So many things relating to this subject have been already treated at considerable length under the article Acoustics, that what we now propose shall be chiefly confined to the propagation of sound in the atmosphere, which, as Laplace observes, affords the most important application which has yet been made of the theory of elastic fluids. In this we shall first give the results of the principal attempts which have been made with the view of determining the velocity of sound experimentally; and shall notice more particularly some of them which are of a later date, and were made with far greater precautions than any of those mentioned under the article just cited. We shall then advert to various important oversights and imperfections which still attach to this department of science, and to certain improvements of which it seems to be susceptible.

Results of Various Experiments on the Velocity of Sound.

| Observers | Date | Country | Distance in Feet | Velocity | Temperature | |--------------------|--------|-------------|------------------|----------|-------------| | Florentine Academicians | 1699 | Italy | 5,906 | 1148 | | | Cassini, Huygens, &c.| | France | 9,239 | 1151 | | | Flamsteed and Halley| | England | 13,840 | 1142 | | | Derham | 1704 | England | 5,280 to 63,369 | 1142 | | | French Academicians | 1738 | France | 13,744 to 102,624| 1106 | 43° F. | | Cassini and Lacaille| 1739 | France | 144,114 | 1110 | | | Bianconi | 1740 | Italy | 76,740 | 1043 | | | La Condamine | 1740 | Quito | 87,400 | 1112 | | | La Condamine | 1744 | Cayenne | 128,560 | 1175 | | | T. F. Mayer | 1778 | Germany | 3,412 | 1105 | | | G. E. Müller | 1787 | Germany | 8,530 | 1102 | | | Espinoza and Bauza | 1824 | Chili | 53,626 to 14,071 | 1222-3 | 74°+7 | | Benzenberg | 1829 | Germany | 29,764 | 1093 | 32 | | Goldingham | 1821 | India | 13,932 to 29,547 | 1086-7 | 32 | | Myrbeck | 1822 | Germany | 32,615 | 1092-1 | 32 | | Arago, Matthieu, &c.| 1823 | France | 61,064 | 1096-1 | 32 | | Mull, Van Beek, &c. | 1823 | Holland | 87,971-2 | 1089-5 | 32 | | Gregory | 1823 | England | 2,700 to 13,469 | 1098-1 | 32 | | Parry and Foster | 1825 | Polar Regions| 12,892-9 | 1035-2 | 177 |

The reduction of such results to 32° Fahrenheit is usually made conformably to the expansion of air given by Dalton, Gay Lussac, &c., which is nearly at the rate of 1'4 foot for each degree. The expansion of Rudberg, about to be noticed, being rather smaller, would slightly lessen the reduction. From a close examination of the twenty-eight principal experiments made in Holland in 1823, Professor Miller has, in the Philosophical Magazine for July 1839, deduced the following as the correct means of the results. The mean interval of the time in which sound travelled 17,669-28 metres is 51-9873 seconds. The mean temperature during these experiments was 11°-01 cent. The mean pressures of the atmosphere, and of the vapour in it, were 74618 and 00889 metres at 0° cent. respectively. Now, from Rudberg's experiments (Poggendorff's Annalen, xlii. 558 and xliiv. 119), which perhaps require to be repeated by others, it appears that on heating dry air from 0° cent. to 100° under a constant pressure, its volume was increased from 1 to 1-365; hence, at the temperature 4 cent., the pressure of the atmosphere and of the vapour in it being p and p' respectively, the velocity of sound in English feet per second will be:

\[ \frac{109077}{\sqrt{\left(1 + \frac{0.03654}{1 - 0.375p}\right)}}. \]

This, owing to its depending on a smaller rate of expansion, exceeds 1089-5, the number in the table; but it is chiefly to be regarded as a general expression for the experimental results. The theoretical formula will be found further on.

With the exception of Benzenberg, who had long ago used the species of clock described below with a conical pendulum, the most of the experimenters were formerly very ill provided with any adequate means of accurately measuring the intervals of time which elapse between the flash and report of the guns, as observed at the opposite station. In 1822, the French academicians employed the stop-watch of Breguet and the chronograph of Ricusse, a species of watch, one of whose hands revolves in one second, and can, without needing to stop, be made to touch with its extremity the dial-plate at any instant, so as to leave there a dot of ink, merely by suddenly pressing a small lever. In the experiments made in Holland in 1823, a clock was used with a conical pendulum which revolved 69-833 times in a minute, and was supposed capable of determining intervals to the 100th of a second, by suddenly arresting the index without stopping the clock. By means of these machines it was supposed quite practicable to determine the interval between the sight of the flash and the arrival of the report of a gun, with such precision as to obviate all material error which might arise from this cause. The importance of this, if realized, is evident from the circumstance that each tenth of a second corresponds to 110 feet of distance.

But there is too good reason to suspect that all such ideas of attaining great accuracy in any direct measurements of the intervals of time are perfectly illusory, and that observations made with these machines can scarcely be depended on within a tenth of a second. The machines themselves may be possessed of all the perfection ascribed to them; but it would seem that the man has yet to be made who would be competent to use them. In Holland, only one centrifugal clock being employed at each station, we have no means of judging of the correctness of the observations made with it. But whoever examines the tables of results given in the Connaissance des Temps for 1825 (pages 364-367) will find that three and as often four machines, with as many first-rate observers, were generally employed simultaneously in determining the interval for each shot; and that instead of their results agreeing within an inconsiderable fraction, as the above ideas of great correctness would imply, they often differ by three, sometimes by four, and in one case by five tenths of a second. Something farther illustrative of this will be found under the article CLOCK AND WATCH WORK (vol. vi. p. 783), where a machine has been described for determining, by indirect means, but to almost any degree of exactness, the interval in which sound passes over a small distance. The principal part is a strong clock, without compensation or other refinement, and requiring no extraordinary sort of being to observe with it.

Although many eminent philosophers have laboured much to improve and perfect the various reductions which of it is considered necessary to make on the experimental results, there is still a mistake regarding the exact influence of the wind. It seems always to be supposed that the effects of a steady wind are entirely obviated or compensated by taking the mean of the velocities of such sounds as are simultaneously produced at the two opposite stations, and also reciprocally observed at these respectively. This is in effect the view of the matter taken by Professor Moll, (Philosophical Transactions for 1824, p. 426.) Many other philosophers express the same opinion, and in particular Sir John Herschel, who recommends that all experiments on the velocity of sound be made, if possible, either in calm weather or in a direction at right angles to that of the wind. But so far is the result under this last arrangement from being entirely unaffected by the wind, that the error to which we now allude is then a maximum; and indeed, in every case in which the direction of the wind is inclined to the base line, there is still a small error, which cannot be obviated by the reciprocal method.

The earlier experimenters gave themselves no concern about the influence of the wind; but its effect was in some degree eliminated by the arrangements adopted by the French academicians in 1738, though in so imperfect a manner as to leave considerable doubt regarding the accuracy of their result. In order to clear up this point, the Marquis Laplace requested the board of longitude to repeat the experiments, with the precaution of exciting sounds more nearly at the same instant from both ends of the base, in the expectation that the effects of the wind would be thereby completely obviated. The experiments were accordingly repeated in June 1823, though with no greater precision in this respect than that the times of firing the opposite shots still differed about five minutes from being simultaneous. In the experiments made in Holland in 1823, matters were so much better arranged, that the difference in the times scarcely ever amounted to a second. However, M. Arago has remarked, that when the wind is very unsteady, or comes in sudden gusts, it may still affect the mean result, especially since a sudden gust may interfere with the one sound, and yet miss the other altogether. But even when the wind is perfectly steady, the reciprocal method cannot be quite correct, unless the wind also blows directly from the one station to the other. In every other case it is more or less inaccurate, owing to the circumstance that sound is not propagated in parallel lines, but issues from its source in lines which radiate or diverge to every side; so that the same sonorous ray which, during a calm, would pass directly from the one station to the other, will not reach the latter at all when deflected by the wind into a different direction. Had our limits permitted, we should have endeavoured to show that, by applying to this the well-known principles of the composition of forces, the mean of the velocities of the reciprocal sounds will, owing to the action of the wind, come short of the true velocity, by a quantity which will in all cases be nearly expressed by

\[ \frac{W^2}{2S} \sin^2 A; \] where W is the velocity of the wind, S that of sound in still air, and A the angle which the direction of the wind makes with the base line. This correction, though generally very small, may often be several times greater than some which are usually taken into account, especially that arising from the effect of the difference of latitude and of the height of the place above the sea, on the force of gravity. For such we beg to refer to the Philosophical Trans- actions for 1824, 1828, and 1830. The pendulum, however, seems a more certain means of determining the differences in the force of gravity, than any merely theoretical formula, which cannot be expected to provide for particular local attractions. Besides, we suspect that during experiments on the velocity of sound, attention has not always been paid to the velocity and direction of the wind; and still more rarely have these been registered and published, so as to afford data for applying the requisite correction.

In the Reports of the British Association for 1833 and 1836, we find two able and interesting articles by Professor Challis, on the mathematical theory of fluids. Among other things, he treats at considerable length on the theory of sound, giving an analysis and notices of various essays on that subject. But these the learned professor seems, after all, never to have thoroughly examined, otherwise he could not have failed to discover that, in the mathematical investigations employed in several of the papers on which he has bestowed the largest share of commendation, there are various fundamental errors and inconsistencies, which naturally enough have led their authors into results which, independently of every other consideration, are more or less erroneous, because palpably incompatible with each other and with the premises from which they are professedly deduced. Fallacies of the sort in question have been pointed out in the Edin. Phil. Journ. (for Oct. 1826, p. 335, and July 1827, p. 153), as also in various numbers of the Quarterly Journal of Science for 1829. Some of the same errors have been likewise noticed, though not so early as in the first-mentioned journal, by an eminent Italian philosopher, Mr Avogadro, in the Memorie dell' Accademia reale della Scienze di Torino (vol. xxxiii. p. 237), and also in a separate tract of his. Several notable instances of the like sort have been discussed and exposed in the article Hygrometry; but it is only one or two examples which we can notice here.

With the view of accounting for the excess of the actual velocity of sound over that given by Newton's formula, it was very ingeniously suggested, in a general way, by the Marquis Laplace, about the beginning of the present century, that, in the propagation of sound, the minute rises and falls of temperature occasioned by the alternate condensations and dilatations of the air, should tend to augment the disturbance in the equilibrium of the pressure, and consequently to accelerate the transmission of sound. He did not, however, till long after, assign the form or amount of the correction for such acceleration. The first attempt toward this seems to be that of Biot in 1802 (Journal de Physique, tome iv. p. 178). But M. Poisson both claims and gets the no small credit of having very considerably anticipated Laplace in giving the precise form of the correction. This he is alleged to have done in a formula for the velocity of sound, communicated to the Institute as early as the year 1807, and shortly after published in the Journal de l'Ecole Polytechnique (cahier xiv.), and which is meant to consist of Newton's formula multiplied by a constant factor $\sqrt{1+k}$. But whilst it must be admitted that no one has written to better purpose on the theory of sound than M. Poisson has done, there seems to be a very general mistake regarding that factor. The formula is

$$\frac{\sqrt{gh}}{D}(1 + \frac{aw}{(1+ad)\gamma}) = \sqrt{\frac{gh}{D}}(1+k).$$

In this he defines $g$ to represent the force of gravity; $h$ the height of the barometer; $D$ the ratio of the density of the air to that of mercury; $\delta$ the temperature of the air; $w$ the augmentation of that temperature occasioned by a condensation $\gamma$, and so small as to be considered proportional to $\gamma$; also $a = 0.0375$ for the co-efficient of the dilatation of the gases. But these evidently cannot make $k$ constant, as the sequel requires it to be. After introducing the same formula and explanation of symbols in another memoir inserted in Annales de Chimie for May 1823, he proceeds to show that the factor $1 + \frac{aw}{(1+ad)\gamma}$, which for greater brevity he there calls simply $k$, is equal to the ratio of the specific heat of air under a constant pressure to its specific heat under a constant volume; which would identify $\sqrt{k}$ with the above-mentioned correction given by Laplace himself, for the first time only, in the same Annales for November 1816, and even then without demonstration. But it is worthy of particular notice, that the reason why the factor $k$ in this case comes to be a correct expression for the ratio of the specific heats, is entirely owing to M. Poisson's there using $\gamma$ as proportional, not to $\gamma$, as above defined, but to $\frac{aw}{1+ad}$. This will be readily seen on examining the Annales for May 1823 (p. 8 and 14), where, having put $\gamma = \frac{dz}{z}$ and $w = adz$, he deduces

$$\frac{adz}{1+ad} = (k-1)\frac{dz}{z};$$

and hence $k = 1 + \frac{aw}{(1+ad)\gamma} = 1 + \frac{aw}{(1+ad)\gamma}$. Indeed, otherwise, $\sqrt{k}$ could neither have been constant, nor coincided with the correction of Laplace.

The leading principle in these formulæ therefore is, that as long as the total heat in the air undergoes no change (which may safely be supposed to be the fact, during a much longer interval than the theory of sound requires), it is not $\gamma$ and $w$, but the differentials of log. $(1+ad)$ and of log. $z$ which are always held to be proportional; and, consequently, any changes, whether great or small, in these two logarithms themselves, must also be proportional; so that the general expression for the ratio of the specific heats, whether the simultaneous changes of $z$ to $z'$, and of $t$ to $t'$, be great or small, will be

$$1 + \log_{\frac{z'}{z}}(1+ad') - \log_{\frac{t'}{t}}(1+ad) = k.$$ where \( p \) is the pressure, \( \rho \) the density, and \( q \) the difference between the total quantity of heat which a given mass of air may then contain, and that which it contains at a temperature and pressure chosen arbitrarily. Also \( k \) is still a constant, expressing the ratio of the specific heats. We here state the matter in something like the more simple language and notation of M. Poisson, who, in the Annales de Chimie for August 1823, has readily deduced that equation from the two leading principles employed for it by Laplace, and which alone are essential to it, viz. the law of Mariotte, and the constancy in the ratio of the specific heats.

From integrating the preceding equation on general principles only, and without having regard to the subject in hand, it could only be inferred that \( q \) is some indeterminate or arbitrary function of \( \frac{p}{\rho} \). But such eminent mathematicians as Laplace and Poisson, instead of being content with this general view of the matter, ought to have perceived that the conditions of the question, particularly the two principles just specified, and with which they had set out for the purpose of finding \( q \) in terms of \( p \) and \( \rho \), necessarily required \( q \) to be of the form \( A + B \left( \frac{1}{k} \log_p - \log_{\rho} \right) \), where \( A \) and \( B \) are constants; because this expression, independently of its being the first to present itself in the integration, is the only form or function of the above general integral which would be free from inconsistency or compatible with those two leading principles. So far, however, were they from attending to or being aware of this circumstance, that supposing themselves perfectly at liberty to adopt any particular function of \( \frac{p}{\rho} \), they made choice of \( q = A + B (266.67 + \theta) \frac{1}{k} - 1 \); where \( A \) and \( B \) are arbitrary constants, and \( \theta \) is the temperature, because \( \rho \) has been eliminated by the law of Mariotte. This form was adopted by these philosophers, that it might agree, as they supposed, with the hypothesis that the expansions of air under a constant pressure are proportional to the corresponding increments of absolute heat. But it is easily shown that such a hypothesis is quite incompatible, not only with the two leading principles, but with that very form which was on its account purposely given to \( q \), as was first pointed out in the Edinburgh Philosophical Journal (for October 1826, p. 335), and afterwards more explicitly in article Hygrometry (vol. xii. p. 113), where a similar question has been consistently solved by a far more simple process of reasoning.

The precise value of the above-mentioned ratio of the specific heats is considered an important element in the theory of sound. The experiments of Desormes and Clement give 1.354; those of Gay Lussac and Welter, 1.3748; and those described under the article Hygrometry, 1.3333; all of which seem to be smaller than what would reconcile the theory with the actual velocity of sound. But we shall afterwards see reason for supposing that there is still another source of acceleration, which has hitherto been overlooked. In the Annales de Chimie for June 1829, and Mémoires de l'Académie (tome x. p. 147), M. Dulong treats at considerable length on the specific heat of elastic fluids; and with the view of obtaining the ratio in question for each of them, he assumes as demonstrated, that the actual velocity of sound, in any elastic fluid whatever, has to the velocity computed by Newton's formula the subduplicate ratio of the specific heat of that fluid under a constant pressure to its specific heat under a constant volume. But since it is only in air that the velocity of sound can be obtained by direct means, Dulong availed himself of a method previously employed by others, though by no means with complete success, as is evident from the discordance of their results. It consists in determining the velocity of propagation from the musical note rendered by a given cylindrical tube, and from the measured distance between two consecutive nodal sections or positions of minimum vibration. The pitch of the note gives the number of vibrations in a given time, and consequently the time of propagation over the measured interval, and therefore the velocity of the sound. By following a particular process, Dulong was enabled to give great precision to this method, which he practised first on atmospheric air, with the view of testing the soundness of the scheme; and by many successive trials he obtained a series of results, each of which, instead of exceeding, fell short of the velocity determined by direct observation. On finding therefore that this method failed in affording quite so great a ratio as he had expected or wished, Dulong abandoned it altogether, and finally adopted 1.421 as the ratio for air. This he obtained merely from taking the mean of a great number of the direct experiments which have been made on the velocity of sound in the open atmosphere, though it is usually quoted by others as the result of Dulong's own experiments with the musical notes from tubes. The same empirical mode of determining the ratio, which, however, does nothing for clearing up the theory, has been prosecuted still more closely by Dr Simons, in the Philosophical Transactions for 1830. When the number thus obtained is substituted in the formula, it certainly answers admirably; and no wonder, for it is just obtained by reasoning in a circle, which is fit to reconcile every sort of discordance.

There can be no question that great ingenuity and superior analytical skill have frequently been displayed in source of theoretically investigating the velocity of sound; but an important oversight, and which probably is the principal source of the remaining discrepancy between the observed and the theoretical velocities, seems to attach to all the researches of this kind with which we are acquainted, in that they are conducted upon two assumptions which are so very incompatible with each other that both cannot be true, if indeed any of them is strictly so. In the first place, it is assumed that the particles of the air, at least during a calm, vibrate accurately in the direction in which the sound is propagated. Secondly, that the air, during these vibrations, preserves or acts strictly in its fluid character. That the investigations involve the first of these assumptions, is what no one will for a moment dispute; and that they also proceed upon the second, is evident from the circumstance that the pressure of the air in such researches is always taken as

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1 In the Philosophical Magazine for April 1827, it has been attempted to deduce the same equation from these two principles, incongruously coupled with a third, but with which they are utterly incompatible; and that is the needless assumption, that, under a constant pressure, the variations of absolute heat in air are proportional to those of its temperature by the common scale. But so much does the resulting equation assure of the incongruity of the data, that it has been abandoned with an admission of its being only true in one particular state of the variables, and therefore totally unfit for integration. All this might have been allowed to pass, had it not been done with the view of pushing the same fatal defect upon the production of Laplace and Poisson, which however is of an essentially different character. For the equation which has been deduced from the two leading principles alone by the most consistent method of Poisson, as above cited, and also given in the Philosophical Magazine for November 1823, neither involves the needless assumption, nor has any dependence on a particular state of the variables, and so is equally true in every case. Nor was it till after having legitimately obtained that equation, that Poisson adopted any such assumption at all on this occasion, though, as we shall presently see, he unfortunately has done so in the sequel of his memoir, and thereby introduced all manner of inconsistencies. strictly proportional to some particular power of the density, viz., either to the simple density, as was done by Newton, or to some higher power of it, as was in effect done by Laplace and his followers. Now, such proportionality of the fluid pressure to any uniform power of the density, would obviously require the particles constantly to observe the same uniform arrangement during their vibrations, so as to have the like distances between the adjacent particles of any small portion of the air everywhere to vary inversely as the cube root of the density, with, perhaps, the mere exception of as much difference of arrangement as is unavoidably due to the density being different in different parts of a wave of air. But it is easy to see that the two assumptions are so inconsistent that only one of them could be strictly true; because vibrations performed accurately in the direction of the sound necessarily imply that (abstracting from the very insignificant change of distance due to the sound's being propagated in lines which are slightly diverging, because radiating from the source of sound) the distances of the particles of the air, estimated perpendicularly to the direction of the sound, would not be altered by or during these vibrations; and consequently that the density of any minute volume of the air, instead of varying alike in all its three dimensions, would vary in only one of them, namely, in the direction alone in which the sound is propagated through it.

But, independently of the incompatibility of the two assumptions, the extreme rapidity or short duration of the vibrations, whilst it gives some probability to the first, throws as much doubt on the second. Nay, anything like a strict fulfilment of the second assumption seems, on this and various other accounts, to be extremely improbable, particularly because, in order to preserve a uniform arrangement, it would require the most of the particles, notwithstanding the rapidity of their vibrations, to describe lines deviating so much and in all manner of ways from the direction in which a clear sound is understood to be propagated; that, granting such interwoven vibrations to be possible, they could be expected to produce nothing but a very confused noise. Nor is it easy to conceive how, on that supposition, two different sounds could ever cross each other's paths without seriously interfering, which does not appear to be the fact. On the other hand, if we may not strictly adhere to the first assumption, yet, by taking a mean between the two, it would follow that the increase of the repulsion due to condensation, and the diminution of it due to dilatation, being both of them principally if not entirely confined to the direction of the sound, they would both be so much the greater in that direction; and by thus constituting an increased disturbance in the equilibrium of the repulsion reckoned in that same direction, they would have a precisely similar effect on the result, as if the pressure of the air in its otherwise supposed strictly fluid character had followed the ratio of a higher power of the density than it really does. Now, such an effect would obviously tend to shorten the duration of the vibrations, or to increase the theoretical velocity of sound, which is just the thing required in order to make it agree with observation.

The preceding remarks become far more striking with respect to the attempt which Poisson has always been considered to have made successfully (Journal Polytechnique, tome vii. p. 364), to show that the velocity of sound would still be the same, although the vibrations were considerable, and confined to a single straight line.

Doubts have been started if the air really consists of particles, and whether it may not rather have some undefinable constitution; but such vague considerations cannot materially alter the case, whilst there can be no question that the air, like all other matter, has inertia when at rest, and momentum when in motion; and also that as long as it is considered to preserve or act strictly in its fluid character, the matter of which any minute volume of it consists must be uniformly distributed. For it is upon properties essentially the same with these that the above remarks are founded.

in Geography, denotes in general any strait or inlet of the sea between two headlands. The name is given by way of eminence to the strait between Sweden and Denmark, joining the German Ocean to the Baltic, being about three miles over.