## FANDOM

167 Pages

Let us consider a fluid stirred so vigorously that the stirring action creates shocks. In order to get the power spectrum of the ensuing turbulence we examine the Fourier transform of the spatial profile of the velocity for a single shock. That spatial profile has the form

$v \left( x \right ) \propto \theta \left(x \right )$

where $\theta \left( x \right)$ is the Heaviside step function. Its Fourier transform is

$\tilde{v} \left( k \right) \propto \int_{-\infty}^{\infty} \theta \left( x \right) \exp \left(i k x \right) dx \propto \frac{1}{k}$

The power spectrum is therefore proportional to

$P \propto \tilde{v}^2 \propto k^{-2}$

The spectrum deviates from this form when the width of the shock becomes comparable to the wavelength. For example, if the fluid is not ideal, but has some viscosity $\nu$, then the upper limit on the wavenumber $k$ is

$k < \frac{}{} v_0/\nu$

where $v_0$ is the velocity of the shock.