in Wave Propagation, 279 



This is a condition of affairs which has the advantage of 

 being easily conceived with distinctness. We can see the 

 beams on their outward journey becoming more and more 

 disentangled from one another as they advance, and can 

 picture them on their return journey as converging, so as 

 more and more to overrun one another, and thus produce a 

 more complex disturbance in the medium the farther they 

 advance inwards, which, although it will fall short of pre- 

 senting the actual disturbance anywhere outside the sphere, 

 becomes the actual disturbance (with motions reversed) 

 everywhere within that space. What takes place is sub- 

 stantially of the same kind as what occurs in the simpler case 

 of a straight string set vibrating ; where the motion what- 

 ever it is may be represented by waves of fixed forms simul- 

 taneously travelling in opposite directions over the length of 

 the string. (See Chapter V. of Donkin's ' Acoustics,' where 

 the case of transverse vibration in one plane is worked out.) 



We can also picture to ourselves what will occur if there 

 be also sources of disturbance outside the sphere. Every 

 such addition will make an alteration* throughout all the un- 

 dulations, and therefore in those portions of them included 

 in the beams that travel across our sphere ; but will never- 

 theless not change in the least the effect produced by their 

 coexistence within that space. 



. It is evident, on this and other accounts, that a resolution 

 into undulations of plane waves may occur in numberless 

 ways. The choice between them which nature will make in 

 any particular case is, as in applications of Fourier's theorem 

 &c, to be ascertained by considerations affecting the energy 

 received by, stored in, and escaping from, the system ; as, 

 for example, the necessity of complying with the Principle of 

 Least Action. It may be of interest to mention that the 

 " crumples " which Helmholtz observed in the motion of 

 violin strings, but left unexplained (see ' Sensations of Tone,' 

 part i. chap. v. § 4), can be accounted for and their forms 

 determined by including in the investigation the considera- 

 tions here referred to. 



Extension to other titan monotropic 'media. — Hitherto we 

 have supposed the medium to be monotropic. If it be of 



* This alteration has a twofold source. It is primarily due to direct 

 contributions to the wave at AB made by the new sources of disturbance; 

 but it is also due in part to the changed circumstances of the puncta 

 within the sphere, since the motion within the sphere which is occasioned 

 by the sources of disturbance outside it has now to be deducted before 

 we determine what contributions the puncta within the sphere will make 

 to the wave at AB. See footnote, p. 275. 



