a wave oscillating in two time dimensions with light speed in space. Obviously,
\(c^2\cfrac{\partial^2\psi}{\partial\,x^2}=\cfrac{\partial^2\psi}{\partial\,t^2}\)
We replace one of the space dimension by a time dimension \(t_g\),
\(c^2\cfrac{\partial^2\psi}{\partial\,x\partial\,t_g}=\cfrac{\partial^2\psi}{\partial\,t^2}\)
This relationship holds because,
\(c^2\cfrac{\partial}{\partial\,x}\left\{\cfrac{\partial\psi}{\partial\,t_g}\right\}=\cfrac{\partial\,t_g}{\partial\,t}.\cfrac{\partial\,x}{\partial\,t}\cfrac{\partial}{\partial\,x}\left\{\cfrac{\partial\psi}{\partial\,t_g}\right\}=\cfrac{\partial}{\partial\,t}\left\{\cfrac{\partial\psi}{\partial\,t}\right\}=\cfrac{\partial^2\psi}{\partial\,t^2}\)
\(\cfrac{\partial\,x}{\partial\,t}=c\)
and assuming that,
\(\cfrac{\partial\,t_g}{\partial\,t}=c\)
Furthermore,
\(t_g=it_c\)
\(\cfrac{\partial\,t_g}{\partial\,t_c}=i\)
and
\(\cfrac{\partial\,x}{\partial\,t_c}=\cfrac{\partial\,x}{\partial\,t_g}=ic\)
where \(i\) is used to express orthogonality explicitly. We replaces \(t\),
\(\cfrac { \partial \, t }{ \partial \, t_{ g } } .\cfrac { \partial \, t_{ g } }{ \partial \, t } .\cfrac { \partial \, x }{ \partial \, t } \cfrac { \partial }{ \partial \, t_{ g } } \left\{ \cfrac { \partial \psi }{ \partial \, x } \right\} =\cfrac { \partial \, t }{ \partial \, t_{ g } } .\cfrac { \partial }{ \partial \, t } \left\{ \cfrac { \partial \psi }{ \partial \, t } \right\} \)
\( \cfrac { \partial \, t }{ \partial \, t_{ g } } .\cfrac { \partial \, x }{ \partial \, t } \cfrac { \partial }{ \partial \, t_{ g } } \left\{ \cfrac { \partial \psi }{ \partial \, x } \right\} =\cfrac { \partial \, t }{ \partial \, t_{ g } } .\cfrac { \partial }{ \partial \, t_{ g } } \left\{ \cfrac { \partial \psi }{ \partial \, t } \right\} \)
\( \cfrac { \partial \, x }{ \partial \, t_{ g } } \cfrac { \partial }{ \partial \, t_{ g } } \left\{ \cfrac { \partial \psi }{ \partial \, x } \right\} =\cfrac { \partial ^{ 2 }\psi }{ \partial \, t_{ g }^{ 2 } } \)
\( ic\cfrac { \partial ^{ 2 }\psi }{ \partial \, x\partial \, t_{ g } } =\cfrac { \partial ^{ 2 }\psi }{ \partial \, t_{ g }^{ 2 } } \)
which is the same wave equation as when \(\psi\) is travelling down \(t_g\) at light speed and oscillating on \(x\). This wave however, has light speed in space, the other can be stationary. In both cases, \(\psi\) wraps around the particle as a stationary wave, in a sphere; refer to post "Standing Waves, Particles, Time Invariant Fields" dated 20 Nov 14.
This wave although in space at light speed has energy oscillating in two time dimensions. These energies is coupled to the space dimension only via other particles that is oscillating in one of the respective time dimensions and the other in space, or by collisions. During collisions, energy oscillating in the two time dimensions is transferred/shared by the colliding particles.
This is highly speculative, but none the less, specific properties/behaviors of these waves, in general, depend on which dimensions the energy oscillates between and which dimension the wave is at light speed. These properties are not captured by the wave equations.
If we do not assume
\(\cfrac{\partial\,t_g}{\partial\,t}=c\)
but let
\(\cfrac{\partial\,t_g}{\partial\,t}=u\)
then the final expression for the wave is simply,
\( i\cfrac{c^2}{u}\cfrac { \partial ^{ 2 }\psi }{ \partial \, x\partial \, t_{ g } }=\cfrac { \partial ^{ 2 }\psi }{ \partial \, t_{ g }^{ 2 } } \)
where \(c\) is replaced by the factor \(\cfrac{c^2}{u}\).
It seems that swapping the oscillation dimension with the light speed dimension of the wave does not change the form of the wave equation. Both types of wave with its distinct characteristics share the same wave equation.
