UNDULATION. 151 



AC2=EC=^+AE2=(DC + EDf+AE2=(a + a'cos^)2+(a'sm^)2=A2. 

 Also, AE=a'sintf=A.smACD. And CE=a+«'cos<?=A.cosACD. 



Therefore, ACD3=w, or tlie angle between the diagonal and the side denoting 

 the force corresponding to the component which is least advanced in phase, is the 

 measure of the interval of time between that component and the resultant, or the 

 time earlier than a, at which A must be applied in order to produce alone pre- 

 cisely the same succession of vibrations in phase and in force which is produced 

 by the combined action of a and a' . 



If the second impulse be oblique to the first, it may be decomposed into two 

 components, one acting in the direction of a, and the other at right angles to a. 

 The effects of these may be successively estimated according to the principles 

 already illustrated. And the same principles may be applied to the d(^t(>rmina- 

 tion of the resultant of any number of impulses, acting in all possible directions. 

 The important conclusion is that, though the form of the path and the ampli- 

 tude of the oscillations of the body may be altered, yet they are, in all cases, 

 capable of determination, and the time of the vibration will be invariable. 



§11. UNDULATION. 



Hitherto we have supposed that the vibrating body imparts none of its motion 

 to surrounding bodies. This is a case which can riever be experimentally real- 

 ized, and, if it could, it is not the case which concerns us af present. We wish 

 to show the connexion between vibration and undulation, and, to this end, we 

 must suppose the body in vibration to be immersed in an elastic fluid, whose 

 particles are set in motion by it. Such a fluid is the atmosphere ; find it is mat- 

 ter of common knowledge that sound is an effect of undulations produced in the 

 air by vibrating bodies. It is also well known that sound does not attend all 

 movements in the air, not even all movements of vibration. A certain rapidity 

 is required, for a reason which will presently appear. 



An elastic fluid may be described as one whose particles tend to recede from 

 each other, or have the same mechanical relations as if they possessed the 

 property of mutual repulsion. The distances between the particles, in the case 

 of such elastic fluids as actually exist, are probably very g)-eat compared with 

 the magnitude of the particles themselves. When the fluid is at rest, each par- 

 ticle is held in equilibrium by the repulsions of its U'cighbors. If a slow move- 

 ment be excited among these parti-cles, they will not, to any material degree, 

 alter their distances from each other; but, if any one particle, or any stratum of 

 particles, be driven toward those adjacent, with such suddenness that the inertia 

 of the latter is brought sensibly into play, then the distances will be momenta- 

 rily diminished, or there will occur a local compression of the fluid at that place. 

 This impulse, it is evident, must come from some body foi-eign to the fluid, for 

 by the definition it must appear that the fluid is incapai)le of unequally com- 

 pressing itself. Suppose, then, the compressiiig body, after making its sudden 

 advance, to stop with equal suddenness. The repelling force between the first 

 and second strata will exceed that between the second and third; but the first 

 stratum cannot recede on account of the obstacle. The second, therefore, will 

 advance, diminishing its distance from the third, and so calling a greater repul- 

 sive force into activity between those strata also. The third will then advance, 

 then the fourth, then the fifth, and so on. It thus appears that the movement 

 originally communicated to the fii'st stratum will pass from stratum to stratum 

 through the whole fluid. Each stratum, moreover, will come to a state of per- 

 manent rest the instant the next has taken up the movement. This is in accord- 

 ance with the well known law of impact of equal elastic bodies. There is, 

 tberefore, no vibration in this case. But the progress of the movement is uni- 

 form throughout the medium, and the velocity of transmission is the same, (or 



