gemmy94
Junior Member level 3
I am estimating the scattering field from these below integrals by asymptotic approximation (saddle point method) for H and E polarizations, respectively.
\begin{align}
I_{\text{H}} &= 2jka\int_{-\pi/2}^{\pi/2}\cos({\varphi + \phi_0 - \phi}) e^{jka[\cos{\varphi} + \cos({\varphi + \phi_0 - \phi})]} \ d\varphi. \\
I_{\text{E}} &= -2jka\int_{-\pi/2}^{\pi/2}\cos{\varphi}~e^{jka[\cos{\varphi} + \cos({\varphi + \phi_0 - \phi})]} \ d\varphi.
\end{align}
\[ \phi_0 \] is already known.
Then the results are compared with one by using Simpson rule. The difference between two methods is calculated in the range of \[\phi = [0^{\circ},360^{\circ}]\] as following
\begin{align}
\text{Error} = \big|\text{uI} - \text{nI}\big|^2,
\end{align}
in which \[\text{uI}\] is the result estimated by asymptotic solution, \[\text{nI}\] is the result estimated by Simpson rule. The \[\text{Error}\] values for H and E cases as in the figure.
The figure shows the average value for the range \[\phi = [0^{\circ},360^{\circ}]\] with each \[ka\].
\begin{align}
\text{Error}_{average} = \frac{\text{Error}_{0^{\circ}}+\text{Error}_{1^{\circ}}+...+\text{Error}_{360^{\circ}}}{361}
\end{align}
I have no idea why different integrals give us the same \[\text{Error}\] as calculating following the third equation.
I have checked the data and see that for each value of observation angle \[\phi\] the \[\text{Error}\] between two polarization are always opposite in sign.
For example
\begin{align}
\text{Error}_H = \text{uI}_H - \text{nI}_H &= a + bj, \\
\text{Error}_E = \text{uI}_E - \text{nI}_E &= -a - bj.
\end{align}
Could someone please explain for me why the \[\text{Error}\] between H and E polarizations are always like that.
Thank you very much!
\begin{align}
I_{\text{H}} &= 2jka\int_{-\pi/2}^{\pi/2}\cos({\varphi + \phi_0 - \phi}) e^{jka[\cos{\varphi} + \cos({\varphi + \phi_0 - \phi})]} \ d\varphi. \\
I_{\text{E}} &= -2jka\int_{-\pi/2}^{\pi/2}\cos{\varphi}~e^{jka[\cos{\varphi} + \cos({\varphi + \phi_0 - \phi})]} \ d\varphi.
\end{align}
\[ \phi_0 \] is already known.
Then the results are compared with one by using Simpson rule. The difference between two methods is calculated in the range of \[\phi = [0^{\circ},360^{\circ}]\] as following
\begin{align}
\text{Error} = \big|\text{uI} - \text{nI}\big|^2,
\end{align}
in which \[\text{uI}\] is the result estimated by asymptotic solution, \[\text{nI}\] is the result estimated by Simpson rule. The \[\text{Error}\] values for H and E cases as in the figure.
The figure shows the average value for the range \[\phi = [0^{\circ},360^{\circ}]\] with each \[ka\].
\begin{align}
\text{Error}_{average} = \frac{\text{Error}_{0^{\circ}}+\text{Error}_{1^{\circ}}+...+\text{Error}_{360^{\circ}}}{361}
\end{align}
I have no idea why different integrals give us the same \[\text{Error}\] as calculating following the third equation.
I have checked the data and see that for each value of observation angle \[\phi\] the \[\text{Error}\] between two polarization are always opposite in sign.
For example
\begin{align}
\text{Error}_H = \text{uI}_H - \text{nI}_H &= a + bj, \\
\text{Error}_E = \text{uI}_E - \text{nI}_E &= -a - bj.
\end{align}
Could someone please explain for me why the \[\text{Error}\] between H and E polarizations are always like that.
Thank you very much!
