\( m\cfrac { d^{ 2 }x }{ dt^{ 2 } } =-2{ mc^{ 2 } }ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } x))+\psi _{ d }\)
\( \cfrac { d^{ 2 }x }{ dt^{ 2 } } =-2{ c^{ 2 } }ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } x))+\cfrac { \psi _{ d } }{ m } \)
Compared to,
\(m\cfrac{d^2x}{dt^2}=-kx\)
it is not linear. So, consider,
\((2{ mc^{ 2 } }ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } x)))^{'}=c{ \sqrt { { 2m } } }G.tanh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } x)\)
\(\lim\limits_{x\to\,large}{c{ \sqrt { { 2m } } }G.tanh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } x)}=c{ \sqrt { { 2m } } }G\)
Then we have,
\( m\cfrac { d^{ 2 }x }{ dt^{ 2 } } =-c{ \sqrt { { 2m } } }G.x+\psi _{ d }\)
and so, the approximate natural oscillating frequency is,
\(f_{osc}=\sqrt{\cfrac{c{ \sqrt { { 2m } } }G}{m}}=\left(\cfrac{2}{m}\right)^{{1}/{4}}\sqrt{{cG }}\)
where \(m\) is the mass density of the particle and \(G\) is the constant of integration from solving \(F_{\rho}\). \(c\) is the speed of light.
\(\left(f_{osc}\right)^4=\cfrac{2c^2G^2}{m}\)
It is expected that since the lower gradient values are less than or equal to the approximate value, as we let \(x\to\,large\), the actual oscillation frequencies given a large population of particles, spread to higher values of \(f_{osc}\). In other words, blue.
