\(\cfrac { \partial ^{ 2 }\psi }{ \partial \, { t }^{ 2 } } ={ c }^{ 2 }\cfrac { \partial ^{ 2 }\psi }{ \partial \, { x }^{ 2 } } \)
A valid solution is,
\( \psi =\psi _{ o }e^{ i\left( wt+kx \right) }=\psi _{ o }e^{ iwt }e^{ ikx }\)
where a particle at \(x=x_o\) experiences \(e^{ iwt }\). As each particle has an exclusive solution to \(x\) then each particle experiences \(t\) differently.
When we have a wave in 3D time,
\( \cfrac { \partial ^{ 2 }\psi }{ \partial \, { x }^{ 2 } } ={ c }^{ 2 }_{ t }\cfrac { \partial ^{ 2 }\psi }{ \partial \, { t }^{ 2 } } \)
A similar solution is,
\( \psi =\psi _{ o }e^{ i\left( wx+kt \right) }=\psi _{ o }e^{ iwx }e^{ ikt }\)
where a particle in time at \(t=t_o\) experiences \(e^{ iwx }\). As each particle has an exclusive solution to \(t\) then each particle experiences \(x\) differently. In this case, time standstill and space \(x\) passes by.
If
\(\cfrac{\partial\,x}{\partial\,t}=\cfrac{1}{\left(\cfrac{\partial\,t}{\partial\,x}\right)}\)
then,
\(c=\cfrac{1}{c_t}\)
assuming all space and time dimensions are equivalent, then both wave equations are equivalent.
What if not all space and time dimensions are equivalent?
What if?
