Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 321 / 3 / b / Solution

Let ${\mathbf{d}}_{jn}=h(j-n){\mathbf{e}}_y$ and $\Delta \mathbf{\xi }={\mathbf{\xi }}_j-{\mathbf{\xi }}_n$, with ${\mathbf{\xi }}_n=(x_n^{\prime },y_n^{\prime })$. Linearizing the Newton's law of universal gravitation usesFor an azimuthal separation, the matrix is $\mathrm{diag}(1,-2)$. After canceling the equilibrium forces, the linearized shearing sheet equations areThe transverse gravitational response tends to restore a radial displacement, while the longitudinal response reinforces an azimuthal density concentration. Both sums converge for the bounded Fourier modes considered next.