Let's prove that the region where the density and velocity of fluid changes discontinuously exists in the
flow field where supersonic flow exist using CFD technique.

The Definition of Discontinuity Function
at x = a,

where

then the function f(x) is discontinuity function at x = a

Differentiability and continuity
If function f(x) is continous at x = a then the function f(x) is differentiable at x = a
If function f(x) is not continuous at x = a, the the function f(x) is not diffentiable at x = a
where, the shock wave region has discontinuous flow region, so the the Navier-Stokes equation
is not differentiable at discontinuous region in the shock wave region where has dicontinuous flow region.

If we vibrate a steel plate slowly, then the plate make a sound which is due to the vibration of the plate itself,
but the sound is not due to the collision of the separate steel. In other words, the stucture born noise is
generated from the vibration of steel not from the collision of the steels. For example, the sound which is
generated from the blade of the helicopter is generated from the vibraton of the blade not from the collision
of the other component of the helicopter parts.
By the way, the vibration of the blade which is the cause of the structure born nois of the helicopter is generated
from the collision of the blade and fluid. That's why we have to study the shock wave!

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