My Own Wave Equation

From the post "Not A Wave But Work Done!",

$\ddot { x } \left( 2-i\cfrac { \dot { x }  }{ c }  \right) \cfrac { \partial \, \psi  }{ \partial \, t_{ c } } =ic\left( 1+i\cfrac { \dot { x }  }{ c }  \right) \dot { x } ^{ 2 }\cfrac { \partial ^{ 2 }\psi  }{ \partial \, x^{ 2 } } +2\ddot { x } \cfrac { \partial V\,  }{ \partial \, t_{ c } }$ --- (*)

derived under the assumption that $\dddot{x}=0$, a time invariant field.

Multiply (*) by $i$,

 $i\ddot { x } \left( 2-i\cfrac { \dot { x }  }{ c }  \right) \cfrac { \partial \, \psi  }{ \partial \, t_{ c } } =-c\left( 1+i\cfrac { \dot { x }  }{ c }  \right) \dot { x } ^{ 2 }\cfrac { \partial ^{ 2 }\psi  }{ \partial \, x^{ 2 } } +2i\ddot { x } \cfrac { \partial V\,  }{ \partial \, t_{ c } }$

Multiply by $\left( 1-i\cfrac { \dot { x }  }{ c }  \right)$,

$i\ddot { x } \left( 2-i\cfrac { \dot { x }  }{ c }  \right) \left( 1-i\cfrac { \dot { x }  }{ c }  \right) \cfrac { \partial \, \psi  }{ \partial \, t_{ c } } =-c\left( 1-\cfrac { \dot { x } ^{ 2 } }{ c^{ 2 } }  \right) \dot { x } ^{ 2 }\cfrac { \partial ^{ 2 }\psi  }{ \partial \, x^{ 2 } } +2i\left( 1-i\cfrac { \dot { x }  }{ c }  \right) \ddot { x } \cfrac { \partial V\,  }{ \partial \, t_{ c } }$

$c\left( 1-\cfrac { \dot { x } ^{ 2 } }{ c^{ 2 } }  \right) \dot { x } ^{ 2 }\cfrac { \partial ^{ 2 }\psi  }{ \partial \, x^{ 2 } } =-i\ddot { x } \left( 2-\cfrac { \dot { x } ^{ 2 } }{ c^{ 2 } } -3i\cfrac { \dot { x }  }{ c }  \right) \cfrac { \partial \, \psi  }{ \partial \, t_{ c } } +2\left( \cfrac { \dot { x }  }{ c } +i \right) \ddot { x } \cfrac { \partial V\,  }{ \partial \, t_{ c } }$

$c\left( 1-\cfrac { \dot { x } ^{ 2 } }{ c^{ 2 } }  \right) \dot { x } ^{ 2 }\cfrac { \partial ^{ 2 }\psi  }{ \partial \, x^{ 2 } } =-\ddot { x } \left\{3\cfrac { \dot { x }  }{ c } +i(2-\cfrac { \dot { x } ^{ 2 } }{ c^{ 2 } } ) \right\} \cfrac { \partial \, \psi  }{ \partial \, t_{ c } } +2\left( \cfrac { \dot { x }  }{ c } +i \right) \ddot { x } \cfrac { \partial V\,  }{ \partial \, t_{ c } }$

Equating Real terms,

$c\left( 1-\cfrac { \dot { x } ^{ 2 } }{ c^{ 2 } }  \right) \dot { x } ^{ 2 }\cfrac { \partial ^{ 2 }\psi  }{ \partial \, x^{ 2 } } =-3\cfrac { \dot { x }  }{ c } \ddot { x } \cfrac { \partial \, \psi  }{ \partial \, t_{ c } } +2\cfrac { \dot { x }  }{ c } \ddot { x } \cfrac { \partial V\,  }{ \partial \, t_{ c } }$

$c\left( 1-\cfrac { \dot { x } ^{ 2 } }{ c^{ 2 } }  \right) \dot { x } \cfrac { \partial ^{ 2 }\psi  }{ \partial \, x^{ 2 } } =\cfrac { \ddot { x }  }{ c } \left\{-3\cfrac { \partial \, \psi  }{ \partial \, t_{ c } } +2\cfrac { \partial V\,  }{ \partial \, t_{ c } }  \right\}$

Let's define

$\cfrac{1}{\gamma^2}=\left( 1-\cfrac { \dot { x } ^{ 2 } }{ c^{ 2 } }  \right)$

We have,

$\cfrac{c^2}{\gamma^2} \dot { x } \cfrac { \partial ^{ 2 }\psi  }{ \partial \, x^{ 2 } } ={ \ddot { x }  } \left\{- 3\cfrac { \partial \, \psi  }{ \partial \, t_{ c } } +2\cfrac { \partial V\,  }{ \partial \, t_{ c } }  \right\}$ --- (**)

Equating Imaginary terms,

$-\ddot { x } (2-\cfrac { \dot { x } ^{ 2 } }{ c^{ 2 } } )\cfrac { \partial \, \psi  }{ \partial \, t_{ c } } +2\ddot { x } \cfrac { \partial V\,  }{ \partial \, t_{ c } } =0$

$\ddot{x}\ne 0$

$(2-\cfrac { \dot { x } ^{ 2 } }{ c^{ 2 } } )\cfrac { \partial \, \psi  }{ \partial \, t_{ c } }=2\cfrac { \partial V\,  }{ \partial \, t_{ c } }$

$\cfrac { \partial \, \psi  }{ \partial \, t_{ c } } +\left( 1-\cfrac { \dot { x } ^{ 2 } }{ c^{ 2 } }  \right) \cfrac { \partial \, \psi  }{ \partial \, t_{ c } }=2\cfrac { \partial V\,  }{ \partial \, t_{ c } }$

$\left(1+\cfrac{1}{\gamma^2} \right)\cfrac { \partial \, \psi  }{ \partial \, t_{ c } }=2\cfrac { \partial V\,  }{ \partial \, t_{ c } }$ --- (***)

Substitute the above into (**),

$\cfrac { c^{ 2 } }{ \gamma ^{ 2 } } \dot { x } \cfrac { \partial ^{ 2 }\psi  }{ \partial \, x^{ 2 } } =-{ \ddot { x }  }\left\{ 3\cfrac { \partial \, \psi  }{ \partial \, t_{ c } } -\left( 1+\cfrac { 1 }{ \gamma ^{ 2 } }  \right) \cfrac { \partial \, \psi  }{ \partial \, t_{ c } }  \right\}$

$\cfrac { c^{ 2 } }{ \gamma ^{ 2 } } \dot { x } \cfrac { \partial ^{ 2 }\psi  }{ \partial \, x^{ 2 } } =-{ \ddot { x }  }\left\{ 2-\cfrac { 1 }{ \gamma ^{ 2 } }  \right\} \cfrac { \partial \, \psi  }{ \partial \, t_{ c } }$ --- (1)

${ \ddot { x }  }\cfrac { \partial \, \psi  }{ \partial \, t_{ c } } =-\cfrac { c^{ 2 } }{ \left\{ 2\gamma ^{ 2 }-1 \right\}  } \dot { x } \cfrac { \partial ^{ 2 }\psi  }{ \partial \, x^{ 2 } }$

Equivalently from (1),

${\left( 1+\cfrac { \dot { x } ^{ 2 } }{ c^{ 2 } }  \right) \ddot { x }  } \cfrac { \partial \, \psi  }{ \partial \, t_{ c } } =-c^{ 2 }\left( 1-\cfrac { \dot { x } ^{ 2 } }{ c^{ 2 } }  \right) \dot { x } \cfrac { \partial ^{ 2 }\psi  }{ \partial \, x^{ 2 } }$

This suggests that when $\dot{x}=c$, $\cfrac{1}{\gamma^2}=0$

${ 2\ddot { x }  }\cfrac { \partial \, \psi  }{ \partial \, t_{ c } }=0$

Then either,

 $\ddot{x}=0$ and $\cfrac { \partial \, \psi  }{ \partial \, t_{ c } }= \cfrac { \partial \, V }{ \partial \, t_{ c } }$

or,

$\cfrac { \partial \, \psi  }{ \partial \, t_{ c } }=0$,

that the total energy of the system is a constant in time $t_c$ and is at an extrema.  In this case $\psi$ has a stable point when $\dot{x}=c$.  From (***),

$\cfrac { \partial \psi  }{ \partial \, t_{ c } }=2\cfrac { \partial V\,  }{ \partial \, t_{ c } }=0$

when $\dot{x}=c$, and

$\cfrac { \partial^2 \psi  }{ \partial \, t^2_{ c } }=2\cfrac { \partial^2 V\,  }{ \partial \, t_{ c }^2 }$

$\psi$ is minimum when $V$ is at its local maximum.  This shows that the system, as far as $\psi$, the total energy is concerned, can be stable at light speed, $\dot{x}=c$.  At that point $V$ is at its local maximum.

We now consider the case when $\dot{x}\lt\lt c$,

${ \ddot { x }  } \cfrac { \partial \, \psi  }{ \partial \, t_{ c } } =-c^{ 2 } \dot { x } \cfrac { \partial ^{ 2 }\psi  }{ \partial \, x^{ 2 } }$

${m \ddot { x }  } \cfrac { \partial \, \psi  }{ \partial \, t_{ c } } =-c^{ 2 }m \dot { x } \cfrac { \partial ^{ 2 }\psi  }{ \partial \, x^{ 2 } }$

With $F=m\ddot{x}$  and $p=m\dot{x}$,

$F\cfrac { \partial \, \psi  }{ \partial \, t_{ c } } =-c^{ 2 }p \cfrac { \partial ^{ 2 }\psi  }{ \partial \, x^{ 2 } }$

Since, $-\cfrac { \partial \, \psi  }{ \partial x } =F=m\ddot { x }$

$\cfrac { \partial \, \psi  }{ \partial x } \cfrac { \partial \, \psi  }{ \partial \, t_{ c } } =c^{ 2 }p\cfrac { \partial ^{ 2 }\psi  }{ \partial \, x^{ 2 } }$

with $t_c=\cfrac{1}{\sqrt{2}}t.e^{-i\pi/4}$,

$\cfrac { \partial \, \psi  }{ \partial x } \cfrac { \partial \, \psi  }{ \partial \, t } =\cfrac{c^{ 2 }p}{\sqrt{2}}.\cfrac { \partial ^{ 2 }\psi  }{ \partial \, x^{ 2 } }.e^{-i\pi/4}$

in time $t$.

There should be more implications from this equation to prove its validity or disprove it.