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It is widely accepted that supersonic, magnetised turbulence plays afundamental role for star formation in molecular clouds. It produces theinitial dense gas seeds out of which new stars can form. However, the exactrelation between gas compression, turbulent Mach number, and magnetic fieldstrength is still poorly understood. Here, we introduce and test an analyticalprediction for the relation between the density variance and theroot-mean-square Mach number in supersonic, isothermal, magnetised turbulentflows. We approximate the density and velocity structure of the interstellarmedium as a superposition of shock waves. We obtain the density contrastconsidering the momentum equation for a single magnetised shock and extrapolatethis result to the entire cloud. Depending on the field geometry, we then makethree different assumptions based on observational and theoretical constraints:B independent of density, B proportional to the root square of the density andB proportional to the density. We test the analytically derived densityvariance--Mach number relation with numerical simulations, and find that for Bproportional to the root square of the density, the variance in the logarithmicdensity contrast, $\sigma_{\ln\rho/\rho_0}^2=\ln[1+b^2\mathscr{M}^2\beta_0/(\beta_0+1)]$, fits very well tosimulated data with turbulent forcing parameter b=0.4, when the gas issuper-Alfv\'enic. However, this result breaks down when the turbulence becomestrans-Alfv\'enic or sub-Alfv\'enic, because in this regime the turbulencebecomes highly anisotropic. Our density variance--Mach number relationssimplify to the purely hydrodynamic relation as the ratio of thermal tomagnetic pressure $\beta_0$ approaches infinite.

The Density Variance--Mach Number Relation in Supersonic Turbulence: I. Isothermal, magnetised gas