Forced Vibrations of Electromagnetic Systems. 381 



of waves, zonal harmonics are more useful, besides leading to 

 the solution of more practical problems. It is then not difficult 

 to generalize results to suit any kind of spherical harmonic. 



17. The simplest Spherical Waves. — Let the lines of H be 

 circles, centred upon the axis from which 6 is measured, and 

 let r be the distance from the origin. We have no concern 

 with <£ (longitude) as regards H, so that the simple specifi- 

 cation of its intensity H fully defines it. Under these 

 circumstances the equation (93) becomes 



= q 2 ~R, say, 



} • (95) 



where the acute accent denotes differentiation to r, and the 

 grave accent to cos 6 or /x, whilst v stands for sin 6. The 

 inductivity will be now /ul , to avoid confusing with the fi of 

 zonal harmonics. Equation (95) also defines q in the three 

 forms it can assume in a conductor, dielectric, and conducting 

 dielectric, 



Now try to make of rH an undistorted spherical wave, i. e. 

 H varying inversely as the distance, and travelling inward or 

 outward at speed v. Let 



rR = Af(r-vt) (96) 



where A is independent of r and t. Of course we must have 

 k = 0, making q—pjv, Now (96) makes 



v\r-R)"=r P m; (97) 



which, substituted in (95), gives 



<vHf = 0; (98) 



therefore 



Aj^A^ + Bi (99) 



From these we find the required solutions to be 



H=E/^= ^F„'(r-4 . . . (100) 



F = fi v^F (r-vt); (101) 



where F is any function, A x and B x constants, E and F the 

 two components of the electric force, F being the radial com- 

 ponent out, and E the other component coinciding with a line 

 of longitude, the positive direction being that of increasing 6, 

 or from the pole. Similarly, if the lines of E be circular 



