Lagrangian perturbation of a fluid variable ($\Delta q=\delta q+\xi \cdot \nabla q$)

The Lagrangian perturbation compares a variable on the same moving fluid element before and after a displacement. Taylor expansion of the background value at the displaced position gives the displayed relation to an Eulerian perturbation. For an adiabatic ideal gas, $\Delta (p/{\rho }^{\gamma })=0$ even when the equilibrium entropy varies spatially.