502 Displacement Currents [CH. xvn 



SOLUTIONS OF -^ = a 2 V 2 u. 



Solution for spherical waves. 



578. The general solution of the equation of wave-propagation is best 

 approached by considering the special form assumed when the solution u 

 is spherically symmetrical. If u is a function of r only, where r is the 

 distance from any point, we have 



d 2 u a? d ( du\ 



- = a 2 V 2 u = -T- I r 2 -=- } , 

 dt 2 r 2 dr \ dr) 



which may be transformed into 



jH=^? (535)> 



and the solution is 



where f and <> are arbitrary functions. 



The form of solution shews that the value of u at any instant over a 

 sphere of any radius r depends upon its values at a time t previous over 

 two spheres of radii r at and r + at. In other words, the influence of any 

 value of u is propagated backwards and forwards with velocity a. For 

 instance, if at time t = the value of u is zero except over the surface of 

 a sphere of radius r } then at time t the value of u is zero everywhere except 

 over the surfaces of the two spheres of radii r at ; we have therefore two 

 spherical waves, converging and diverging with the same velocity a. 



General solution (Liouville). 



579. The general solution of the equation can be obtained in the 

 following manner, originally due to Liouville. 



Expressed in spherical polars, r, 6 and <, the equation to be solved is 

 ! d?u 



Let us multiply by sin 6d6d$ and integrate this equation over the surface 

 of a sphere of radius r surrounding the origin. If we put 



the equation becomes 



i d*\ i d f a axx 



a 2 dP~v*dr \ T dr)' 



the remaining terms vanishing on integration. The solution of this equation 

 (cf. equation (536)) is 



r .................. (538). 



