If we consider all the particles at one moment, that is, if we make t constant, the displacement of different particles follows the law

From the last mentioned characteristics, this law. of disturbance has received the name of the stationary

wave.

The variable part of the elastic force of the air dX

(Article 21) is -D. dx'

ог

2bfD. sin (neaf. t ++*), sin (f+).

2

This, for any one particle, has for coefficient

2bfD. sin (fx +oo).

This coefficient is greatest where

On comparing this with the preceding statements, it appears that those points of the air which have no displacement have the greatest change of density, and those which have the greatest displacement have no change of density.

The reader will at once perceive that the theory of the echo, in Article 41, applies to this case; supposing

Stationary waves are formed, in like manner, by the combination of converging and diverging waves in the cases supposed in Articles 44, 46, 50, 51. The reader will have no difficulty in verifying this. It will

be necessary to distinguish carefully the signs of as derived through u and as derived through v.

dR

dr

59. Deficiences still existing in the mathematical theory of atmospheric vibrations, as applied to important cases occurring in practice.

The first deficiency to which we shall advert is in the general treatment of the reflection of waves of air. We have seen in Article 41, &c. that the reflection of ordinary plane waves of air at a plane surface is treated theoretically without difficulty; and if we should use a similar process for such plane waves as occur in Article 42 (the formulæ of that Article being so altered as to represent two directions of motion of wave inclined to the axis z, in order to exhibit the wave in the generality of inclination), or for such diverging waves as occur in Articles 44, 46, 50, and 51 (with due alteration for representing the places of the two centers of divergence and the two directions of oscillation), we should find no difficulty, provided we assume that the surface of reflection (that is, the surface along which the motions of the particles produced by the combination of two waves are at all times parallel to the portions of surface which they touch) is a plane.

But if we assume that surface to be curved, we meet with difficulties. It might be supposed that, with a parabolic surface, the movements of particles, produced by a spherical wave diverging from the focus, and by a plane wave moving in the direction of the axis, would be so related as to shew that, by reflection at the parabolic surface, one of these waves would be the consequence of the other. This, however, has not been proved algebraically, and appears to be doubtful.

In like manner, it might be supposed that, with a prolate spheroidal surface, the movements of particles in waves diverging from one focus and converging to the other focus would possess the relation proper for reflection; but this is equally in doubt. And these doubts apply to reflection at a curved surface generally.

The second deficiency is in the investigation of the motions of the particles at the junction of two containing vessels. Suppose, for instance, we consider a large tube stopped at one end and communicating at the other end with the open air. There is no difficulty in understanding that there may be a stationary wave in the tube (the stopped end being one of the points of vanishment of motion in Article 58), and that there may be a stationary divergent wave in the open air. But, if so, where will be the next surface of vanishment of motion? or that of vanishment of variability of pressure? Theory has not yet answered these questions.

We commend these problems to the attention of the student.

SECTION V.

TRANSMISSION OF WAVES OF SONIFEROUS VIBRATIONS THROUGH DIFFERENT GASES, SOLIDS, AND FLUIDS.

60. Velocity of waves through gases.

The investigation of Article 21 applies in the same manner to all gases as to atmospheric air, excepting that we are not so well acquainted with the effects of change of temperature and sudden contraction or expansion. Omitting these, that is, putting 1 for no, we find as in Article 21, that about the particle whose original ordinate was x, the density of the gas is represented

dX

by D-D; and by Boyle's Law, Article 10, which

is found to apply to all gases, the elastic pressure of the gas about that point is therefore

K(D-D),

K being a constant of whose value we shall speak very soon. Consequently, the elastic pressure about the point whose original ordinate was x + k is

K (D-DdX - DdXk);

and the excess of the former mentioned pressure, tending to move the included mass of gas forwards, is

K