300 BELL SYSTEM TECHNICAL JOURNAL 



In this very familiar form, 7^ stands for amplitude and n for lir times 

 the frequency, and 5 for something which is commonly called the 

 phase; bul it will be better to reserve this name for the entire argument 

 of the sine-function: 



Phase = <p = nt - b = s^rc sin (//F), (42) 



and I shall use it henceforth in this sense. 



Now, in general, both the amplitude F and the phase (p vary from 

 point to point — the latter because 5 is often a function of position. 

 Consequently / is a function of x, y, z and /; and it is immediately 

 important to find out whether/ satisfies the wave-equation. One who 

 is familiar chiefly with the standard one-dimensional case of waves on 

 a string is likely to think that this is true as a matter of course. How- 

 ever, on differentiating / twice with respect to x or ^^ or 2 and taking 

 due account of the dependence of both F and 8 upon these variables, 

 one sees directly that in general it is not true — not unless the functions 

 F and 8 conform to definite and sharply restrictive conditions.^ 



This appears a rather disconcerting result. However, if instead of 

 /we envisage///^ — the value of the displacement at each point referred 

 to its amplitude there as a unit, or the sine of the phase — it turns 

 out that the condition under which f/F obeys the wave-equation 

 is far less drastic. Forming the derivatives, we obtain: 



d^ f 



_^= -n'sm<p, (43) 



df" F 



V'^(VV)cos.-[(|.)V(^^)^(|y 



sin (f. (44) 



Evidently, in order that the function f/F shall conform to the wave- 

 equation, it suffices that 



VV = 0, (45) 



This condition is fulfilled by a variety of functions, Including all which 

 are linear in x, y, and z — the case of "plane waves," which as we shall 

 see is one of those permissible when the wave-speed is everywhere the 

 same, as I have been assuming. 



Next we will evaluate the speed of the phase-waves, and incidentally 

 we shall be led to a new aspect of wave-motion. When the phase 



' The general expression for \'-fis (y-F — F| v^P) sin <f> + i2vF-V<t> + ■f"V-<>) cos <f). 

 The coefficient of cos <j) must vanish, if / is to satisfy the wave-equation with real 

 phase-speed. By introducing the notion of "imaginary phase-speed" one may 

 continue to regard the function / as conforming to the wave-equation, even though 

 the coefficient of the cos-term does not vanish. 



