Page:EB1911 - Volume 25.djvu/458

We may find here the value of this when we have a train of waves in which the displacement is represented by a sine curve of amplitude a, viz. y＝a sin 2π/λ(x − Ut). For a discussion of this type of wave, see below.

and

$\int _{0}^{\lambda }$ ρu²dx＝ρ4π²U²a²/λ² $\int _{0}^{\lambda }$cos²2π/λ(x−Ut)dx

＝2ρπ2U2a2/λ

(12)

The energy per cubic centimetre on the average is

2ρπ2U2a2/λ2

(13)

and the energy passing per second through 1 sq. cm. perpendicular to the line of propagation is

2ρπ2U3a2/λ2

(14)

The Pressure of Sound Waves.—Sound waves, like light waves, exercise a small pressure against any surface upon which they impinge. The existence of this pressure has been demonstrated experimentally by W. Altberg (Ann. der Physik, 1903, 11, p. 405). A small circular disk at one end of a torsion arm formed part of a solid wall, but was free to move through a hole inthewall slightly larger than the disk. When intense sound waves impinged on the wall, the disk moved back through the hole, and by an amount showing a pressure of the order given by the following investigation:—

Suppose that a train of waves is incident normally on the surface S (fig. 11), and that they are absorbed there without reflection.
Fig. 11. Let ABCD be a column of air 1 sq. cm. cross-section. The pressure on CD is equal to the momentum which it receives per second. On the whole the air within ABCD neither gains nor loses momentum, so that on the whole it receives as much through AB as it gives up to CD. Fig. 11. If P is the undisturbed pressure and P+S the pressure at AB, the momentum entering through AB per second is $\int _{0}^{1}$(P+ῶ+ρu² )dt. But $\int _{0}^{1}$Pdt = P is the normal pressure, and as we only wish to find the excess we may leave this out of account.

The excess pressure on CD is therefore $\int _{0}^{1}$(ῶ+ρu²)dt. But the values of ῶ+ρu² which occur successively during the second at AB exist simultaneously at the beginning of the second over the distance U behind AB. Or if the conditions along this distance U could be maintained constant, and we could travel back along it uniformly in one second, we should meet all the conditions actually arriving at AB and at the same intervals. If then dξ is an element of the path, putting dt＝dξ/U, we have the average excess of pressure

p = $\int _{0}^{1}$(ῶ+....

Here dξ is an actual length in the disturbance. We have fi and u expressed in terms of the original length dx and the displacement dy so that we must put dξ＝dx+dy＝(1 + dy/dx)dx, and

p＝. . .

We have already found that if V changes to V−v

ῶ＝....

since r/V＝− dy/dx.

We also have ρu² = ρ₀u²/(1 + dy/dx). Substituting these values and neglecting powers of dy/dx above the second we get

But $\int _{0}^{\text{U}}{\frac {dy}{dx}}dx=0$ since the sum of the displacements＝0. Then putting (dy/dx)² = (u/U)² , we have

＝1/2(γ + 1) average energy per cubic centimetre,

(15)

a result first published by Lord Rayleigh (Phil. Mag., 1905, 10, p. 364).

If the train of waves is reflected, the value of p at AB will be the sum of the values for the two trains, and will, on the average, be doubled. The pressure on CD will therefore be doubled. But the energy will also be doubled, so that (15) still gives the average excess of pressure.

Experimental Determinations of the Velocity of Sound.

An obvious method of determining the velocity of sound in air consists in starting some sound, say by firing a gun, and stationing an observer at some measured distance from the gun. The observer measures by a clock or chronometer the time elapsing between the receipt of the flash, which passes practically instantaneously, and the receipt of the report. The distance divided by the time gives the velocity of the sound. The velocity thus obtained will be affected by the wind. For instance, William Derham (Phil. Trans., 1708) made a series of observations, noting the time taken by the report of a cannon fired on Blackheath to travel across the Thames to Upminster Church in Essex, 125 m. away. He found that the time varied between 55J seconds when the wind was blowing most strongly with the sound, to 6$ seconds when it was most strongly against the sound. The value for still air he estimated at 1142 ft. per second. He made no correction for temperature or humidity. But when the wind is steady its effect may be eliminated by " reciprocal " observations, that is, by observations of the time of passage of sound in each direction over the measured distance.

Let D be the distance, U the velocity of sound in still air, and w the velocity of the wind, supposed for simplicity to blow directly from one station to the other. Let T and T 2 be the observed times of passage in the two directions. We have U+w = D/T₁ and U−w＝D/T₂. Adding and dividing by 2

U＝D/2(1/T₁+1/T₂).

If T₁ and T₂ are nearly equal, and if T = 1/2(T₁+T₂), this is very nearly U = D/T.

The reciprocal method was adopted in 1738 by, a commission of the French Academy (Mémoires de l'Académie des sciences, (1738). Cannons were fired at half-hour intervals, alternately at Montmartre and Montlhery, 17 or 18 m. apart. There were also two intermediate stations] at which observations were made. The times were measured by pendulum clocks. The result obtained at a temperature about 6° C. was, when converted to metres, U＝337 metres/second.

The theoretical investigation given above shows that if U is the velocity in air at t° C. then the velocity U₀ at 0° C. in the same air is independent of the barometric pressure and that U₀＝U/(1+0·00184t), whence U₀ = 332 met. /sec.

In 1822 a commission of the Bureau des Longitudes made a series of experiments between Montlhery and Villejuif, 11 m. apart. Cannons were fired at the two stations at intervals of five minutes. Chronometers were used for timing, and the result at 15·9° C. was U = 340·9 met. /sec, whence U₀=330·6 met. /sec. (F. J. D. Arago, Connaissance des temps, 1825).

When the measurement of a time interval depends on an observer, his “personal equation” comes in to affect the estimation of the quantity. This is the interval between the ariival of an event and his perception that it has arrived, or it may be the interval between arrival and his record of the arrival. This personal equation is different for different observers. It may differ even by a considerable fraction of a second. It is different, too, for different senses with the same observer, and different even for the same sense when the external stimuli differ in intensity. When the interval between a flash and a report is measured, the personal equations for the two arrivals are, in all probability, different, that for the flash being most likely less than that for the sound. In a long series of experiments carried out by V. Regnault in the years 1862 to 1866 on the velocity of sound in open air, in air in pipes and in various other gases in pipes, he sought to eliminate personal equation by dispensing with the human element in the observations, using electric receivers as observers. A short account of these experiments is given in Phil. Mag., 1868, 35, p. 161, and the full account, which serves as an excellent example of the extra- ordinary care and ingenuity of Regnault's work, is given in the Mémoires de l'académie des sciences, 1868, xxxvii. On page 459 of the Mémoire will be found a list of previous careful experiments on the velocity of sound.

In the open-air experiments the receiver consisted of a large