صور الصفحة
PDF

The principal object of this supplement has been to put the equations due to the surface of junction of two media, and belonging to light polarized perpendicular to the plane of incidence, under a more simple form. The resulting expressions have here been made to depend on those before given in our paper on Sound, and thus the determination of the intensities of the reflected and refracted waves becomes in every case a matter of extreme facility. As an example of the use of the new formulae, the intensities of the refracted waves have been determined for both kinds of light: the consideration of which waves had inadvertently been omitted in a former communication.

Perhaps I may be permitted on the present occasion to state, that though I feel great confidence in the truth of the fundamental principle on which our reasonings concerning the vibrations of elastic media have been based, the same degree of confidence is by no means extended to those adventitious suppositions which have been introduced for the sake of simplifying the analysis.

Let us here resume the equations of the paper before mentioned, namely,

dy ~dx~ dy

dtf> dyfr _^ d<f>t <fyj
dy dx dy dx

d*<ff> _ d'<ff>,
fd? g*dt

d'jr _ dtyfrl

where u and v, the disturbances in the upper medium parallel to the axes * and y, are given by

u _ ^ _|_ *ty
dx dy'

[ocr errors]
[ocr errors]
[ocr errors][ocr errors]

«, and vt the disturbances in the lower medium being expressed by similar formulae in $t and ^r

The two last equations of (17) give, since

9 t 9. 1,

and being accented for a moment to distinguish between the particular values belonging to the plane (yz) and their mora general values

and

The correctness of these values will be evident on referring to the Memoir, formulae (20), (21), and recollecting that

Hence the first equation gives, since x = 0,
, A' - - and

i^+i) *~ 6o*sTir^"

Also the second equation may be written,

dty ,_(?<!> d<f>; _ (/t'-l)8 gfc
f/x tfas djy rfy J(/i*' + l) dy*

And since we may differentiate or integrate fhe equations (17) relative to any variable except x, we get for the conditions requisite to complete the determination of and

z=df,_ (n*-iy dSjt, > (when a? - 0) (29).

dx dx 0**+1) b dx* J

Or neglecting the term which is insensible except for highly refracting substances,

[ocr errors]

•where u = —ia the index of refraction. 7,

These equations belong to light polarized in a plane perpendicular to that of incidence, and as <f> and <f>t are insensible at sensible distances from the surface of junction of the two media, we have, except in the immediate vicinity of thi9 surface,

[ocr errors][ocr errors]

When light is polarized in the plane of incidence, the conditions at the surface of junction have been shewn to be

[ocr errors]
[ocr errors]

dw^dwt Wwhen<r- = 0) (32).

dx dx J

Since in these conditions we may differentiate or integrate relative to any of the independent variables except x, we see that the expressions (30) and (32) are reduced to a form equiva

• Though these equations have been obtained on the supposition that the vibrations of the media follow the law of the cycloidal pepdulum, yet as (J) haa disappeared, they are equally applicable for all plans waves whatever.

In fact, instead of using the value

•.)/,—a, sin (apt + by + ct), and corresponding values of the other quantities, we might have taken the infinite ■(tries

ft = 2o/ sin n- {a,x+ by + ct),

where o and n may have any series of values at will. But the last expression isr the equivalent of an arbitrary function of'

a,as+iy+et."

Or the same equations might have been immediately obtained from (IT), without introducing this consideration. The method in the text has been employed for the sake of the intermediate result (29), of which we shall afterwards make use. lent to that marked (A) in our paper on Sound; and the general equations in ^ and w being the same, we may immediately obtain the intensity of the reflected or refracted waves, by merely writing in the simple formula contained in that paper,

A ~ i and A, = 1 for light polarized in the plane of incidence;

or A^^andA,^^ for light polarized perpendicular to the

plane of incidence.

As an example, we will here deduce the intensity of the refracted wave for both kinds of light.

Representing, therefore, the parts of to and w, due to the disturbances in the Incident Reflected and Refracted waves by

f(ax + by + ot), F(-ax + by + ct), and ft (atx + by + ct)

respectively, and resuming the first of our expressions (7) in the paper on Sound, viz.—

/<-J (£+2) *

we get for light polarized in the plane of incidence, where

A = A, = 1,

/ 2 2 _ 2 cos $ sing,

?~ 1 I a'~l 1 COtg'= 8in +'
a cot 0

which agrees with the value given in Airy's Tracts, p. 356*.

For light polarized perpendicular to the plane of incidence

we have A = and A, =\. If, therefore, we here represent the 7 1,

parts of .yjr and ^r, due to the same disturbances by /, JF'and /, we get

/' 2 sing, cos 6 2

f T5 cotd sin6cos6,' cos0sin0 , " *

* * _1_ f ' -L. J

7* cot 6 cos 6t sin 6j

[* Airy, ubi tup. p. 109.]

o, if D be the disturbance of the incident wave in its own plane, and D, the like disturbance in the refracted wave, we have by first equation of (31),

D sin 6 = u = ~ = If (ax + by + ct),

and D, sin 9t = u, - ^ = bf (ax + by 4- ct),

retaining in yjr the part due to the incident wave only.

Thus by writing the characteristics merely,

D, sin 6 /' cos g 2
2) sin /' cos 6t' cos sin 6 ^
cos # sin 6.

[ocr errors][ocr errors][ocr errors][ocr errors][merged small][ocr errors]

which agrees with the formula in use. (Vide Airy's Tracts, p. 358*.)

In our preceding paper, the two media have been supposed to terminate abruptly at their surface of junction, which would not be true of the luminiferous ether, unless the radius of the sphere of sensible action of the molecular forces was exceedingly small compared with X, the length of a wave of light.

In order, therefore, to form an estimate of the effect which would be produced by a continuous though rapid. change of state of the ethereal medium in the immediate vicinity of the surface of junction, we will resume the conditions (29), which belong to light polarized in a plane perpendicular to that of Reflexion, viz.

and instead of supposing the index of refraction to change suddenly from 0 to ft, we will conceive it to pass through the

[* Airy, ubi tup. p. 110.]

« السابقةمتابعة »