Page:EB1911 - Volume 25.djvu/457

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VELOCITY]
441
SOUND


approximately. Newton found 979 tt./sec. But, as we shall see, all the determinations give a value of U<> in the neighbourhood of 33,000 cm./sec., or about 1080 ft./sec. This discrepancy was not ex- plained till 18 16, when Laplace {Ann. dechimie, 1 816, vol. iii.) pointed out that the compressions and extensions in sound waves in air alternate so rapidly that there is no time for the temperature inequalities produced by them to spread. That is to say, instead of using Boyle's law, which supposes that the pressure changes so exceedingly slowly that conduction keeps the temperature constant, we must use the adiabatic relation p = kpy, whence

dp/dp = yhpT 1 = yp/p, and U = V (yp/p) [Laplace's formula]. (8)

If we take 7 = 1-4 we obtain approximately for the velocity in dry air at 0° C .

Uo = 33.i5° cm./sec, which is closely in accordance with observation. Indeed Sir G. G. Stokes {Math, and Phys. Papers, iii. 142) showed that a very small departure from the adiabatic condition would lead to a stifling of the sound quite out of accord with observation.

If we put p = kp{i+at) in (8) we get the velocity in a gas at t°C,

U« = V{7*(l+ai)}. At o° C. we have U = V {yk), and hence

U« = U„V(i+aO

= Uo(i +0-00184/) (for small values of t). (9) The velocity then should be independent of the barometric pressure, a result confirmed by observation.

For two different gases with the same value of y, but with densi- ties at the same pressure and temperature respectively pi and p2, we should have

Ui/U 2 = V(Wpi), (10)

another result confirmed by observation.

Alteration of Form of the Waves when Pressure Changes are Con- siderable. — When the value of dy/dx is not very small E is no longer constant, but is rather greater in compression and rather less in extension than 7 P. This can be seen by considering that the relation between p and p is given by a curve and not by a straight line. The consequence is that the compression travels rather faster, and the extension rather slower, than at the speed found above.

We may get some idea of the effect by supposing that for a short

time the change in form is negligible. In the momentum equation

(4) we may now omit X and it becomes

w+p(U-«) 2 = poU 2 .

Let us seek a more exact value for 01. If when P changes to P+w

volume V changes to V-v then (P+w)(V-»)? = PVv,

. t>( v , 7(7 + 1) v 2 \ Pd / 7 + 1 v \ whence M = P^+ ^J =y ^ \ l +^~ y) â–

We have U — « = U(i Also since p(V— v) VpoU 2 (i-z./V).

Substituting in the momentum equation, we obtain

7P/ T 1 7 + 1 «\

If U = V(7P/po) is the velocity for small disturbances put Uo for U in the small term on the right, and we have


2 V;

-«/L t )=U(i-p/V), since «/U = -dy/dx = vfV.

p V, or p = p /(l-p/V), then p(U-w) 2

whence


U 2 =


we may


"M^ti


or U=Uo+i(7+i)K- (11)

This investigation is obviously not exact, for it assumes that the form is unchanged, i.e. that the momentum issuing from A (fig. 9) is equal to that entering at B, an assumption no longer tenable when the form changes. But for very small times the assumption may perhaps be made, and the result at least shows the way in which the velocity is affected by the addition of a small term depending on and changing sign with u. It implies that the different parts of a wave move on at different rates, so that its form must change. As we obtained the result on the supposition of unchanged form, we can of course only apply it for such short lengths and such short times that the part dealt with does not appreciably alter. We see at once that, where u — o, the velocity has its " normal " value, while where u is positive the velocity is in excess, and where u is negative the velocity is in defect of the normal value. If, then, a (fig. 10) represents the displacement curve of a train of waves, b will represent the pressure excess and particle velocity, and from (11) we see that while the nodal conditions of 6, with =Em/U from equation (3) Then $ffl»/V = iEM 2 /U 2 = ipw 2 from equation (6) Then in the whole wave the potential energy equals the kinetic Energy and the total energy in a complete wave in a column I sq. cm.

cross-section is W

= I puHx.