  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.
