### Frequency Spectrum of Pulse  
The Fourier transform of Equation (1) is  
\[ \begin{array}{ll}E(\omega,0)&=2E_{c}\omega_{\mathrm{c}}\frac{\sin\omega N \frac{T_{\mathrm{c}}}{T}}{\omega^{2}-\omega_{\mathrm{c}}^{2}}\\ &=2E_{c}\omega_{\mathrm{c}}\frac{\sin(\omega_{\mathrm{c}}^{2})}{\omega^{2}- \omega_{\mathrm{c}}^{2}}\end{array}  \tag{2}\]  
where \(\mathrm{T_{c}}\) is the period of the wave. The magnitude spectrum \(|\)E(\(\omega\),0)\(|\) at the surface consists of a main lobe at 10 GHz, with an infinite number of sidelobes spaced 1 GHz apart that progressively decrease in magnitude with increasing distance from the main lobe. The attenuation of different frequency components of the pulse can be analyzed in terms of the frequency components of the wave packet. We consider a plane electromagnetic wave of radian frequency \(\omega\) as it propagates through an absorbing medium. The electric field E (\(\omega\),z) of the wave at depth z is related to that at the surface E (\(\omega\),0) via  
\[E(\omega,z)=E(\omega,0)e^{-(\alpha+j\beta)z}e^{j\omega t} \tag{3}\]  
where  
\[\alpha+j\beta=\frac{j\omega\sqrt{\varepsilon^{*}}}{c} \tag{4}\]  
and \(\alpha\) and \(\beta\) are, respectively, the attenuation and phase constants of the medium, \(c\) is the velocity of light in vacuum, and \(\varepsilon^{*}\) is the complex permittivity of water. We consider related quantities instead: the phase velocity \(\mathrm{v}=\omega/\beta\) and the power penetration depth \(\mathrm{L}=1/(2\alpha)\)(the depth at which the power density of the wave has decreased by a factor \(1/\mathrm{e}\) compared to that at the surface). For water at room temperature at frequencies below \(\approx\)100 GHz, \(\epsilon^{*}\) is closely described using the Debye function  
\[\varepsilon^{*}=\varepsilon_{\infty}+\frac{\varepsilon_{s}-\varepsilon_{\infty} }{1+j\omega/\omega_{c}} \tag{5}\]  
where \(\varepsilon_{\mathrm{s}}\) = 78.6, \(\varepsilon_{\infty}\) = 5.4, and \(\omega_{\mathrm{c}}\) = 2\(\pi\)\(\bullet\)(19 GHz) rad/s at room temperature. The wave velocity and power attenuation depth for water are shown in Figure 1.  
Figure 2 shows the magnitude spectrum of the 1 ns \"ultrashort\" pulse as described by [PERSON] et al. [5; 6] (10 cycles at 10 GHz) with rectangular modulation given in Equation (1). The main lobe is at 10 GHz, corresponding to the carrier frequency, and the lowest frequency sidelobe is close to 0.5 GHz. For water, the power penetration depth (L) ranges from 44 cm for the lowest frequency sidelobe to 1.3 mm at the carrier frequency, decreasing further at higher frequencies. Figure 1: Power penetration depth (**a**) and phase velocity (**b**) of a plane wave propagating through pure water. Figure 2: Frequency spectrum (magnitude) of a 1 ns, square-wave-modulated RF-EMF pulse with carrier frequency of 10 GHz propagating in pure water at z = 0 (surface) () and z = 1 cm below the surface (\(\cdots\)). The wave is normally incident on the surface, and 100% transmission into the medium is assumed. The power penetration depth of a frequency component at 3 GHz in pure water is about 1 cm, meaning that higher frequency components are significantly attenuated even at a depth of 1 cm beneath the surface. Thus, as the pulse propagates through the medium, higher frequency components will be increasingly filtered out of the wave packet, and the evolution of the pulse will increasingly be determined by the attenuation characteristics of its lowest frequency components. The \"deeper penetration depths\" referred to by [PERSON] and [PERSON], and previously by [PERSON], refer to the comparatively deeper penetration of the lowest frequency components of the wave packet, which represent only a minor fraction of the total energy in the packet. Cellular and other communications waveforms are strictly band-limited and consequently lack such \"deeply penetrating\" components. All frequency components in a 5G NR signal are contained within the channel bandwidth and consequently their power penetration depths are similar to that of the carrier.