# The shearing sheet and swing amplification revisited

## TL;DR

This paper revisits the classic shearing sheet and swing amplification analysis, providing clearer derivations and new insights into stellar disc dynamics, including wavepacket behavior and limitations of existing dispersion relations.

## Contribution

It offers a more accessible derivation of key results, explores the group velocity and wavepacket structure, and highlights limitations of the Lin-Shu-Kalnajs dispersion relation in disc dynamics.

## Key findings

- Two wavepackets emerge inside corotation, not one.
- Disturbances persist longer in velocity space than in real space.
- Limitations of the Lin-Shu-Kalnajs dispersion relation are demonstrated.

## Abstract

The principal results of the classic analysis of the shearing sheet and swing amplification by Julian & Toomre (1966) are re-derived in a more accessible way and then used to gain a better quantitative understanding of the dynamics of stellar discs. The axisymmetric limit of the shearing sheet is derived and used to re-derive Kalnajs' 1965 dispersion relation and Toomre's 1964 stability criterion for axisymmetric disturbances. Using the shearing sheet to revisit Toomre's important 1969 paper on the group velocity implied by Lin-Shu-Kalnajs dispersion relation, we discover that two rather than one wavepackets emerges inside corotation: one each side of the inner Lindblad resonance. Although LSK dispersion relation provides useful interpretations of both wavepackets, the shearing sheet highlights the limitations of the LSK approach to disc dynamics. Disturbances by no means avoid an annulus around corotation, as the LSK dispersion relation implies. While disturbances of the shearing sheet have a limited life in real space, they live on much longer in velocity space, which Gaia allows us to probe extensively. C++ code is provided to facilitate applications of winding spiral waves.