\(f \overset{Fourier}{\longleftrightarrow} t\)
In a similar way, energy oscillations in the time dimension stretches out in space dimension as force density from negative infinity to positive infinity. Energy density oscillations in time and force density in space are orthogonal conjugate pair, also using Fourier transform and, assuming that the force is moving at light speed, \(c\). With light speed, time, \(t\) is replace by space, \(x\).
Because,
\(E=F.x=F.ct\)
where \(E\) is energy density, \(F\) is force density, and \(c\) is light speed, a constant. Oscillations in \(E\) is then \(E\) per unit time,
\(\cfrac{E}{t}=F.c\)
\(\cfrac{E}{t}=F.c\)
energy density oscillations in the frequency domain is transformed to force density multiplied by light speed in the time domain. That is
but space \(x\),
\(x=ct\)
So,
\(E\,\, in\,\, f \overset{Fourier}{\longleftrightarrow} F.c^2\,\,in\,\,x\)
and \(F\), force density stretches from negative infinity to positive infinity in space, \(x\). The \(c^2\) constant reminds us of,
\(\cfrac{\partial^2\psi}{\partial\,t^2}=c^2\cfrac{\partial^2\psi}{\partial\,x^2}\)
and for a wave that wraps around \(x\),
\(\cfrac{\partial^2\psi}{\partial\,t^2}=(ic)^2\cfrac{\partial^2\psi}{\partial\,x^2}\)
\(\cfrac{\partial^2\psi}{\partial\,t^2}=-c^2\cfrac{\partial^2\psi}{\partial\,x^2}\)
OK? Fourier transform comes about naturally as we consider oscillation frequency. Time stretches out into space when we assume light speed. Indeed we have previously,
\(E=\psi=-\cfrac{\partial\psi}{\partial x}=F\)when both Fourier and the wave equation are to apply simultaneously. The negative sign is the result of the wave wrapping around \(x\), where \(c\to ic\).
What happened to \(c^2\)? But first, what is Fourier transform? Until next time...
