As pointed out above, the divergence of is zero, so the wave
equation reduces to
The particular geometry I am interested in is the initial condition of a toroidal magnetic flux loop, which is to say, a magnetic field loop situated on a plane, concentrated between a minor and major radius. The magnetic field is always toroidal (i.e., in the direction). Because of this symmetry, I will use cylindrical coordinates (,,). In this arrangement the flux loop is placed in the plane, with uniform magnetic field in the direction. This further simplifies the equations because, at least initially, and are zero. Also, we can assume that derivatives with respect to are zero because the field is azimuthally uniform.
Standard calculus and physics textbooks contain the formulae for
differential operators expressed in cylindrical coordinates. The curl
The evaluation of this formula appears at first glance to be very difficult, but because of the symmetry in the initial conditions, we can set all and components, and all terms to zero. This leaves only two terms remaining after performing the curl operation twice:
Substituting back into the modified wave equation
(8), I arrive at this differential equation,