## 2 Modern Physical Layer
### Underwater Acoustic Channel Characterization
#### 2.1.2 Multipath  
Multipath in underwater channels is mainly caused by two relevant factors: wave reflection at the surface, bottom and any object and sound refraction in the water [5].  
The reflection coefficient is equal to \(-1\) under ideal conditions, while bottom reflection coefficients depend on the type of bottom (hard, soft) and the grazing angle [5].  
The second effect is a consequence of [PERSON]'s law. The propagation speed of an acoustic signal underwater is, on average, approximately 1,500 \(\frac{m}{s}\); however, the actual value depends on the salinity, temperature and pressure of the medium, among other factors. A complete nine-term equation for calculating propagation speed is defined in [11], as shown in Equation (4), where \(T\) denotes the temperature in degrees Celsius, \(D\), the depth in meters, and \(S\), the salinity in parts per thousand of the water.  
\[\begin{split} c=1448.96+4.591T-5.304\times 10^{-2}T^{2}+2.374\times 1 0^{-4}T^{3}\\ +1.340(S-35)+1.630\times 10^{-2}D+1.675\times 10^{-7}D^{2}\\ -1.025\times 10^{-2}T(S-35)-7.139\times 10^{-13}D^{3}\end{split} \tag{4}\]  
Figure 1: Narrow-band signal-to-noise ratio (SNR; \(1/(AN)\)) as a function of frequency for varying transmission distance. Environmental factors used: practical spreading, \(k=1.5\); wind speed, \(w=3\) m/s; and moderate shipping activity, \(s=0.5\).  
Figure 2: Temperature, salinity data and their impact on the sound speed profile. Data retrieved from the Papa station placed in the Pacific ocean (39\({}^{\circ}\)N, 146\({}^{\circ}\)W) in August, 1959.  
a 600-m depth, remaining nearly constant around 4 in deep oceans waters. In subfigure (b), where the individual contributions of temperature, salinity and pressure to sound speed are displayed, shows that deep water sound speed variation mainly depends on the increasing pressure.  
Combination of reflection and refraction can lead to some remarkable effects, as explained for the sonar propagation in [12]:  
* Positive temperature gradient combined with surface reflection. Due to positive gradient, vertical rays are deflected upwards, as shown in Figure 3a. Both deflected and direct rays are reflected on the water surface. Reflected rays are once again deflected, and so on. As a result, acoustic waves barely achieve depth areas under these conditions.
* Combination of positive and negative temperature gradient. Near the surface, vertical rays are deflected upwards as in the previous case. However, acoustic waves that reach negative temperature gradient area are deflected downwards as shown in Figure 3b. As a consequence, there is some area -close to the temperature gradient discontinuity region- in which received acoustic wave intensity is very low.
* Combination of negative and positive temperature gradient. In contrast to the previous case, this combination leads to a concentration of the rays around the gradient discontinuity region. This region is known as the SOFAR channel and it can be used to transmit acoustic waves to long distances.  
Besides, conditions change continuously in both long and short time-scales, e.g., temperature variation and surface waves, respectively. The last effect is especially critical, since acoustic reflection points are continuously displaced, resulting in scattering and Doppler spreading, due to the changing path length [5].  
As a conclusion, the impulse response of an acoustic channel is strongly correlated to the deployment scenario: holography, depth, particular environmental conditions, final nodes placement and even the current time slot have a severe impact on the channel response.  
Figure 3: Acoustic Ray Tracing for Sound Speed Profiles with Positive and Negative Gradients. (**a**) Positive Gradient Sound Speed Profile. (**b**) Negative Gradient Sound Speed Profile.