## 4 Fixed Wavelength Versus Fixed Frequency
Additionally, such modeling tools are well suited for assessing antenna geometry, bandwidth with respect to conductor diameter and impact of the various dielectric permittivities of soil and air when the sensor is buried at depths less than a wavelength. The selected implementation of FDTD is gprMax3D [45], an open source software allowing for antenna impedance characterization (Figure 3, top). A thin-wire dipole geometry exhibits a bandwidth which is too small to efficiently couple with a SAW delay line: the means of widening the operating bandwidth classically involves broadening the radiating element, so we investigated the impact of the radiating element diameter on the bandwidth and impact on the antenna impedance with the aim of meeting the SAW impedance complex conjugate matching condition (Figure 3, bottom). Attempts at replacing the dipole with a bow tie geometry were met with simulation challenges for properly modeling the triangularly-shaped radiating element: impedance results were observed to be dependent on the mesh size of the FDTD grid, making results hardly reliable. Nevertheless, the poor efficiency of the broadband bow tie with respect to the dipole is observed, with hardly any energy coupling within the SAW bandwidth. Since the electromagnetic wave velocity shrinks to the same extent as the square root of the medium permittivity, operating in media whose water content strongly varies--soil, sand--inducing strong permittivity variations [46], already presents a challenge of matching the antenna transfer  
Figure 3: GprMax3D (www.gprmax.com) modeling of antennas using FDTD: (**a**) is a typical geometry, here of a dipole antenna with square cross-section, and (**b**) is the simulation volume excluding the perfectly-matching layer cells on the sides. \(S_{11}\) (**c**) and impedance (**d**) reflection coefficients of various antenna geometries discussed in the text are exhibited, as well as the same characteristics for the SAW transducer acting as antenna load: bottom chart colors match the legend of the top charts. The soil permittivity was set to four with no imaginary part, and the dipole length halved with respect to its dimensions in air when simulated in soil. The imaginary part of the sensor impedance is displayed on the same chart as the opposite of the imaginary part of antenna impedances: matching conditions require that the complex conjugate of the antenna impedance equals the impedance of the sensor in its operation bandwidth. function with the sensor transfer function. Indeed, despite being commercially defined by an operating frequency, GPR antennas actually act at constant wavelength and thus varying frequency. The surface dipole is subject to varying soil permittivities, which vary the central frequency of the broadband pulse filtered by the fixed geometry dipole whose electromagnetic length varies with permittivity. The condition is worse for the buried sensor, where again the antenna geometry is fixed, but the surrounding permittivity varies. More reasonable than the permittivity variation of sand, concrete only exhibits minor permittivity changes [47] with moisture content, keeping the dipole transfer function within the operating band of the SAW transducer. Nevertheless, the antenna mismatch under varying environmental conditions remains an issue even if tuning the antenna properties to match the SAW transducer in a given condition.