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equally one could use the solution of the Langevin equation: $$m\dot{v} = -\zeta v +\delta F(t)$$ witch is: $$v(t)=v(0)e^{-\zeta t \over m }+ \int \limits_0^t dt'e^{-\zeta (t-t') \over m } \delta F(t) $$ doing the ensamble average gives: $$v(t)=\overbrace{\langle 1 \rangle}^{=1} v(0)e^{-\zeta t \over m }+ \int \limits_0^t dt'e^{-\zeta (t-t') \over m } ...


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There isn't a simple formula for fan noise, but the physics can be worked out from the fundamental equations of fluid mechanics and acoustics. It isn't a simple problem however. The noise created by fans is complex and from several fundamental sources, and its amplitude depends on frequency. Here is an example of a fan noise frequency spectrum for a cooling ...



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