The problem is that atmospheric Mach numbers don't follow from first principles, and the Wolfram Alpha values emanate from an empirically derived approximation. An analytical relationship between the speed of sound and air pressure arising from first principles would require a solution to the Navier-Stokes equations, for which solution a million dollar prize lies unclaimed:
Quote: "The Navier–Stokes equations are also of great interest in a purely mathematical sense. Somewhat surprisingly, given their wide range of practical uses, mathematicians have not yet proven that, in three dimensions, solutions always exist (existence), or that if they do exist, then they do not contain any singularity (smoothness). These are called the Navier–Stokes existence and smoothness problems. The Clay Mathematics Institute has called this one of the seven most important open problems in mathematics and has offered a US$1,000,000 prize for a solution or a counter-example."
> There's still a chance that Wolfram Alpha is incorrect ...
At the moment there is no "correct". It's all based on field observations, numerical modeling, and estimation.
http://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations
Quote: "The Navier–Stokes equations are also of great interest in a purely mathematical sense. Somewhat surprisingly, given their wide range of practical uses, mathematicians have not yet proven that, in three dimensions, solutions always exist (existence), or that if they do exist, then they do not contain any singularity (smoothness). These are called the Navier–Stokes existence and smoothness problems. The Clay Mathematics Institute has called this one of the seven most important open problems in mathematics and has offered a US$1,000,000 prize for a solution or a counter-example."
> There's still a chance that Wolfram Alpha is incorrect ...
At the moment there is no "correct". It's all based on field observations, numerical modeling, and estimation.