530 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. 



and inserting these in the equation (5) we have 



f"(R-at) f'(R-at) f(R-at) 



~7R~ ~W~ ~W~ : - 



This is an ordinary differential equation, which, if we put 



u = R at, 

 becomes 



This has the solution /= e Xtt , where 



00 x.-Jx + i-o. x= 



From this we obtain the general solution 



d cos ^TT-* u + M sm ^^ u 



r-at 



). 



representing a damped harmonic spherical wave of wave-length 



Z=47r#/\/3 = 7-255^. 

 The logarithmic decrement is 



7T/V3 = 1-814, 



so that the oscillation almost ceases after a complete vibration. 

 This extreme damping is due to the radiation of the energy, and 

 not at all to the dissipation in the conductor, of which we have 

 taken no account. 



The nature of the field radiated by an oscillation of this sort 

 has been discussed by Hertz*, making certain assumptions. 



The preceding problem is the simplest that can be proposed 

 to represent a practical case of oscillations in a conductor. The 

 above demonstration is given by Poincare'. It is evident, from 

 the investigation of oscillations in the last chapter, that a system 

 has as many possible periods as there are degrees of freedom. In 

 a conductor of three dimensions the currents have an infinite 



* Hertz. "Die Krafte der elektrischen Schwingungen, behandelt nach der 

 Maxwell'schen Theorie." Wied. Ann. 36, p. 1, 1889. Translation, p. 137. 



