Prologue: Laplace, Biot, and Poisson

Abstract

Urged by Citizen Laplace, Citizen Biot¹ undertook “to examine the influence that the variations of temperature which accompany the dilatations and condensations of air might have on the speed of sound…. It is a fact known to the physicists that atmospheric air, when it is condensed, loses a part of its latent heat, which goes into the state of sensible heat, and on the contrary when it is rarefied, it takes back a portion of sensible heat, which it converts into latent heat.” The sonorous condensations must therefore be accompanied by changes of temperature. Since both of these are very small, “we shall regard them as proportional….” Thus Biot assumes that²

$$dot{\theta}=\beta \frac{\dot {\rho}}{\rho},$$

(3A.1)

ft being a coefficient to which he attributes no particular functional dependence. By use of (2C.2)_a we conclude from (1) that³

$$[\dot p = \frac{{\partial p}}{{\partial \rho }}\dot \rho + kp\frac{{\dot \rho }}{\rho },$$

(3A.2)

Where

$$[k \equiv \frac{\beta }{p}\frac{{\partial p}}{{\partial \theta }} = \frac{\beta }{\theta },$$

(3A.3)

the latter expression being appropriate to an ideal gas. By (2B.1) we obtain for the speed of sound the relation

$$[c^2 = \frac{{\dot{p}}}{{\dot{\rho}}} = (1 + \frac{{kp}}{{\rho \frac{{\partial p}}{{\partial \rho }}}})\frac{{\partial p}}{{\partial \rho }},$$

(3A.4)

which for an ideal gas reduces to

$$[c^2 = (1 + k)\frac{p}{\rho }.$$

(3A.5)

Dinanzi parea gente;…… a’ due mie’ sensi faceva dir 1’un “No,” l’altro “Sì, canta.”

Dante, Purgatorio X, 58–60.