## 4 Analysis of Numerical Results
### Distilled Water Resonator
The results for the half-cylinder-based geometries reveal that the WPA inverted half-cylinder, as shown in Figure 5a, exhibits additional notches in \(|S_{11}|\), enabling multiple resonant frequencies. Moreover, all LPA half-cylinder shapes reduce the resonant frequency, indicating their potential for reducing the patch antenna size. The WPA inverted hollow \begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multicolumn{1}{c}{**Geometry**} & \(|\mathbf{S_{11}}|\) & **BW** & **MLM** & **HPBW** & **SLL** \\  & **(dB)** & **(MHz)** & **(dBi)** & **(°)** & **(dB)** \\ \hline Patch antenna & \(-34.30\) & \(60\) & \(2.79\) & \(148.2\) & w/o \\ \hline Rectangular prism & \(-17.03\) & \(55\) & \(3.73\) & \(132.4\) & w/o \\ \hline Hollow rectangular prism & \(-0.86\) & w/o & \(-3.94\) & \(151.4\) & \(-6.2\) \\ \hline Cylindrical hemisphere & \(-14.82\) & \(57\) & \(3.39\) & \(142.6\) & w/o \\ \hline Concave hemisphere & \(-14.46\) & \(29.04\) & \(4.1\) & \(88.1\) & \(-1.4\) \\ \hline Hollow concave hemisphere & \(-10.57\) & \(18\) & \(3.04\) & \(111.4\) & \(-6.2\) \\ \hline Hemisphere & \(-12.41\) & \(1\) & \(-1.46\) & \(48.5\) & \(-1\) \\ \hline Inverted cone & \(-22.06\) & \(34\) & \(5.6\) & \(95.4\) & \(-20.9\) \\ \hline Hollow inverted cone & \(-23.50\) & \(49\) & \(4.62\) & \(119.8\) & \(-10\) \\ \hline Cone & \(-14.44\) & \(55\) & \(3.37\) & \(147\) & w/o \\ \hline LPA Inverted half-cylinder & \(-7.28\) & w/o & \(3.21\) & \(148.4\) & w/o \\ \hline LPA hollow inverted half-cylinder & \(-1.85\) & w/o & \(-2.39\) & \(139.6\) & w/o \\ \hline LPA half-cylinder & \(-4.04\) & w/o & \(0.49\) & w/o & w/o \\ \hline WPA inverted half-cylinder & \(-5.90\) & \(16\) & \(4.07\) & \(116.7\) & \(-8.8\) \\ \hline WPA hollow inverted half-cylinder & \(-12.26\) & \(54\) & \(2.38\) & \(143\) & w/o \\ \hline WPA half-cylinder & \(-11.02\) & \(44\) & \(3.32\) & \(144.7\) & w/o \\ \hline LPA inverted trapezoidal prism & \(-12.44\) & \(51\) & \(5.12\) & \(104.5\) & \(-20.3\) \\ \hline LPA hollow inverted trapezoidal prism & \(-16.10\) & \(52\) & \(5.22\) & \(85.7\) & \(-8.1\) \\ \hline LPA trapezoidal prism & \(-5.72\) & w/o & \(-0.02\) & \(84\) & w/o \\ \hline WPA inverted trapezoidal prism & \(-7.47\) & \(9\) & \(4.73\) & \(117.2\) & \(-9.5\) \\ \hline WPA hollow inverted trapezoidal prism & \(-2.87\) & w/o & \(0.89\) & \(119.5\) & \(-5.4\) \\ \hline WPA trapezoidal prism & \(-12.65\) & \(40\) & \(3.9\) & \(133.7\) & w/o \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Patch antenna with a DW-made resonator: Key parameters values at the resonant frequency. Figure 7: Patch antenna with a DW-made resonator: the radiation pattern plot of the 1D far-field realized gain (\(\phi=0\)). (**a**) half-cylinder-based; and **(b) trapezoidal prism-based. half-cylinder also presents a BW close to the patch antenna without a resonator. For the radiation pattern, as illustrated in Figure 7a and summarized in Table 4, the WPA and LPA inverted half-cylinders and WPA half-cylinder achieve MLM values higher than the patch antenna without resonator. The WPA inverted half-cylinder significantly enhances directionality, achieving substantial reductions in both HPBW and SLL compared to the patch antenna without a resonator. Moreover, the LPA half-cylinder geometry causes a notable alteration in MLD, changing its value from 0\({}^{\circ}\) to 90\({}^{\circ}\), resulting in a decline in performance. Finally, for trapezoidal prism-based geometries, it is evident that most of these geometries reveal additional notches in \(|S_{11}|\), enabling multiple resonant frequencies, except for the WPA hollow inverted trapezoidal prism, shown in Figure 5b. Furthermore, all trapezoidal prism-based geometries show potential for antenna size reduction at lower frequencies, particularly with the LPA half-cylinder.