in the case of a simple spring-particle system, the force from the retaining spring changes direction and reverses the travel direction of the attached particle. The particle's motion is periodic and oscillatory. In the case of a particle held by a field, the nature of the particle's interaction with the field as a particle or as a wave changes the direction of the force. The field is negative and attracts the particle beyond \(\pi\), as the particle returns below \(\pi\) it is repelled by the field as a wave and is eventually pushed out towards \(\pi\) again.
When the particle is oscillating, energy is conserved,
\(\int_{\pi-A_w}^{\pi}{tanh(x)}dx=\int_{\pi}^{\pi+A_p}{tanh(x)}dx\)
Solving,
\(\int _{ \pi -A_{ w } }^{ \pi }{ tanh(x) } dx=\int _{ \pi }^{ \pi +A_{ p } }{ tanh(x) } dx\)
\( \left[ ln(cosh(x)) \right] _{ \pi -A_{ w } }^{ \pi }=\left[ ln(cosh(x)) \right] _{ \pi }^{ \pi +A_{ p } }\)
\( ln(cosh(\pi ))-ln(cosh(\pi -A))=ln(cosh(\pi +A))-ln(cosh(\pi ))\)
\( 2ln(cosh(\pi ))-ln(cosh(\pi -A_{ w }))=ln(cosh(\pi +A_{ p }))\)
\( 2ln(cosh(\pi ))=ln(cosh(\pi +A_{ p })cosh(\pi -A_{ w }))\)
\( cosh^{ 2 }(\pi )=cosh(\pi +A_{ p })cosh(\pi -A_{ w })\)
\(cosh(\pi -A_{ w })=\cfrac { cosh^{ 2 }(\pi ) }{ cosh(\pi +A_{ p }) } \)
A plot of cosh(pi)*cosh(pi)/cosh(pi+x) and cosh(pi-x) gives,
where valid solutions to \(A_w\) and \(A_p\) share a common value on the y-axis.
It is interesting that a valid solution swings the particle within the boundary of \(-\pi\) and \(\pi\) across the center of \(\psi\).
And the thickness of this shield is \(A_w+A_p\).
The actual expression for \(F_{\rho}\) the post "Not Exponential, But Hyperbolic And Positive Gravity!" dated 22 Nov 2014 is,
\(F_{ \rho }=i\sqrt { 2{ mc^{ 2 } } } \, G.tanh\left( \cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } (x-x_{ z }) \right)\)
\(\cfrac{\partial\,F_{\rho}}{\partial\,x}=G^2sech^2(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } }x)\)
where we let \(x_z=0\), ie a point particle, and ignore \(i\),
at \(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } }x=\pi\) or,
\(x=\cfrac{\pi \sqrt { 2{ mc^{ 2 } } } }{G}\)
we approximate \(f_{res}=\cfrac{1}{2\pi}\sqrt{\cfrac{gradient|_{\pi}}{m}}=\cfrac{1}{2\pi}\sqrt{\cfrac{G^2sech^2(\pi)}{m}}\)
\(f_{res}=\cfrac{sech(\pi)}{2\pi}\cfrac{G}{\sqrt{m}}=0.01373\cfrac{G}{\sqrt{m}}\)
where \(m\) is the mass of the particle.
From the same post "Not Exponential, But Hyperbolic And Positive Gravity!" dated 22 Nov 2014, \(G\) has the same dimension (units) as \(\sqrt{2mc^2}\) per meter, the expression for \(f_{res}\) has a consistent unit of per second. But without an estimate for \(G\) it is useless.
If the radius of an electron is
\(a_e=2.8179403267e-15 m\)
and its mass
\(m=9.10938291e-31 kg\)
and that the \(\psi\) of an electron extend up to \(a_e\) then,
\(a_e=\cfrac{\pi \sqrt { 2{ mc^{ 2 } } } }{G}\)
\(G=\cfrac{\pi \sqrt { 2{ mc^{ 2 } } } }{a_e}\)
\(G=4.511e8\)
So, in the case of an electron,
\(f_{res}=0.01373\cfrac{G}{\sqrt{m}}=0.01373\cfrac{4.511e8}{\sqrt{9.10938291e-31}}\)
\(f_{res}=6.489e21 Hz\)
We can still achieve resonance at integer division of this number although slow, but this shield is just the size of an electron.
In the case of Earth as one big gravity particle,
\(a_E=6371e3m\)
and mass
\(m_E=5.972e24 kg\)
\(G\)=pi*sqrt(2*5.972*(299792458)^2*10^(24))/(6371e3)
\(G=5.109e14\) and
\(f_{res}\)=0.01373*5.109e14/sqrt(5.972e24)
\(f_{res}=2.870\,\,Hz\)
on the surface of earth.
This frequency can be reproduced.
Good luck! And I dream of anime...ZZZ...
Note: Newton Gravitational Constant \(G=6.67408*10^{-11}\),
\(GM_E=6.67408*10^{-11}*5.972*10^{24}\)
\(GM_E=3.9858*10^{14}\)
