This might sound trivial or a simple misunderstanding, but please bear with me as I'm not a Math major.


I want to investigate some aspects of [PCA](https://en.wikipedia.org/wiki/Principal_component_analysis) in homogeneous directions and needed simple analytical functions. General solution to the one dimensional wave equation with periodic boundary condition seems reasonable as it's homogeneous and I can use sine and cosine waves in the general form:


$u(x,t)=\sum\_i{a\_i\sin(f\_i(x-ct))+b\_i\cos(f\_i(x+ct))}$


where $a\_i,b\_i$ determine amplitudes, $f\_i$ represents frequencies and $c$ is the propagation speed.


Now here's my confusion, shouldn't the standard deviation in a homogeneous direction have transitional symmetry? if I use $u(x,t)=sin(3(x-2t))$ this is satisfied, but if I add another term then the std will be a harmonic function of space which is not homogeneous!
I cannot figure out what I'm missing here.



Confusion about homogeneity
So after more readings and talking with my friends in Math department, I was confusing homogeneous equations (zero forcing function) and statistical homogeneity. The general solution is valid for homogeneous wave function, but it is not necessarily statistically homogeneous.


