TEモードの円形導波菅の半径\(a\)の管壁\(C\)上での境界条件は， \begin{gather} \left. \frac{\partial \Psi^\mathrm{TE}}{\partial \rho} \right| _{\rho =a} = 0 \end{gather} よって， \begin{gather} J_m '(k_c a) = 0, \ \ \ \ \ k_c = \frac{\chi '_{mn}}{a} \end{gather} ただし，\(\chi '_{mn}\) は\(J_m'(\chi) = 0\) を満たす\(n\)番目の零点を示す．
境界条件より，TEモードのスカラ関数\(\psi^\mathrm{TE}\)は， \begin{gather} \psi^\mathrm{TE} = \Psi^\mathrm{TE} (\rho,\phi) \mathcal{Z}^\mathrm{TE}(z) \end{gather} ここで， \begin{eqnarray} \Psi^\mathrm{TE} &=& A_{[mn]} J_m \left( \frac{\chi '_{mn}\rho}{a} \right) \begin{matrix} \sin \\ \cos \end{matrix} \ m \phi \\ \mathcal{Z}^\mathrm{TE} &=& e^{-jk_{z,[mn]}z} \end{eqnarray} ただし，\(m=0\) のとき，\(\sin m\phi = 0\) ゆえ，上式の下側のみである．また，\(A_{[mn]}\)はTE\(_{mn}\)モードの正規化係数を示す．ここで， \begin{eqnarray} k^2 &=& \left( \frac{\chi '_{mn}}{a} \right)^2 + k_{z,[mn]}^2 \nonumber \\ &\equiv& k_{c,[mn]} ^2 + k_{z,[mn]}^2 \end{eqnarray} ただし， \begin{gather} k_{c,[mn]} = \frac{\chi '_{mn}}{a}, \ \ \ \ \ J_m' \left( \chi'_{mn} \right) =0 \end{gather} このとき，伝搬定数\(\gamma _{[mn]}\)は， \begin{eqnarray} \gamma _{[mn]} &=& jk_{z,[mn]} \nonumber \\ &=& \left\{ \begin {array}{ll} j\beta _{[mn]} = j\sqrt{k^2-k_{c,[mn]}^2} & (k > k_{c,[mn]}) \\ \alpha _{[mn]} = \sqrt{k_{c,[mn]}^2-k^2} & (k < k_{c,[mn]}) \end{array} \right. \end{eqnarray} また，管内波長\(\lambda_{g,[mn]}\)（伝搬モード），遮断波長\(\lambda_{c,[mn]}\)は， \begin{eqnarray} \lambda_{g,[mn]} &=& \frac{2\pi}{\beta_{[mn]}} \\ \lambda_{c,[mn]} &=& \frac{2\pi}{k_{c,[mn]}} = \frac{2\pi}{\displaystyle{\frac{\chi '_{mn}}{a}}} \nonumber \\ &=& \frac{2\pi a}{\chi '_{mn}} \end{eqnarray}
磁界のモード関数\(\boldsymbol{h}_{[mn]}\)は， \begin{eqnarray} \boldsymbol{h}_{[mn]} &=& - \nabla _t \Psi^{\mathrm{TE}} \nonumber \\ &=& -\left( \frac{\partial \Psi^{\mathrm{TE}}}{\partial \rho} \boldsymbol{a}_\rho + \frac{1}{\rho} \frac{\partial \Psi^{\mathrm{TE}}}{\partial \phi} \boldsymbol{a}_\phi \right) \nonumber \\ &=& -A_{[mn]} \left[ \frac{d}{d\rho} \left\{ J_m \left( \frac{\chi'_{mn} \rho}{a} \right) \right\} \begin{matrix} \sin \\ \cos \end{matrix} \ m \phi \ \boldsymbol{a}_\rho \right. \nonumber \\ &&\left. + \frac{1}{\rho} J_m \left( \frac{\chi'_{mn} \rho}{a} \right) \frac{\partial}{\partial \phi} \left( \begin{matrix} \sin \\ \cos \end{matrix} \ m \phi \right) \boldsymbol{a}_\phi \right] \nonumber \\ &=& A_{[mn]} \left[ -\frac{\chi'_{mn}}{a} J_m' \left( \frac{\chi'_{mn} \rho}{a} \right) \begin{matrix} \sin \\ \cos \end{matrix} \ m \phi \ \boldsymbol{a}_\rho \right. \nonumber \\ &&\left. \mp \frac{m}{\rho} J_m \left( \frac{\chi'_{mn} \rho}{a} \right) \begin{matrix} \cos \\ \sin \end{matrix} \ m \phi \ \boldsymbol{a}_\phi \right] \nonumber \\ &\equiv& h_{\rho,[mn]} \boldsymbol{a}_\rho + h_{\phi,[mn]} \boldsymbol{a}_\phi \end{eqnarray} また，電界のモード関数\(\boldsymbol{e}_{[mn]}\)は， \begin{eqnarray} \boldsymbol{e}_{[mn]} &=& \boldsymbol{h}_{[mn]} \times \boldsymbol{a}_z \nonumber \\ &=& \left( h_{\rho,[mn]} \boldsymbol{a}_\rho + h_{\phi,[mn]} \boldsymbol{a}_\phi \right) \times \boldsymbol{a}_z \nonumber \\ &=& -h_{\rho,[mn]} \boldsymbol{a}_\phi + h_{\phi,[mn]} \boldsymbol{a}_\rho \nonumber \\ &=& h_{\phi,[mn]} \boldsymbol{a}_\rho -h_{\rho,[mn]} \boldsymbol{a}_\phi \nonumber \\ &=& A_{[mn]} \left[ \mp \frac{m}{\rho} J_m \left( \frac{\chi'_{mn} \rho}{a} \right) \begin{matrix} \cos \\ \sin \end{matrix} \ m \phi \ \boldsymbol{a}_\rho \right. \nonumber \\ &&\left. +\frac{\chi'_{mn}}{a} J_m' \left( \frac{\chi'_{mn} \rho}{a} \right) \begin{matrix} \sin \\ \cos \end{matrix} \ m \phi \ \boldsymbol{a}_\phi \right] \nonumber \\ &\equiv& e_{\rho,[mn]} \boldsymbol{a}_\rho + e_{\phi,[mn]} \boldsymbol{a}_\phi \end{eqnarray}