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The number of terms computed by logarithmic process for this part of the integral is about 900.
I could have wished to extend the computation as far as m = + 6-0, so as to include perhaps one or two more maxima of the values of intensity. Indeed, if it had been possible to foresee the approximate places &c. of maxima, I should probably have commenced the computations to that extent. The trouble however would be great, as the summation must be extended as far as w = 2.5. Should any person be disposed to go to that extent, I would recommend that the summation of the values computed in this Paper should be also extended, for the values of m beginning with + 3-0.
I subjoin a Table of the different sections of the summation and remaining integration for all the values used in this Paper.
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 formulae 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
a quantity of the same order as X = inch, and the unit of time
I 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 A – B the intensity of the incident vibration T 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 ~ +. 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 (yz), and let the axis of x be parallel to the line of intersection of the plane front of the wave with (yz), the axis of a being supposed vertical for instance, and directed downwards; then, if A and Al are the densities of the two media under the constant pressure P and s, s, the condensations, we must have |. (1 + 8) = density in the upper medium, A, (1 + s 1) = density in the lower medium. P(1 + As) = pressure in the upper medium, §§ + 4, 8,) = pressure in the lower medium.
Also, as usual, let p be such a function of a, y, z, that the resolved
parts of the velocity of any fluid particle parallel to the axes, may be represented by
dop dop dop dr’ dy' do. . In the particular case, here considered, p will be independent of x, and the general equations of motion in the upper fluid will be