x

, with convention

v (k_{x}) = \int v (x) e^{- i k_{x} x} d x

v^{''} = - k_{x}^{2} v

and

x^{2} v = - v^{''}

turn the forced equation into

S^{2} k_{y}^{2} \frac{d ^{2} v}{d k _{x}^{2}} + [κ_{r}^{2} + v_{s}^{2} (k_{x}^{2} + k_{y}^{2})] v = i [(2Ω - S) k_{x} ψ - S k_{y}^{2} \frac{d ψ}{d k _{x}}] .

For the unforced equation,

d k_{x} / d t = S k_{y}

implies

\frac{d ^{2}}{d t ^{2}} = S^{2} k_{y}^{2} \frac{d ^{2}}{d k _{x}^{2}} .

It is therefore exactly the shearing-wave oscillator found in part c, now parametrized by radial wavenumber rather than time.

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