When was the speed of sound first “correctly” calculated?

I am not a student of thermodynamics, but I will reproduce some equations and discussion from Wikipedia to highlight the principles involved in obtaining a fairly accurate — one percent perhaps — value for the speed of sound in air.

From the article Speed of sound:

$$c=\sqrt{\frac{K_s}{\rho}}$$

where $K_s$ is "...a coefficient of stiffness, the isentropic bulk modulus (or the modulus of bulk elasticity for gases)" and the isentropic bulk modulus $K_s=\gamma p$ where $p$ is the pressure, and the heat capacity ratio $\gamma$ for a diatomic gas is equal to $1+2/5$ or 1.4.

The article continues: "For general equations of state, if classical mechanics is used, the speed of sound c is given by":

$$c=\sqrt{\left( \frac{dp}{d\rho} \right)_s}$$

where again $p$ is the pressure, $\rho$ the density, and the derivative is taken isentropically, that is, at constant entropy $s$.

I'm wondering if there is any well recognized first accurate calculation of the speed of sound from modern principles? I realize the history of science is a continuum and it's likely concepts were added in stages, but it's possible there was a moment when a theory was first developed such that it produced a nearly-correct speed of sound, and there was an "aha!"

• Have you checked out the first sentence? "... obtaining a fairly accurate — one percent perhaps — value for the speed of sound in air." I have a hunch that there was a strong transition. Without the understanding of molecular degrees of freedom, the missing $\gamma=7/5$ would have resulted in a 40% error, right? It might just be in this case that one can point to a particular time. I didn't choose this problem at random. – uhoh Jul 15 '17 at 8:58