particle is resisted by some frictional force which acts in proportion to its velocity. The path being rectilinear, the equation of motion is z + 2kż + n2 z = 0, having for solution z= Aekt cos{√n2 - k2. t — a}........ ..(19), where A and a are two constants of integration to be determined by the initial conditions of motion. It will be noticed that the period of oscillation is increased by the friction and that each maximum of displacement is smaller than its predecessor. The motion is no longer simply periodic, and could not give rise to homogeneous waves. If the friction be so great that n2 - k2 is negative, the form of the solution alters and the motion becomes "aperiodic," but for our present purpose, we may leave this case out of account. When is small, the period may be expressed in terms of a series which shows that the most important term involving k depends on its square, so that even though we may take account of effects depending on the first power of k, the period is not affected by friction if we may neglect the second power. Equation (19) represents the motion which the particle assumes when unacted on by external forces, and is therefore called the free vibration. Let the same particle be now subject to an additional periodic force of period 2π/w. Its equation of motion becomes ż + 2kż + n2z = Ecos wt, where, m being the mass, Ecos wt/m is the force. The complete solution now is The second term represents the free vibration which gradually dies out, leaving permanently the "forced" vibration which is represented by and which must now be investigated somewhat more closely. .(22), If n>w, i.e. if the forced period be greater than the natural period, e lies in the first quadrant, and the forced vibration is, as regards phase, behind the force. If, on the other hand, n<w, the forced vibration is accelerated as compared with the force. If the forced and free vibrations have the same period, n = w and E Here the motion is a quarter of a period behind the force and the amplitude becomes very great for small values of k. If there is very little friction, we may put, neglecting higher powers, The friction now only affects the phase. For vanishing k, the phase is in complete agreement with that of the force when n>w and in complete disagreement when n<w. As a suggestive example of forced vibrations, neglecting friction, we may work out the case of one pendulum having mass m, and length 1, suspended from another pendulum of mass M and length L. For the equations of motion of m, we have, neglecting friction and confining ourselves to small motions, 9 where x and x are the displacements of m and M respectively. If 27/n be the free period of m, when M is stationary, the equation may be written To form the equations of motion of M, we may take the tension of the lower string to be mg to the degree of accuracy aimed at. Hence writing a for the ratio of the masses, and n2 for the free period of M when the lower string is not attached, the equation of motion becomes 2 2 X1 + n12 x1 + n2a (x, − x) = 0 We shall not attempt to obtain a general integral of these equations but confine ourselves to that particular solution in which each pendulum can perform a simple periodic motion. Writing therefore we see at once by substitution that (24) cannot be satisfied unless € = 0. Substituting (25) and (26) into (23) and (24) we obtain the equations of condition 2 w2 + n‚ ̧2 + n2a (1 − r) = 0, 1 -rw2 + n2 (r-1)=0, from which rand o may be obtained. When M, the mass of the upper pendulum, is great compared with m, a is small and then becomes nearly equal to n1, i.e. the period of the combined pendulum is nearly equal to the period of the upper pendulum, as is indeed to be expected. Substituting w=n1 in the terms involving a, we obtain as a second approximation The combined period is therefore longer than that of the upper pendulum when n>n1, i.e. when the upper pendulum has already a longer period than the shorter one. We must draw therefore the unexpected conclusion that the combined period does not lie between the two free periods, but is greater or less than that of the heavy upper pendulum according as that pendulum has already a greater or less period than the lower one. If a is small, r approaches the value n2/(n2 - n,2), and there is agreement or opposition of phase according as n is greater or smaller than n1, i.e. according as the upper pendulum has the longer or shorter period. The relative positions of the pendulum in the two cases are represented in Fig. 166, the time being such that both pendulums are at their points of greatest deviation. Fig. 166. z 147. Passage of light through a responsive medium. We now consider light to pass through a medium, the particles of which are subject to forces, capable of giving rise to free vibrations of definite periods. We consider plane waves propagated in the a direction, the displacements being in the direction. In order to obtain a simply periodic motion for the free vibrations of the particles, we may imagine each to be attracted to a fixed centre by a force varying as the distance. This centre of force we take to form part of the medium to which it is rigidly attached. If be the displacement of the medium, and that of the particle, the equation of motion of the particle is If p, is the mass of the particle, n'p, is the force of attraction at unit distance from the centre of force. The reaction of that force has X to be taken into account in forming the equations of motion of the medium. At each centre, the medium is acted on by a force n3p1 (1 − 5), and if there are a great many particles within the distance of a wavelength, we may average up the effects and imagine all the forces to be uniformly distributed. Let p be the inertia of that portion of the medium which contains on the average one and only one particle. Then the equation for the propagation of the wave is : where we have written B = p/p and V stands for the velocity of propagation when there are no particles or when n=0. If the wave is of the simple periodic type and if the motion has continued without disturbance for a sufficiently long time for the free vibrations of the particle to have died out, their position is expressed by S1 = r cos (ax - wt) .(30). a is a parameter which is constant for each particle, but varies from particle to particle. By substituting (29) and (30) into (28) and (27), two equations to determine a and r are obtained: w/a is the velocity (v) of transmission of the wave having a frequency This is Sellmeyer's equation, by means of which he first showed that the velocity of light must depend on the periods of free vibration of the molecules embedded in the æther. 148. General investigation of the effect of a responsive medium. It will be useful to introduce here a more general investigation, which we shall base on the electromagnetic theory. In Art. 134 we had expressed the total current as the sum of a polarization or displacement current and the conduction current. Το this, we may now add the convection currents. If there are N positive electrons in unit volume, each carrying a charge e and moving with velocity in the z direction, then Ne1 is the ≈ component of the convection current, and to this we must add the convection current of negative electricity - Net2. We may include both currents in the expression Neġ if έ denote the relative velocity of positive and negative electricity. The conduction current is also due to the convection of electrons, but we leave it in the form CE, because we want to distinguish between the current subject to ohmic resistance which forms a system depending only on one variable, and that which is due to oscillations of electric charges within the molecule. Confining therefore to the velocity of these charges, we have for the z component of the total current in place of (17) Chapter x., The last of equations (12) Chapter x. gives with the help of (13) and putting the magnetic permeability equal to one, The last term vanishes in isotropic media, and we may in that case, eliminating w, write for the equation of motion in the z direction 'R d KdR dt dt + 4πCR + 4π Ne¿) = ▼3R ....... ..(35). Always assuming the disturbance to be simply periodic, the displacement may be divided into two portions, one of which is in phase with R and the other in phase with dR/dt. Writing therefore Comparing with this (1), we see that the investigation of Art. 144 applies to this case, writing G = K+ NAe, F= 4πC+NBe. |