CONTEMPORARY ADVANCES IN PHYSICS 299 



We have spoken of the value of 5 at the origin of coordinates, for 

 mathematical convenience; but in reahty the "origin" is any point P, 

 and 5 is any surface enclosing that point, and Sq is the value of 5 in the 

 point P at any moment /, and U is the value of 5 in any point distant 

 by r from P, evaluated at the moment {t — r/c). Hence (29) may be 

 written thus: 



4x5 = \ dS \ cos {n, r) ' ' 



^M 



dr \ r r dn 



(30) 



The task is achieved. It has been proved that when a function con- 

 forms to "the wave-equation" it conforms also to the first-suggested 

 definition of a wave-motion, in that its value at any time and place is 

 determined by its anterior values and those of its derivatives over a 

 surface completely enclosing the place. Moreover the actual formula 

 has been derived whereby the value at any point and moment can be 

 computed when the values all over any surrounding surface are known 

 at the appropriate prior moments. 



Introduction of the Ideas of Frequency and Wave-length 

 Hitherto I have spoken chiefly of an extremely abstract "some- 

 thing," denoted by a symbol s, and possessed of the property that its 

 value at any point and moment is built up out of contributions des- 

 patched at earlier moments from all of the area-elements of any con- 

 tinuous surface which encloses the point; these contributions being 

 borne as it were by messengers, who travel to the point at a finite speed 

 from the various area-elements whence they depart. Only one physical 

 constant has been introduced, and this is the speed of these messengers. 

 This is the constant which appears in the wave-equation (6), being 

 there denoted by c. It is commonly called the speed of the waves; 

 but, for various reasons which will eventually appear, it had better be 

 called the phase-speed. Now there are two other constants familiar 

 in our experience with water-waves and sound; they are frequency 

 and wave-length. Let us try to import them into the general theory. 

 At any point of a water-surface over which uniform ripples are 

 passing, the elevation is a periodic function of time; so also are the 

 pressure and the density at any point of a gas through which uniform 

 sound is flowing, or the displacement of either prong of a steadily- 

 humming tuning-fork. Any periodic function of time is either a sine- 

 function, or a composite of sine-functions. It is suitable therefore to 

 begin by analyzing the case in which the function is a sine. Using/ 

 to denote any of the quantities above mentioned or anything behaving 

 like them — say displacement, for example— let us write: 



/ = Fsin (nt - 8) = F sin ^p. (41) 



