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Standing Wave Solution

With the original partial differential equation (11) separated into three separate ordinary differential equations, the problem becomes much simpler. Each of these equations has a solution in terms of known functions:
$\displaystyle T(t)$ $\textstyle =$ $\displaystyle A \cos(\omega t + a)$ (19)
$\displaystyle Z(z)$ $\textstyle =$ $\displaystyle B \cos(kz + b)$ (20)
$\displaystyle R(r)$ $\textstyle =$ $\displaystyle C J_1(\lambda r) + D Y_1(\lambda r)$ (21)

where $J_1(x)$ and $Y_1(x)$ are the Bessel functions of the first kind. I can discard the $Y_1(x)$ term because it diverges at $r=0$, which is not physical for this particular problem (it would be if there were a current passing along the $z$-axis).

I can further eliminate some other variables. First, I know that the field is symmetric about the $xy$-plane, so $b$ must be zero. If $b$ were not zero, the function would be asymmetric about $z=0$. Similarly, I assert that at time $t=0$, the magnetic field is maximum, so $a$ 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

B = (ABC)\cdot\cos(\omega t)\cdot\cos(kz)\cdot{J_1(\lambda r)}
\end{displaymath} (22)

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 $z=0$ 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 $k$ and $\lambda$ will provide valid solutions. (Recall that $\omega = c\sqrt{k^2 + \lambda^2}$, so $\omega$ 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:

B(r,z,t) = \sum_{ij}{c_{ij}\cos(\omega_{ij} t)\cos(k_i z)J_1(\lambda_j r)}
\end{displaymath} (23)

where $c_{ij}$ are constants determined by the initial conditions and
\omega_{ij} = c\sqrt{k_i^2 + \lambda^2_j}
\end{displaymath} (24)

Thus, the problem is essentially solved, with the only really difficult thing remaining to accomplish is to determine the constants $c_{ij}$ to match the initial conditions of a flux loop.

next up previous
Next: Determining the Initial Conditions Up: Maxwell's Equations and a Previous: Solution by Separation of
Craig Markwardt 2001-07-30