long+short wavelength velocity perturbations

We saw before that if the initial conditions are cold, the $r^{-1/2}$ scaling behavior is observed, for a number of different initial velocity structures. We also saw that if there is jitter in the initial positions and/or velocities, this scaling behavior is lost. In this study, I have added more structure to the initial velocities of the particles, which now have both a long and short wavelength mode. The positions and long wavelength portion of the velocities are

shot.X(n) = n*d-Pih;
shot.V(n) =-V*sin(shot.X(n));

Here pih is $\pi/2$. To this I have added a term with one tenth the amplitude, but higher frequency, i.e. multiple wavelengths in the initial velocity structure of the particles. For exmple:

shot.V(n)+=-0.1*V*sin(1000*shot.X(n));

I have studied $2\lambda$, $10\lambda$, $100\lambda$ and $1000\lambda$. Below are the resulting plots.



As the wavelength of the hight frequency mode approaches the interparticle spacing, we see that the $r^{-1/2}$ scaling is lost. This is consisent with what we see with the warm simulations, where the jitter could be interpreted as lots of high frequency components to the initial velocity structure. The plots above are the average of 6 runs, each of roughly 10k particles. Just to make sure, I also ran 1k particles, with the $100\lambda$ hight frequency velocity component. I suspected that this should look qualitatively similar to the $1000\lambda$ case with 10k particles. Here it is:


It's kindof hard to tell because fewer particles means a smaller dynamic range we have to look at, but qualitatively it seems to be departing from power law more than the 10k particle case.