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- - e 6 - - in which equation # expresses the variation of temperature corres
ponding to a change of height, so far as it varies independently of change of density. If also p be the pressure where the density is p, and g be the force of gravity, we have d # = – go, (2). Lastly, we have the known relation between the pressure, density, and temperature, given by the equation
in which the temperature 6 is supposed to be reckoned in degrees of the centigrade thermometer, a” is the pressure where p = 1 and 6 = 0, and a is the numerical coefficient 0,00375. With respect to the equation (2) we may remark that though it is in strictness applicable only to the air at rest, it is very nearly true when the atmosphere is in motion; for the direction of winds is necessarily nearly parallel to the Earth's surface, and consequently the effective accelerative force
in the vertical direction is very small. Hence #. is nearly equal to
the impressed accelerative force, that is, to the force of gravity.
Hence by means of (2) we get,
a;---THEand by substituting this value of # in (1), it will be found that d?__&p_.48 (#)-Ho, &
The solution of the problem requires, therefore, the knowledge of
expressions for the partial differential coefficients # and #. There
are at present no means of finding these by a method entirely a priori; and recourse must consequently be had to experiment and observation. To obtain the value of ; we shall refer to the experimental determination of the velocity of sound, beginning, first, with some
Propositions for finding the velocity theoretically.
PRop. I. To find an expression for the velocity with which a given state of density is propagated in any medium.
The motion is supposed to be in parallel lines. Take an axis parallel to the direction of motion, and let v, p, be the velocity and density of a particle in motion at the distance a from a fixed origin, and at the time t. Then we have the equation,
The differential coefficients are partial with respect to time and space. Let now p’ be the density at the same time at the distance a + 3a. Then, d
After the small time 3t let the density at the distance a + 8a become p. Consequently,
Now ; is the rate at which the density at the distance a is transferred to the distance a + 3a, and is equal to the velocity of the particles + the velocity of propagation.
dw Therefore the velocity of propagation = #. pda.
If the given state of density be propagated with the uniform velocity b, it follows that go - b. 2p. da: pda.”
an equation applicable to uniform propagation under whatever circumstances it takes place. By integration, v = b . Nap. log p, assuming that v = 0, when p = 1.
Supposing the medium to be such that p = bop, the propagation is
known to be uniform and equal to b. Therefore for this medium
do — a de
Jr. T b. pda' Hence the relation between the velocity and density is the same in any medium which propagates a given state of density with the uniform velocity b, as in the medium defined by the equa
tion p = bop, provided the motion in both be subject to the condition that w = 0 where p = 1.
By help of what precedes the following Propositions may be solved.
PROP. II. To find the impressed accelerative force which will alter
the rate of uniform propagation in a medium whose density varies as the pressure.
Let the medium be such that p = ap, and let b be the altered rate of propagation. It has been shewn that the motion is the same as in a medium for which p = bop, no impressed force acting. Hence the effective accelerative force is the same. Hence d’a: bodp dź - T -1.
PROP. III. To find the relation between the pressure and the density in a medium which propagates a given state of density uniformly.
It is here supposed that there is no impressed force. The two equations following are therefore to be applied:
Now by supposition a given state of density is propagated with a uniform velocity. Hence if b equal the velocity of propagation, by Proposition I, * - 5.4. da: T pda. '
and introducing the condition that w = 0 wherever p = 1, which can be satisfied when, as we suppose, the propagation is in a single direction only, it follows that p (t) = 0. Hence, differentiating with respect to time only,
* – 5.4p
Substituting these values of #. and # in the equations (a) and (b), we obtain,
Multiplying the first of these by b and subtracting, the result is,
We can now find an expression for the velocity of the propagation of sound in the atmosphere, assuming the velocity to be uniform. Let 9, be the temperature of the air when at rest. Experiments shew that by sudden compression the temperature is increased, and by sudden dilatation diminished. Let 9, + p be the temperature corresponding to