{
l^2(z)=
\frac{N^2(z)}{[\overline u(z)-c]^2}
-\frac{\overline u_{zz}(z)}{\overline u(z)-c}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 333 / 3 / c / Solution

{c}
{wiki}

The Taylor–Goldstein equation governs linear two-dimensional disturbances of an inviscid, vertically sheared, stably stratified parallel flow. For horizontal <phase velocity> $c$, base velocity $U(z)$, <buoyancy frequency> $N(z)$, and horizontal <wavenumber> $k$,
$$
\widehat w''+\left[
\frac{N^2}{(U-c)^2}
-\frac{U''}{U-c}-k^2
\right]\widehat w=0.
$$

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