The TM and TE modes of a hollow-tube waveguide are determined by the two-dimensional Helmholtz equation [∇²_⊥ + γ²]ψ = 0 with boundary conditions ψ|_S = 0 and ∂ψ/∂n|_S = 0, respectively. The same equation and boundary conditions apply when ψ(x, y) is the wave function of a free particle in a two-dimensional box with infinite or finite potential walls, and when ψ(x, y) is the vibrational amplitude of a drumhead whose perimeter is held fixed or left free.
(a) Produce an argument based on the behavior of a quantum particle-in-a-box to argue that one type of mode (either TE or TM) always has the lowest (non-zero) frequency for a hollow-tube waveguide with an arbitrary cross sectional shape.
(b) Produce an argument based on the behavior of an elastic drumhead to reach the same conclusion as in part (a).

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Hollow shell Flexible membrane