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A smooth Maxwell eigenmode in a rectangular PEC cavity

Example

Let $Ω = (0, a) \times (0, b) \times (0, d)$ with $a, b, d > 0$ , constant scalar $ϵ, μ > 0$ , zero conductivity, zero impressed current $J_{s r c} = 0$ , and zero volume charge, and choose positive integers $m, n$ and a real amplitude $E_{0} \neq 0$ . Put $k_{x} = m π / a$ , $k_{y} = n π / b$ , $ω = (ϵ μ)^{- 1 / 2} \sqrt{k_{x}^{2} + k_{y}^{2}}$ , and $Φ = E_{0} \sin (k_{x} x) \sin (k_{y} y)$ . Define the real fields $E = (0, 0, Φ \cos (ω t)), B = \frac{E_{0}}{ω} (- k_{y} \sin (k_{x} x) \cos (k_{y} y), k_{x} \cos (k_{x} x) \sin (k_{y} y), 0) \sin (ω t) .$

Facts & Assumptions

Given: The constant-coefficient source-free lossless Maxwell equations and the PEC condition $n \times E = 0$ .

[F1]

In a homogeneous source-free lossless medium, $ϵ E_{t} = curl (μ^{- 1} B)$ , $B_{t} = - curl E$ , and PEC means zero tangential electric trace. (Transient Maxwell system with piecewise materials and boundary conditions)

Verification

1.1F1givenalgebra

The tangential electric field vanishes on $x = 0, a$ and $y = 0, b$ because $Φ = 0$ there; it also vanishes on $z = 0, d$ because the only component of $E$ is normal to those faces. Direct differentiation gives $div B = 0$ and $div (ϵ E) = 0$ . Thus both divergence constraints and the PEC trace hold.

2.1F1givenalgebra∎

Differentiation gives $B_{t} = - curl E$ . The $z$ component of $curl B$ is $- (k_{x}^{2} + k_{y}^{2}) Φ \sin (ω t) / ω$ , with the other two components zero. Since $ϵ ω^{2} = (k_{x}^{2} + k_{y}^{2}) / μ$ , the expression $- (k_{x}^{2} + k_{y}^{2}) Φ \sin (ω t) / (μ ω)$ equals $- ϵ ω Φ \sin (ω t) = ϵ E_{t}$ . Hence $ϵ E_{t} = curl (μ^{- 1} B)$ . At $t = 0$ , $x = a / (2 m)$ and $y = b / (2 n)$ , the electric component is $E_{z} = E_{0} \neq 0$ , so this is a nonzero cavity mode. The angular frequency is therefore $ω = (ϵ μ)^{- 1 / 2} \sqrt{(m π / a)^{2} + (n π / b)^{2}}$ .

Remarks

The same formulas with $E_{0} = 0$ give the zero Maxwell solution, which is not an eigenmode.

Depends on

Transient Maxwell system with piecewise materials and boundary conditions

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

John B. Schneider, Understanding the Finite-Difference Time-Domain Method (standard reference, not scraped)