CONTEMPORARY ADVANCES IN PHYSICS 677 



values! Such equations need not be more difficult to solve than the 

 conventional wave-equation in which u? stands for a positive constant; 

 but the image of the elastic medium becomes unsatisfying. In the 

 one-dimensional case, so long as «" remains a positive function of x, 

 one can visualize a string of which the density varies along its length ; 

 but when it- passes through zero and becomes negative, the wave- 

 speed attains zero and is superseded by an imaginary quantity. One 

 may speak, in such a case, of a "string" or a "fluid" characterized 

 by an "imaginary wave-speed." So speaking, one comes perilously 

 close to the verge of using words devoid of physical meaning; but 

 otherwise, there is no verbal language with which to relieve the mon- 

 otony of the procession of equations. 



The differential equation of the type of (151), with a constant 

 negative value of the coefficient u^, is not a difficult one. Confining 

 ourselves to one dimension, we find for one of the solutions of the 

 equation for a "string with constant imaginary wave-speed" this 

 expression : 



^ = {A cos mUt ^ B sin mUt)(Ce"'' -\- De-"""), (152) 



in which U stands for the (real) square root of — «^. This is a much 

 less tractable function than the product of sine-functions which serves 

 when M^ is positive. One cannot, for instance, find Eigenwerte for 

 the constant m whereby the function can be made to vanish at all 

 times at two distinct points upon the "string"; or rather, one can 

 find only the value m = 0, which fulfils this familiar boundary- 

 condition by destroying the function. Similarly, one cannot force 

 ^ to remain finite everywhere except by annulling either m or else 

 both A and B, again destroying the function. Vibrations which are 

 sine-functions of time are, however, permitted by the differential 

 equation. 



Consider now the equation 



(Pyldx'' = (a - bx^)d^yldt^ (153) 



which may be regarded as the wave-equation of a string of which the 

 wave-speed varies with x along its length as the function (a — bx^)~^/^, 

 being therefore real over the central part from x = — -"^ajb to 

 X = + Va/^, and imaginary from each extremity of this central 

 range outward to infinity. In the usual way, we derive the equations: 



y = f{x)si^)' § = ^ cos vt -\- B sin vt, 

 dj/dx'" + v^(a - bx^)f = dj/dx^ + (C - .r^)/ = 0, (154) 



and it is incumbent upon us to solve the equation ^' for/(x). 



" The constant v^b has been equated to unity, which entails no loss of generality. 



