Question

(a) Suppose that $\nabla^2 \psi_p = \lambda_p \psi_p$ and $\nabla^2 \psi_q = \lambda_q \psi_q$ in a two-dimensional domain A where either $\psi|_{C} = 0$ or $\partial \psi/\partial n|_{C} = 0$ on the perimeter C of A. Use one of Green's identities to show that $\int_A d^2 r \psi_p(\mathbf{r}) \psi_q(\mathbf{r}) = 0$ $\qquad (\lambda_p \neq \lambda_q)$. (b) Let $\mathbf{E}_p(\mathbf{r}, t) = \mathbf{E}_p(x, y)e^{i(h_p z - \omega t)}$ and $\mathbf{B}_p(\mathbf{r}, t) = \mathbf{B}_p(x, y)e^{i(h_p z - \omega t)}$ be the fields associated with the pth mode of a cylindrical waveguide with cross sectional area A. When $p \neq q$ and both modes are TE, use the results of part (a) to show that non-degenerate modes satisfy $\int_A d^2 r \mathbf{E}_p \cdot \mathbf{E}_q = 0 = \int_A d^2 r \mathbf{B}_p \cdot \mathbf{B}_q$. (c) Repeat part (b) when both modes are TM. (d) Repeat part (b) when one mode is TE and one mode is TM.

          (a) Suppose that $\nabla^2 \psi_p = \lambda_p \psi_p$ and $\nabla^2 \psi_q = \lambda_q \psi_q$ in a two-dimensional domain A where either $\psi|_{C} = 0$ or $\partial \psi/\partial n|_{C} = 0$ on the perimeter C of A. Use one of Green's identities to show that
$\int_A d^2 r \psi_p(\mathbf{r}) \psi_q(\mathbf{r}) = 0$  $\qquad (\lambda_p \neq \lambda_q)$.
(b) Let $\mathbf{E}_p(\mathbf{r}, t) = \mathbf{E}_p(x, y)e^{i(h_p z - \omega t)}$ and $\mathbf{B}_p(\mathbf{r}, t) = \mathbf{B}_p(x, y)e^{i(h_p z - \omega t)}$ be the fields associated with the pth mode of a cylindrical waveguide with cross sectional area A. When $p \neq q$ and both modes are TE, use the results of part (a) to show that non-degenerate modes satisfy
$\int_A d^2 r \mathbf{E}_p \cdot \mathbf{E}_q = 0 = \int_A d^2 r \mathbf{B}_p \cdot \mathbf{B}_q$.
(c) Repeat part (b) when both modes are TM.
(d) Repeat part (b) when one mode is TE and one mode is TM.

(a) Suppose that ∇^2 = and ∇^2 = in a two-dimensional domain A where either ψ|C = 0 or ∂ψ/∂ n|C = 0 on the perimeter C of A. Use one of Green's identities to show that
d^2 r (𝐫) (𝐫) = 0 (≠).
(b) Let 𝐄p(𝐫, t) = 𝐄p(x, y)e^i(hp z - ω t) and 𝐁p(𝐫, t) = 𝐁p(x, y)e^i(hp z - ω t) be the fields associated with the pth mode of a cylindrical waveguide with cross sectional area A. When p ≠ q and both modes are TE, use the results of part (a) to show that non-degenerate modes satisfy
d^2 r 𝐄p ·𝐄q = 0 = d^2 r 𝐁p ·𝐁q.
(c) Repeat part (b) when both modes are TM.
(d) Repeat part (b) when one mode is TE and one mode is TM.

Added by Heidi B.

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