# On Modeling Eavesdropping Attacks in Underwater Acoustic Sensor Networks +
## 3 Transducers  
In UASNs, transducers are not only used to implement the conversion of acoustic energy into electrical energy or vice versa, but also used to transmit/receive acoustic signals. Transducers used as receivers are called hydrophones, and those used for transmitting are called transmitters or projectors [39]. In this paper, we focus on hydrophones. Conventional UASNs are commonly equipped with a single isotropic hydrophone which collects acoustic signals uniformly in all directional. Different from a single isotropic hydrophone, modern transducers are comprised of an array of hydrophones which receive signals more effectively in some directions (_i.e._, improving signal noise ratio in some directions). Moreover, with the evolution of VLSI, modern hydrophones are able to form beams in desired directions. Thus, this section will introduce these two types of hydrophones. An isotropic hydrophone collects acoustic signal uniformly in all directions. In particular, Figure 2 shows the beam pattern of an isotropic hydrophone. An array hydrophone consists of an array of isotropic hydrophones. To model the characteristics of an array hydrophone, we introduce _Array Gain_, which is the improved gain on signal-to-noise ratio (SNR) compared with an isotropic hydrophone. The array gain can be expressed as follows [39]:  
\[AG=\frac{\sum_{i=1}^{M}\sum_{j=1}^{M}a_{i}a_{j}\beta_{s}^{ij}}{\sum_{i=1}^{M} \sum_{j=1}^{M}a_{i}a_{j}\beta_{n}^{ij}} \tag{7}\]  
where \(M\) is the number of isotropic hydrophone elements of an array hydrophone, \(a_{i}\) and \(a_{j}\) are the amplitude shading coefficients of the \(i\)-th hydrophone element and the \(j\)-th hydrophone element, respectively. If array hydrophones are unshaded, \(a_{i}=a_{j}=1\). \(\beta_{s}^{ij}\) is the correlation of the received signals between the \(i\)-th hydrophone element and the \(j\)-th hydrophone element and \(\beta_{n}^{ij}\) is the correlation of the noise between the \(i\)-th hydrophone element and the \(j\)-th hydrophone element. One of most commonly used array hydrophones is an unshaded uniform linear array transducer, which consists of \(M\) isotropic hydrophones evenly spaced in a line, as shown in Figure 3. In this structure, \(l\) is the distance between two neighboring elements, and \(\lambda\) is the wavelength of acoustic  
Figure 2: The beam pattern of an isotropic hydrophone. wave radiated from signal source (denoted by the blue star in Figure 3). In underwater communication, a signal wave is usually considered as a plane wave, meaning that signals can be received by each hydrophone element in the same direction \(\theta\). Although plane waves cannot be generated in practice, both spherical and cylindrical waves approximate plane waves when they are sufficiently far from signal source [39]. Therefore, we consider acoustic signal as a plane wave in this paper. Given this unshaded uniform linear array hydrophone, \(\beta_{s}^{ij}\) is described as follows:  
\[\beta_{s}^{ij}=\cos(\frac{2\pi}{\lambda}\left|i-j\right|l\sin\theta) \tag{8}\]  
where \(\left|\cdot\right|\) denotes the absolute value. In this paper, we consider that the array hydrophone uses a location identification scheme to acquire the direction of signal source and then adjust their angles to transmitters. Thus, we have \(\theta=0\). Then we apply this value in Equation (8) and obtain the correlations of received signal \(\beta_{s}^{ij}=1\). In general, we consider omni-directional noise, meaning that noise surround each hydrophone with constant power in all directions. Then, the correlations of received noise \(\beta_{n}^{ij}\) is expressed by  
\[\beta_{n}^{ij}= \begin{cases}1&i=j\\ \frac{\sin(\frac{2\pi}{\lambda}\left|i-j\right|l)}{\frac{2\pi}{\lambda}\left| i-j\right|l}&i\
eq j\end{cases}  \tag{9}\]  
A common choice of the distance between the two neighboring isotropic hydrophone elements is \(l=\frac{\lambda}{2}\). Applying the value in Equations (8) and (9) and combining with Equation (7), we then get \(AG=M\).