It is widely accepted that supersonic, magnetised turbulence plays a fundamental role for star formation in molecular clouds. It produces the initial dense gas seeds out of which new stars can form. However, the exact relation between gas compression
, turbulent Mach number, and magnetic field strength is still poorly understood. Here, we introduce and test an analytical prediction for the relation between the density variance and the root-mean-square Mach number in supersonic, isothermal, magnetised turbulent flows. We approximate the density and velocity structure of the interstellar medium as a superposition of shock waves. We obtain the density contrast considering the momentum equation for a single magnetised shock and extrapolate this result to the entire cloud. Depending on the field geometry, we then make three different assumptions based on observational and theoretical constraints: B independent of density, B proportional to the root square of the density and B proportional to the density. We test the analytically derived density variance--Mach number relation with numerical simulations, and find that for B proportional to the root square of the density, the variance in the logarithmic density contrast, $sigma_{ln rho/rho_0}^2=ln[1+b^2mathscr{M}^2beta_0/(beta_0+1)]$, fits very well to simulated data with turbulent forcing parameter b=0.4, when the gas is super-Alfvenic. However, this result breaks down when the turbulence becomes trans-Alfvenic or sub-Alfvenic, because in this regime the turbulence becomes highly anisotropic. Our density variance--Mach number relations simplify to the purely hydrodynamic relation as the ratio of thermal to magnetic pressure $beta_0$ approaches infinite.
