# Acoustic Parametric Signal Generation for Underwater Communication
## 4 Results
### Amplitude Ratio Between the Secondary and Primary Beams  
In this section we compare the amplitude ratio between the secondary beam and the primary beam \(A_{s}/A_{p}\) obtained experimentally with a theoretical model referenced in [2].  
A 0.28% direct ratio has been measured for the pressure amplitude of both beams, 12,940 Pa for the primary beam and 36.8 Pa for the secondary one.  
With respect to the theoretical predictions, using the first approach, the equation of the complex parametric gain \(G\) (dB) is used, which is defined as a value whose magnitude is the ratio between the primary and secondary beam pressures [2], where.  
\[g\equiv rP(r,0)/R_{o}P_{o} \tag{13}\]  
Figure 8: Directivity pattern of the primary and secondary beams.  
Figure 9: Influence of the noise in the secondary beam for the string of bits (experimental) and the ones adding some white noise of different amplitude: 2, 5, 7 and 10 mV. (a) Directivity; (b) Relative amplitude; (c) Reconstructed bits.  
and in decibels:  
\[G=20\log\lvert g\rvert \tag{14}\]  
where \(r\) is the observer's radial coordinate, \(P(r,0)\) secondary beam pressure at a point \(r\) centered on the emitter (Pa), \(R_{o}=A_{o}f_{o}/c\) is the Rayleigh length (m), \(P_{o}\) the peak face pressure amplitude of the primary component (Pa) at the Rayleigh distance.  
According to the specifications of the emitters and the calibrations done, for our case we have \(g\approx 0.0028\) and \(G\equiv-51\) dB, thus obtaining an amplitude ratio between the secondary beam and the primary beam of 0.28%, so fully agreeing with the measurements. This gain \(G\) corresponds to a value of \(L_{o}^{*}=245\) dB rms source level of the primary component and \(\overline{\alpha}R_{o}\approx 10^{-3}\) dB (0.0012 dB).  
\[Lo=20\log\left(P_{o}R_{o}/\sqrt{2}\right)\text{, and }L_{o}^{*}=L_{o}+20\log f_{o}( \text{db }//1\ \upmu\text{Pa m kHz}) \tag{15}\]  
According to the parametric-gain curves, Figure 2 of Ref. [2], we effectively find values close to \(10^{-3}\) for \(\overline{\alpha}R_{o}\) and \(L_{o}^{*}\ \sim\ 240\) in the shown curves.  
Moreover, by applying Equation (1), the measured sweep is studied with a set of parametric sine waves at the limit frequencies and a centered one; that is, 4 kHz, 20 kHz and 40 kHz frequencies are chosen. The ratios between the secondary beam and the primary beam obtained are 0.141%, 0.35% and 1.41%, respectively, with a primary beam pressure of 12,940 Pa. As expected, the closest ratio to the experimental one is the one for 20 kHz sine wave, which is about the average frequency of the parametric sweep emitted.  
Finally, the measurement is also contrasted according to the operating regime for parametric sources. Following ref. [2], our case is in the regime of absorption limiting in the far spherical zone:  
\[X\ll 1,\quad 2\alpha R_{o}f_{o}/f\ll 1 \tag{16}\]  
where \(\alpha(\text{Np}/\text{m})\) is the absorption, \(R_{o}(m)\) Rayleigh length, \(f_{o}\) (Hz) carrier frequency and \(f\) difference frequency 4, 20 and 40 kHz. Applying the Equation (16) for \(2\alpha R_{o}f_{o}/f\) we obtain 0.014, 0.0028 and 0.0014, respectively, and for \(X\) 0.006, so all values are much smaller than one.  
Subsequently, the parameter \(g\) is calculated through the following equation:  
\[\lvert g\rvert=\frac{X}{2}\ \frac{f}{f_{o}}\ E_{1}\bigg{(}\frac{2\alpha R_{o}f_{ o}}{f}\bigg{)} \tag{17}\]  
where \(E_{1}\) is the exponential integral function. We obtain 0.0002 for 4 kHz, 0.0015 for 20 kHz and 0.0035 for 40 kHz and applying Equation (14) we have \(-73.36\) dB \(-56.27\) dB and \(-49.18\) dB, respectively, validating that for the emitted frequency of 20 kHz the value of \(G=-56.27\) dB is similar to the first analysis studying the pressure received from the primary and secondary beams with \(G=-51\) dB.