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We present some new discoveries on the mathematical foundation of linearhydrodynamic stability theory. The new discoveries are: 1. Linearized Eulerequations fail to provide a linear approximation on inviscid hydrodynamicstability. 2. Eigenvalue instability predicted by high Reynolds numberlinearized Navier-Stokes equations cannot capture the dominant instability ofsuper fast growth. 3. As equations for directional differentials, Rayleighequation and Orr-Sommerfeld equation cannot capture the nature of the fulldifferentials.

Novel discoveries on the mathematical foundation of linear hydrodynamic stability theory