  In the linear theory of hydrodynamic stability up to now there exist examples
of flows for which there is full quantitative distinction, as for cylindrical
Hagen-Poiseuille (HP) flow in a pipe with round section, between theory
conclusions and experimental data on the threshold Reynolds number Reth. In the
present work, we show that to get a conclusion of linear instability of the HP
flow for finite Reynolds numbers Re, it is necessary to abandon the use of
traditional 'normal' form of disturbances which assumes an opportunity of
separation of variables describing disturbances variability depending on radial
and longitudinal (along the pipe axis) coordinates. In the result of the
absence of such variables separation, in the suggested linear theory, it is
proposed to use Bubnov-Galerkin's approximation method modification that gives
an opportunity to account longitudinal variability periods distinctions for
different radial modes defined a priori in the result of standard
Galerkin-Cantorovich's method to the equation of evolution of extremely small
axially symmetric velocity field tangential component disturbances. We found
that when considering even two linearly interacting radial modes for the HP
flow, linear instability is possible only when there exists mentioned above
conditionally periodic longitudinal along the pipe axis disturbance variability
when Re_th(p) very sensitively depends on the ratio p of two longitudinal
periods each of which describes longitudinal variability for its own radial
mode only. Obtained for the HP flow linear instability realization minimal
value Re_th=448 (when p=1.527) quantitatively agrees with the Tolmin-Shlihting
waves in the boundary layer emergence, where also Re_th=420. We get
quantitative agreeing of the phase velocity values of the considered
disturbances with experimental data on the fronts of the turbulent 'puffs'
spreading in the pipe.
