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π

TABLE of the different parts of cos (w-m.w), from 0 to x infinity.

2

Value

of

1 .

Values of the integral found by actual summation.

Values of

the remaining integral.

From x=0.00 From x=1:00 From x=1-26 From x=144 From x=1.58 From x=1.70 From x=192 Term depend-Term depend-
to x=1.00.
to x=1.26. to a=1.44. to x=1.58. to a=1.70. to x=1-92.
ing on sin u. ing on cos u.

to x=2.00.

Sum.

-4-0 +0929154-1621545 +12802070757713-0036556-0150365 +0374784 0000000 +0011788 +0029754 −3.8 +0899076-1537134 +1071428-0318602 |-·0507057 +0210294-0021156 +0236322 +0009898 +0043069 -3.6 +0783010-1250013 +0663779 +0200012 0852129 +0538698-0412717 +0387234 +0003926 +0061800 -34+0591033-0787271 +0125367 +06837060979422 +0721698-0655350 +0392214-0004079 +*0087896 -3.2 +0342832-0200615-0450110 +1023750-0852309 +0684171-0658255 +0245557-0011101 +0123920 -30+0066555 +0439416 -0959701 +1141579 0500345 +0418171-0418367 00000000014269 +0173039 -28-0202801 +1051981-1309732 +1006947-0012464-0008297 -0022201-0252118-0012011 +0239304 −2·6-0426027 +1556300 -1433068 +0645600 +0485494-0467686 +0385325 0413456-0004776 +0327706| -24-0562773 +1882352-1301966 +0134390 +08632950810887 +0653953 -0419123+0004974 +0444215 -2-2-0574983 +1980667-0934064-0415027 +10204230915539 +06834970262631+0013569 +0595912 |−2·0-0430375 +1829652-0390242 ~·0880664 +0912823-0729091+0461214 0000000 +0017488 +0790805 -1.8 -0105661 +1439504+0235453 -1157328+0565228-0291543 +0067189 +0270131+0014761 +1037734 16 +0410858 +0852018 +0832149-1180453 +0065563 +0271234-0354520 +0443405 +0005884 +1346138 |−1·4 +1117252 +0136169 +1291863 09411340457342 +0783376-0648587 +0449910-0006146 +1725361 -12 +1996990-0620069 +1529169-0488959 0866990 +10740100705636 +0282196 0016814+2183897 -10 +3019144-1320681 +1497066 +0078255-1054914 +1034152-0502948 00000000021732 +2728342 −0·8 +4139711-1873967 +1196276 +0635668 0969417 +0656568 0113461-0290842-0018395 +3362141 -06 +5303921-2204699 +0675952 +1058949-0629782 +0044341 +0320494-0477897-0007356 +4083923 -0.4 +6449407-2264423 +0025723 +1252092-01220170616677 +0639213-0485430+0007707 +4885595 -0.2 +75104002038501-0639887 +1169208 +0422963-1114242 +0724424-0304813+0021152 +5750704 00 +8422086-1548861-1201662 +0825417 +0863120-1278013 +0543205 0000000 +0027427 +6652719 +0.2 +9125150-0852037-1557205 +0294203 +1082374-1037569 +0160657 +0314867 +0023295 +7553735 +04 +9570127-0032555-1639662 0308393 +1021275-0450631-0283460 +0517992 +0009347 +8404040 +0.6 +9721117 +0807460-1430312-0848876 +0693085 +0309509-0625827+0526798 0009826 +9143128| +0.8 +9558650 +1561216-0962649-1206499 +0181032 +1005011-0739646 +0331201-0027067 +9701249 +10 +9081446 +2131038-0317049-1299775-0382828 +1408098-0581618 ⚫0000000 -0035225 +10004087 +12 +8306993 +2441213 +0393109-1105332-0851711 +1375753-0208408-0343016-0030028 +*9978573 +14 +7270641 +2448168 +1041461-0663663-1102398 +0899919 +02436700565062-0012095 +9560641 +16+6023587 +2146627 +1511250-0070683-1067638 +0115856 +0608471-0575460 +0012765 +8704775 +1.8 +4629601 +1570802 +1716481 +0543138 0754212-0736006+0751107-0362310 +0035295 +7393896

0000000 +0046115+*5649030

+20 +3160989 +0790543 +1617709 +10415490241758-1383955 +0617838 +2-2 +1693705-0097331 +1229644 +1312871 +0337502 -1611914+0256330 +0376336+0039467+3536610 +24+0302520-0981192 +0619006 +1295037+0832918-1331501-0201404 +0620890+0015960 +1172234 +26-0944017-1748910-0107004 +0989853 +1114670-0615225-0587219 +0633290-0016907 -1281469 +2.8-1987892-2302290-0819605 +0463043 +1107817 +0321479-0758654 +0399339-0046921-3623684 +3·0 -2785332-2569831-1391417-0169919 +0812275 +1185093 | -*0651540 00000000061508-5632179 +32-3309399-2516145-1719364 0769201 +0303310 +1696336-0304033-0416103-0052779-7087378 +34-3551338-2146670-1743344-1201569-0287648 +1680727 +0156965-0687550-0021380-7801807 +3.6-3520567-1507154-1457300-1370101-0807012 +1128004 +0562188-0702322+0022653-7651611 +38-3243480-0677906-0910614-12359601119003 +0200155 +0762169-0443470+0062719-6605390 +40 -2761063 +0236462-0199552-0827357-1141216-0815972 +0682405 ·0000000 +0081698-4744595

ROYAL OBSERVATORY, GREENWICH,

March 12, 1838.

G. B. AIRY.

XVIII. On the Reflexion and Refraction of Sound. By G. GREEN, Esq.,

of Caius College, Cambridge.

[Read December 11, 1837.]

THE object of the communication which I have now the honour of laying before the Society, is to present, in as simple a form as possible, the laws of the reflexion and refraction of sound, and of similar phenomena which take place at the surface of separation of any two fluid media when a disturbance is propagated from one medium to the other. The subject has already been considered by Poisson, (Mém. de l'Acad., &c. Tome X. p. 317, &c.) The method employed by this celebrated analyst is one that he has used on many occasions with great success, and which he has explained very fully in several of his works, and recently in a digression on the Integrals of Partial Differential Equations. (Théorie de la Chaleur, p. 129, &c.) In this way, the question is made to depend on sextuple definite integrals. Afterwards, by supposing the initial disturbance to be confined to a small sphere in one of the fluids, and to be everywhere the same at the same distance from its centre, the formulæ are made to depend on double definite integrals; from which are ultimately deduced the laws of the propagation of the motion at great distances from the centre of the sphere originally disturbed.

The chance of error in every very long analytical process, more particularly when it becomes necessary to use Definite Integrals affected with several signs of integration, induced me to think, that by employing a more simple method we should possibly be led to some useful VOL. VI. PART III.

3 F

result, which might easily be overlooked in a more complicated investigation. With this impression, I endeavoured to ascertain how a plane wave of infinite extent, accompanied by its reflected and refracted waves, would be propagated in any two indefinitely extended media of which the surface of separation in a state of equilibrium should also be in a plane of infinite extent.

The suppositions just made simplify the question extremely. They may also be considered as rigorously satisfied when light is reflected. In which case the unit of space properly belonging to the problem is 1 a quantity of the same order as λ= inch, and the unit of time 50,000 that which would be employed by light itself in passing over this small space. Very often too, when sound is reflected, these suppositions will lead to sensibly correct results. On this last account, the problem has here been considered generally for all fluids whether elastic or non-elastic in the usual acceptation of these terms; more especially, as thus its solution is not rendered more complicated. One result of our analysis is so simple that I may perhaps be allowed to mention it here. It is this: If A be the ratio of the density of the reflecting medium to the density of the other, and B the ratio of the cotangent of the angle of refraction to the cotangent of the angle of incidence. Then for all fluids the intensity of the reflected vibration the intensity of the incident vibration

=

A-B
A+B'

If now we apply this to the reflexion of sound at the surface of still water, we have A > 800, and the maximum value of B<1. Hence the intensity of the reflected wave will in every case be sensibly equal to that of the incident one. This is what we should naturally have anticipated. It is however noticed here because M. Poisson has inadvertently been led to a result entirely different.

When the velocity of transmission of a wave in the second medium, is greater than that in the first, we may, by sufficiently increasing the angle of incidence in the first medium, cause the refracted wave in the second to disappear. In this case the change in the intensity of the

reflected wave is here shown to be such that, at the moment the refracted wave disappears, the intensity of the reflected becomes exactly equal to that of the incident one. If we moreover suppose the vibrations of the incident wave to follow a law similar to that of the cycloidal pendulum, as is usual in the Theory of Light, it is proved that on farther increasing the angle of incidence, the intensity of the reflected wave remains unaltered whilst the phase of the vibration gradually changes. The laws of the change of intensity, and of the subsequent alteration of phase, are here given for all media, elastic or non-elastic. When, however, both the media are elastic, it is remarkable that these laws are precisely the same as those for light polarized in a plane perpendicular to the plane of incidence. Moreover, the disturbance excited in the second medium, when, in the case of total reflexion, it ceases to transmit a wave in the regular way, is represented by a quantity of which one factor is a negative exponential. This factor, for light, decreases with very great rapidity, and thus the disturbance is not propagated to a sensible depth in the second medium.

Let the plane surface of separation of the two media be taken as that of (yx), and let the axis of ≈ be parallel to the line of intersection of the plane front of the wave with (yx), the axis of a being supposed vertical for instance, and directed downwards; then, if ▲ and A, are the densities of the two media under the constant pressure P and s, s, the condensations, we must have

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Also, as usual, let be such a function of x, y, z, that the resolved parts of the velocity of any fluid particle parallel to the axes, may be represented by

do do аф
dx' dy' dz

In the particular case, here considered, will be independent of ≈, and the general equations of motion in the upper fluid will be

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The above are the known general equations of fluid motion, which must be satisfied for all the internal points of both fluids; but at the surface of separation, the velocities of the particles perpendicular to this surface and the pressure there must be the same for both fluids. Hence we have the particular conditions

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neglecting such quantities as are very small compared with those retained, or by eliminating s and s,, we get

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The general equations (1) and (2), joined to the particular conditions (A) which belong to the surface of separation (yx), only, are sufficient

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