# Acoustic Parametric Signal Generation for Underwater Communication
## 1 Introduction  
Communications in underwater environments have become a field of research of great interest in recent years. The transmission of information in underwater media can be based on acoustic systems, which present the advantage that the acoustic waves have lower absorption than the electromagnetic ones. However, the underwater acoustic channel has important limitations because of limited bandwidth, extended multipath, severe fading, and refractive properties of the medium. Therefore, it is quite difficult to have clean, direct, and private acoustic communication in underwater environments. To deal with some of these limitations, new methods of communication are proposed based on the non-linear parametric effect. With this technique, directive communication can be achieved by using directive high frequency transducers to produce a low-frequency secondary beam in the medium that can travel over long distances. With this, several advantages are foreseen: to communicate in the desired direction, being more robust against the wanted dissemination of information, or reducing reflections or multi-path effects that could worsen the quality of the communication.  
The non-linear parametric effect is observed when a high-intensity acoustic beam with given frequencies is propagated so that secondary frequencies (such as the addition or difference of the primary frequencies) are formed and also propagated. This was first studied by [PERSON] [1] and later developed and applied under different conditions [2; 3]. The parametric effect has become one of the most popular research areas in underwater acoustics for several decades with multiple applications.  
In general, if a modulated emitted wave has a high carrier frequency (primary beam), it self-interferes and is rapidly absorbed in the medium allowing the formation of low frequencies (secondary beam) that propagate at greater distances. As it is well known, it is easier to generatedirectional beams for high frequencies than for low ones, the latter being usually more omni directional like. However, one of the fundamental characteristics of the parametric effect is that low frequencies, when generated parametrically, have a rather narrow directivity, comparable to that of the primary beam [3].  
Theoretical studies have determined that the shape of the secondary beam signal is the second derivative of the square of the emitted signal envelope, its amplitude being proportional to the square of that of the primary beam. The waveform of the secondary beam is determined by the following equation:  
\[p(x,t)=\left(1+\frac{B}{2A}\right)\frac{p^{2}S}{16\pi\rho c^{4}ax}\ \frac{\delta^{2}}{\delta t^{2}}\ \left[f \Big{(}t-\frac{x}{c}\Big{)}\right]^{2}\sim\frac{\delta^{2}}{\delta t^{2}}\ f^{2} \tag{1}\]  
where \(S\) is the area of the vibrating surface of the transducer, \(f(t-x/c)^{2}\) is the envelope of modulation, \(x\) is the distance to the source and \(t\) is time, \(B/A\) is the nonlinearity parameter of the medium, \(\rho\) the density, \(c\) the velocity of sound and \(\alpha\) the absorption coefficient in the medium. Therefore, the resulting wave \(p(x,t)\) is proportional to the second derivative of the envelope of the emitted signal squared [4; 5; 6].