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Mathematics and Scientific Representation$

Christopher Pincock

Print publication date: 2012

Print ISBN-13: 9780199757107

Published to Oxford Scholarship Online: May 2012

DOI: 10.1093/acprof:oso/9780199757107.001.0001

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(p.305) Appendix C Speed of Sound

(p.305) Appendix C Speed of Sound

Source:
Mathematics and Scientific Representation
Publisher:
Oxford University Press

(p.305) Appendix C

Speed of Sound

We consider a gas at rest with uniform pressure p and initial density ρ 0 and imagine a plane wave that arises due to the compression of the air.1 The change in density associated with the wave can be given by the condensation σ where

ρ ρ 0 = 1 + σ
(C.1)

If we can calculate the speed at which this disturbance σ moves through the gas, then we will have the speed of sound for that medium. Beginning with equations for the conservation of mass and the balance of momentum, it is possible to derive the plane wave equation

2 σ σ t 2 = c 2 2 σ σ x 2
(C.2)

A crucial step in this derivation is to treat the pressure p as a function of the density ρ and to expand p into the truncated series

p ( ρ ) = p ( ρ 0 ) + c 2 ( ρ ρ 0 )
(C.3)

where

c 2 = ρ p ( ρ 0 )
(C.4)

But it is known that for this plane wave equation, the velocity of the wave is given by c, so the problem reduces to finding ρ p ( ρ 0 ) .

Here, assumptions about the relationship between density, pressure, and heat come into play. In the Newton-Euler derivation, the temperature θ was assumed to be constant, so the equation of state p = Rρθ or p ρ = K 0 for some constant K 0 yielded

(p.306)

c 2 = ρ p ( ρ 0 ) = p ρ
(C.5)

If we allow the temperature to vary but assume that there is no net heat flow into the area in question, then we must adjust the equation of state to p = K 1 ργ where γ is the ratio of specific heat at constant pressure to specific heat at constant volume:

γ = C p C V
(C.6)

Making this change results in a different value for c 2:

c 2 = ρ ( K 1 ρ γ )
(C.7)

= γ K 1 ρ γ 1
(C.8)

= γ K 1 ρ γ ρ
(C.9)

= γ p ρ
(C.10)

The basic idea of this adjustment is that an increase in density will correspond to an even greater increase in pressure because of the associated increase in temperature. As the plane wave equation shows, this brings about a faster propagation of the compression wave.

Notes:

(1.) Lin and Segel (1988, §15.4). Historical issues are pursued in Fox (1971, pp. 161–162), drawing on an 1822 clarification of an 1816 paper by Laplace, and unpublished work by Bryan Roberts.