= Solution
The <shearing sheet> is a local Cartesian expansion about a reference circular orbit in a differentially rotating disk. One works in a frame rotating at $\Omega$, identifies $x,y,z$ with radial, azimuthal, and vertical directions, retains the linear background shear and radial tidal acceleration, and neglects curvature and background variation across a patch much smaller than the orbital radius. The incompressible version also takes constant reference density and imposes $\nabla\cdot\mathbf u=0$, filtering sound waves.
It represents local three-dimensional vortical motion, epicyclic motion, shear dynamics, <inertial waves>, and, after adding buoyancy, <internal gravity waves> and convection. It cannot represent global geometry or boundaries, order-radius structures, density-changing compressible motion, or acoustic waves.
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