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 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:
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.