290 BELL SYSTEM TECHNICAL JOURNAL 



in P Sit t. Therefore, if there is any closed surface in space, every'where 

 over which the wave-motion is known for all times, it is possible to 

 compute the wave-motion at any point in the volume which that sur- 

 face encloses.* 



This is a feature common to all the familiar examples of wave- 

 motion, and it is suitable for a tentative basis for a general definition 

 of waves. 



To formulate it strictly, let 5 be used as the symbol for any quantity 

 which is propagated in waves. Examples of such a quantity are: 

 the twist of a taut and twisted wire — the lateral displacement of a 

 taut wire or a tense membrane — the excess of the pressure in the air 

 over its average value — a component of the electric field-strength 

 or the magnetic field-strength in a vacuum — the entirely imperceptible 

 and hypothetical entity denoted by "^ in wave-mechanics. 



We write 5 as a function of x, y, z, and /: 



5 = s{x, y, z, t). (1) 



Fewer than three dimensions of space will suffice in some cases (e.g., 

 those of the wire and the membrane) ; in certain problems of wave- 

 mechanics, more than three may be required ; but in dealing with sound 

 in air and light in vacuo, three are usually necessary and sufficient. 

 For the time being I will suppose that the speed of the waves is every- 

 where the same. Interesting things will happen when this assumption 

 is discarded. 



What I have loosely called "the state of affairs" in a point P{x, y, z) 

 at a moment / will involve the value of 5 at x, y, z, and /. Also it may 

 involve the first and higher derivatives of 5 with respect to space and 

 time, evaluated at x, y, z, t. Which of these derivatives we are re- 

 quired to know is something which might vary from case to case. For 

 the present, we may consider ourselves required to know 5 and its first 

 derivatives ds/dx, ds/dy, ds/dz, ds/dt. 



We are to evaluate 5 and its derivatives at a point P at a moment /, 

 in terms of the values which 5 and its derivatives possessed at certain 

 earlier moments over a surface 6* enveloping P. 



Let P be made the origin of our coordinate-system ; let x, y, z denote 

 the coordinates of the points on the surface S; let r denote the distance 

 from the origin to any of these points, so that: 



r- = x~ -\- y- + 2^. (2) 



Introduce as an auxiliary the function U, defined thus : 



* Naturally the surface must not be so drawn that it includes sources emitting 

 waves during the time-interval {f — i). 



