Since the disturbance of all the particles, in any front of the incident plane wave, is the same at the same instant, we shall have for the incident wave $=f(ax -+ by + ct), .. retaining b and c unaltered, we may give to the fronts of the reflected and refracted waves, any position by making for them $= F (a's + by + ct), 0.= f, (a,x+by+ct). Hence, we have in the upper medium, $= f (ax + by + et) + F (a’x + bu + ct) ........... (4), and in the lower one 0,=f. (a,+ by + ct) ............................. (5). These, substituted in the general equations (1) and (2), give d = q (a* + B) c=y" /a.'+b) ) Hence, a' = a, where the lower signs must evidently be taken to represent the reflected wave. This value proves, that the angle of incidence is equal to that of reflexion. In like manner, the value of a,, will give the known relation of sines for the incident and refracted wave, as will be seen afterwards. Having satisfied the general equations (1) and (2), it only remains to satisfy the conditions (A), due to the surface of separation of the two media. But these by substitution give af" (by + ct) – aF" (by + ct) = a, fi (by + ct), A{f" (by + ct) + F" (by + ct)} = A f (by + ct), because a' =-a, and x= 0. · Hence by writing, to abridge, the characteristics only of the functions or if we introduce 0, 0,, the angle of incidence and refraction since and therefore " ore F=Ā A cot 0 which exhibits under a very simple form, the ratio between the intensities of the disturbances, in the incident and reflected wave. But the equations (6) give Hence the reflected wave may be made to vanish if nec, and ()* - (,4) have different signs. For the ordinary elastic fluids, at least if we neglect the change of temperature due to the condensation, A is independent of the nature of the gas, and therefore A= A, or g'A=YA, which is the precise angle at which light polarized perpendicular to the plane of reflexion is wholly transmitted. But it is not only at this particular angle that the reflexion of sound agrees in intensity with light polarized perpendicular to the plane of reflexion. For the same holds true for every angle of incidence. In fact, since and the formulæ (7) give sin o tan sin 8, * tan@, which is the same ratio as that given for light polarized perpendicular to the plane of incidence. (Vide Airy's Tracts, p. 356)*. What precedes is applicable to all waves of which the front is plane. In what follows we shall consider more particularly the case in which the vibrations follow the law of the cycloidal pendulum, and therefore in the upper medium we shall have, $= a sin (ax + by + ct) + B sin (- ax + by + ct) .........(8). Also, in the lower one, 0,= 0, sin (a &c + by + ct): and as this is only a particular case of the more general one, before considered, the equation (7) will give If y, >y, or the velocity of transmission of a wave, be greater in the lower than in the upper medium, we may by decreasing a render a, imaginary. This last result merely indicates that the form of our integral must be changed, and that as far as * [Airy on The Undulatory Theory of Optics, p. 111, Art. 129.] regards the co-ordinate x an exponential must take the place of the circular function. In fact the equation, may be satisfied by 0,= 's. B sinys, (where, to abridge, y is put for by + ct) provided c=r(-2" + b*); when this is done it will not be possible to satisfy the conditions (A) due to the surface of separation, without adding constants to the quantities under the circular functions in $. We must therefore take, instead of (8), the formula, $=a sin (ax + by + ct +e) +B sin (- ax + by+ci+e.) .... (9). Hence when x = 0, we get = aa cos (4 +e) – aß cos (+e), A COS on = ca cos (+e) – eß cos (46 +e), don--a:B sin , d=cB COB ll; these substituted in the conditions (A), give and, consequently, ier-e, B=& and tan e=+a, This result is general for all fluids, but if we would apply it to those only which are usually called elastic, we have, because in this case yA=7,A,, es But generally =4"(2* + 6) =we (a + 6) .......... (11); and therefore, by substitution, tan e a t = , Vyana + b) **= A N tano 0 – sect 8, because x =%, and = tan'd. As e=-2,, we see from equation (9), that 2e is the change of phase which takes place in the reflected wave; and this is precisely the same value as that which belongs to light polarized perpendicularly to the plane of incidence ; (Vide Airy's Tracts, p. 362*.) We thus see, that not only the intensity of the reflected wave, but the change of phase also, when reflexion takes place at the surface of separation of two elastic media, is precisely the same as for light thus polarized. * Airy, ubi sup. p. II+, Art. 133. |