I can further eliminate some other variables. First, I know that the
field is symmetric about the -plane, so must be zero. If
were not zero, the function would be asymmetric about .
Similarly, I assert that at time , the magnetic field is maximum,
so must also be zero (this also follows because there is no charge
or current in the system). Thus, a solution of the original partial
differential equation, with the known symmetry conditions, is
Equation (22) can be considered a standing wave, or eigenmode, solution to Maxwell's equations for the toroidal flux loop initial condition. In other words, this solution provides waves are fixed in space and that oscillate for infinite time. However, it is not the same kind of standing wave that Schallger refers to. Specifically, this solution has waves that extend infinitely throughout space, so they do not reflect (as yet) a concentrated flux loop bundle. Schallger conjectures that the magnetic flux can be contained in a tight loop on the plane and essentially nowhere else. Thus, at this step we cannot verify that a flux loop is a valid standing wave solution.
However, we can take the analysis further. Equation (22)
is only one solution of an infinite set of solutions. Any
combination of and will provide valid solutions.
, so is not an
independently choosable variable). Also, because the partial
differential equation is linear, any sum of solutions will also
be a solution. Thus, the most general solution is of the form:
Thus, the problem is essentially solved, with the only really difficult thing remaining to accomplish is to determine the constants to match the initial conditions of a flux loop.