| 33 | | There seems then to be 2 different ways of computing the power spectrum of the kinetic energy. First, we can take the fourier transform of [[latex($ \sqrt{\rho (v_x + v_y + v_z)^2}$)]] and then square this to get the power spectrum. Or, we can take [[latex($\sqrt{\rho v_x^2}$)]] and then sum up the squares of the various fourier transforms for each dimension. |
| | 33 | There seems then to be 2 different ways of computing the power spectrum of the kinetic energy. First, we can take the fourier transform of [[latex($ \sqrt{\rho (v_x + v_y + v_z)^2}$)]] and then square this to get the power spectrum. Or, we can take [[latex($\sqrt{\rho v_x^2}$)]] and then sum up the squares of the various fourier transforms for each dimension. That is, does |
| | 34 | |
| | 35 | [[latex($(F(\sqrt{v_x^2 + v_y^2 + v_z^2} ~))^2 = (F(v_x))^2 + (F(v_y))^2 + (F(v_z))^2 ~?$)]] |
| | 36 | |
| | 37 | Further, which do we want, |
| | 38 | |
| | 39 | |
| | 40 | [[latex($(F(\sqrt{\rho(v^2)} ~))^2 ~or~ (F(\sqrt{\rho}))^2*(F(v))^2 $)]] |
| | 41 | |
| | 42 | as it seems both give the same units.. |
