« السابقةمتابعة »
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
neglecting such quantities as are very small compared with those retained, or by eliminating s and s, we get
The general equations (1) and (2), joined to the particular conditions (A) which belong to the surface of separation (yx), only, are sufficient
for completely determining the motion of our two fluids, when the velocities and condensations are independent of the co-ordinate z, whatever the initial disturbance may be. We shall not here attempt to give their complete solution, which would be complicated, but merely consider the propagation of a plane wave of indefinite extent, which is accompanied
by its reflected and refracted wave.
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
retaining b and c unaltered, we may give to the fronts of the reflected and refracted waves, any position by making for them
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
Hence by writing, to abridge, the characteristics only of the functions
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
the ordinary law of sines.
Hence the reflected wave may be made to vanish if y” – ), and (oy A) – (), A) 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
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
which is the same ratio as that given for light polarized perpendicular to the plane of incidence. (Wide 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,
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 regards the co-ordinate a an exponential must take the place of the circular function. In fact the equation,
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 p. We must therefore take, instead of (8), the formula,