If we consider two points very near together on a wave, it is evident that they rise or fall through the difference of their heights during the time in which the waves advance through their horizontal distance. Hence, the difference of their heights, divided by their horizontal distance, is equal to the velocity of either particle divided by the velocity of propagation of the waves. That is to say, in the language of the differential calculus, because is negative from a to c, where the particles dy are rising, or, in other words, where y is positive. This conclusion can be verified by differentiating equation (10). For, writing that equation in the form the inclination of the curve to the axis of x. 51. Next let the vibrations be longitudinal; that is, let the lines of motion of the particles be coincident with the direction of propagation of the waves. The particles will then remain always in one straight line, but their alternate forward and backward movements will bring them sometimes nearer together and sometimes further apart; so that instead of crests and troughs we shall have compressions and extensions propagated along the line of particles. Equation (10) or (11) still represents the displacement of any particle from its mean position, and we shall regard y as positive when the displacement is forward (understanding by the forward direction the direction in which the waves are propagated, which is the same as the direction of the positive axis of x). Hence, where we had before a crest we shall now have a maximum forward displacement, and where we had before a trough we shall now have a maximum backward displacement. To calculate the compressions and extensions, let x be the mean co-ordinate of one particle, that is, the coordinate of the particle when in its mean position, and x+6x the mean co-ordinate of a particle a little in advance of it. Also let y and y+dy be their respective displacements from their mean positions. Then the distance between the two particles, instead of being the distance or between their mean positions, is dx+dy. The measure of the extension is the increase of distance dy divided by the mean distance dx, or more strictly is the limit of this ratio dy When dy dx dx is positive, it indicates extension, and when negative, compression. From the equation dy = dx we infer that particles which are moving forward (and for which therefore y is positive) are in compression, that particles which are moving backward are in extension, and that the amount of compression or extension is directly as the forward or backward velocity of the particles, being equal to the quotient of this velocity by the velocity of propagation of the waves. It is by longitudinal vibrations that sound is propa. gated through air, and through all gases and liquids. CHAPTER V. COMPOSITION OF TWO SIMPLE HARMONIC UNDULATIONS OF EQUAL WAVE-LENGTH. 52. By calling an undulation the resultant of two others, we mean that the motion of each particle is the resultant of the motions due to these two component undulations. In general the resultant of two displacements is the diagonal of the parallelogram of which the two component displacements are the sides; but we shall for the present confine our attention to the case in which the two components are parallel. The resultant will then be their algebraic sum. In every case we shall suppose the velocity of propagation to be the same for both the component undulations. 53. First, let the direction of propagation be the same for both, and their wave-lengths equal. Since the velocities of propagation are also equal, the periods will be equal, and the difference of phase will be a definite quantity, the same for all the particles and remaining constant. Let this difference of phase be 4, and let the amplitudes of the two undulations be a and b. The motion of each particle is then the resultant of two simple harmonic motions in the same straight line, with a difference of epochs . This resultant is (by § 33) a simple harmonic motion of the same period as the components, and having an amplitude c = √ (a2 + b2 + 2 a b cos p) represented by the diagonal of the parallelogram, whose sides are the two component amplitudes a and b placed at an angle . 54. If the wave lengths of the two component undulations are only approximately equal, while their velocities of propagation are still supposed to be exactly equal, the difference of epoch & will have different values at different points at the same moment, or for the same particle at different moments. The amplitude of vibration of each particle will, on the principles of § 34, alternately increase and diminish, its maximum value being the sum, and its minimum value the difference of a and b. 55. These results can be verified by means of the standard equation (10) of wave motion. Denoting for shortness vt-x 2 by 4, we may write the equations of the two undulations, on the assumption that λ is the same for both, Y1 =a cos 6, Y1⁄2=b cos (0-4), and the equation of the resultant undulation will be y=a cos 4+b cos (0-6) = cos(a+b cos ) + sin 6 (6 sin þ).· |