# 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!"

One of the first theories is due to Newton. He derived a formula for the speed of sound from his wave theory, and compared with experiment. (The experiment was difficult at that time, because of the lack of exact clocks). His theory had a reasonable agreement with the experiment but was not very precise because he did not take thermodynamics into account at all (it did not exist!). With the development of thermodynamics, more and more precise theories were developed. In the last edition of Principia, Newton arrived at the number 1142 feet/sec, which was in good agreement with the latest measurements. The details of the story are described in Westfall, Never at rest, p. 734-735. But one cannot point the exact moment when "accurate" theory was created, while all previous theories were inaccurate or incorrect. The history after (and before!) Newton is described in detail in this article:

http://www3.nd.edu/~powers/ame.20231/finn1964.pdf

The paper is concentrated on Laplace's contribution but many people were involved, and one cannot point the precise moment when the theory "became accurate". But this happened in the early 19th century.

• 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
Commented Jul 15, 2017 at 8:58
• I edited my answer. Newton's theory gives much better than 40% (but worse than 1%). Nevertheless it was in good agreement with the measurements of that time. Commented Jul 15, 2017 at 9:16
• OK I've found an on-line source for that section and am taking a look, thanks! I'm curious what fundamental information was really put into that calculation, so little was known about gases at the time. Was it based on what we would call good science, or more of a lucky guess? I've asked for the calculation not just an undocumented prediction. Now I'm really curious how Newton could have really calculated this. edit: Ah, OK thanks for the edit!
– uhoh
Commented Jul 15, 2017 at 9:24
• Some say that Newton played with numbers to make his theory fit the observation. Of course at that time, a sound theory taking thermodynamics into account could not be created. See the details on the further development in my second reference. Commented Jul 15, 2017 at 9:31
• But this is really great reading - thank you for the link!! I should have said 20% in my first comment; I forgot about the square root. See the continuation of footnote 19 (bottom of page 9) Since CP/CV = 1.42 for air, we can see why Newton's calculation fell 20 per cent short of the experimental value for the speed of sound.
– uhoh
Commented Jul 15, 2017 at 9:40