## 3 Our Contribution
### UWSN Channel Model
#### 3.1.2 Characterization of the Small Scale Acoustic Channel
\begin{array}{l}\mathrm{a_{vp}}=\frac{1}{c}\big{[}[\mathrm{v_{rv}}\cos( \mathrm{\theta_{p}}-\mathrm{\theta_{rv}})-\mathrm{v_{rv}}\cos(\mathrm{\theta_ {p}}+\mathrm{\theta_{rv}})]-[\mathrm{v_{rv}}\cos(\mathrm{\theta_{0}}-\mathrm{ \theta_{rv}})-\mathrm{v_{rv}}\cos(\mathrm{\theta_{0}}+\mathrm{\theta_{rv}})] \big{]}\\ \quad=\frac{1}{c}\Big{[}\mathrm{v_{rv}}\Big{[}-2\sin\Big{(}\frac{\mathrm{ \theta_{p}}+\mathrm{\theta_{p}}-2\mathrm{\theta_{rv}}}{2}\Big{)}\sin\Big{(} \frac{\mathrm{\theta_{p}}-\mathrm{\theta_{0}}}{2}\Big{)}\Big{]}+\mathrm{v_{rv }}\left[2\sin\Big{(}\frac{\mathrm{\theta_{p}}+\mathrm{\theta_{0}}+2\mathrm{ \theta_{rv}}}{2}\Big{)}\sin\Big{(}\frac{\mathrm{\theta_{p}}-\mathrm{\theta_{0}}} {2}\Big{)}\right]\Big{]}\end{array}  \tag{45}\]Considering the similar transmitter and receiver movement across the network region in any direction \(\theta_{\rm{tv}/rv}\), the autocorrelation function with respect to the transceiver motion can be obtained as:  
\[\mathbb{E}\Big{\{}\mathrm{e}^{\mathrm{i}2\pi\mathrm{f}\mathrm{a}_{\rm{vp}}\Delta t }\Big{\}}=\mathrm{J}_{0}\left(2\pi\frac{2\mathrm{v}_{\rm{tv}}\sin\Big{(}\frac{ \theta_{\rm{p}}-\theta_{0}}{2}\Big{)}}{\mathrm{c}}\ \mathrm{f}\Delta t\right)\times \mathrm{J}_{0}\left(2\pi\frac{2\mathrm{v}_{\rm{rv}}\sin\Big{(}\frac{ \theta_{\rm{p}}-\theta_{0}}{2}\Big{)}}{\mathrm{c}}\ \mathrm{f}\Delta t\right) \tag{46}\]  
Our applied channel model considers only vertical surface motion. A signal interrupting jth reflection point along jth path achieves it randomly, at the longitudinal velocity of \(\mathrm{v}_{\rm{w}}\sin\Big{(}\mathrm{y}_{\rm{p}\dot{\rm{j}}}+2\pi\mathrm{f}_{ \rm{w}}\mathrm{t}\Big{)}\), where \(\mathrm{v}_{\rm{p}\dot{\rm{j}}}\sim\mathrm{u}[-\pi,\pi]\) and \(\mathrm{v}_{\rm{w}}=2\pi\mathrm{f}_{\rm{w}}\mathrm{A}_{\rm{w}}\). Thus, the protrusion of the vertical or the longitudinal velocity onto jth path, across entire surface points, generates (47)  
\[\mathrm{v}_{\rm{sp}}=2\mathrm{v}_{\rm{w}}\sin\theta_{\rm{p}}\sum_{\rm{j}=1}^{ \mathrm{n}_{\rm{sp}}}\sin\Big{(}\mathrm{y}_{\rm{p}\dot{\rm{j}}}+2\pi\mathrm{f} _{\rm{w}}\mathrm{t}\Big{)} \tag{47}\]  
In case the surface reflection points are located far apart in the way that \(\mathrm{\psi}_{\rm{p}\dot{\rm{j}}}\) are independent and hence the time correlation can be estimated by taking the expectation over \(\mathrm{a}_{\rm{sp}}=\mathrm{v}_{\rm{sp}}/\mathrm{c}\) with \(\mathrm{\psi}_{\rm{p}\dot{\rm{j}}}\) distributed uniformly over \(2\pi\). Mathematically,  
\[\mathbb{E}\Big{\{}\mathrm{e}^{\mathrm{i}2\pi\mathrm{f}\mathrm{a}_{\rm{wp}} \Delta t}\Big{\}}=\big{[}\mathrm{J}_{0}\big{(}2\pi\mathrm{a}_{\rm{wp}}\mathrm{ f}\Delta t\big{)}\big{]}^{\mathrm{n}_{\rm{sp}}} \tag{48}\]  
where \(\mathrm{a}_{\rm{wp}}=2\mathrm{v}_{\rm{w}}\sin\frac{\theta_{\rm{p}}}{\mathrm{c}}\) and \(\mathrm{n}_{\rm{sp}}\) signify the surface encounters found in the path p. Now, combining (27) and (44)-(48), the cumulative autocorrelation \(\mathrm{\widetilde{y}}_{\rm{p}}(\mathrm{f},\mathrm{t})\) is derived as:  
\[ \begin{split}\mathrm{R}_{\mathrm{\widetilde{y}}_{\rm{p}}}(\Delta t )=&\ \Big{[}\mathrm{\widetilde{y}}_{\rm{p}}^{2}(\mathrm{f})+2\mathrm{ \widetilde{o}}_{\rm{p}}^{2}(\mathrm{f})\mathrm{e}^{-\pi\mathrm{R}_{\rm{p}}( \mathrm{f})\Delta t}\Big{]}\mathrm{J}_{0}^{2}(2\pi\mathrm{a}_{\rm{d}}\ell \Delta t)\\ &\times\mathrm{J}_{0}\left(2\pi\frac{2\mathrm{v}_{\rm{w}}\sin \big{(}\frac{\theta_{\rm{p}}-\theta_{0}}{2}\big{)}}{\mathrm{c}}\ \ell\Delta t\right)\\ &\times\mathrm{J}_{0}\left(2\pi\frac{2\mathrm{v}_{\rm{w}}\sin \big{(}\frac{\theta_{\rm{p}}-\theta_{0}}{2}\big{)}}{\mathrm{c}}\ \ell\Delta t\right)\\ &\times\Big{[}\mathrm{J}_{0}\left(2\pi\frac{2\mathrm{v}_{\rm{w}} \sin\theta_{\rm{p}}}{\mathrm{c}}\ \ell\Delta t\right)\Big{]}^{\mathrm{n}_{\rm{sp}}}\end{split}  \tag{49}\]  
The autocorrelation function derived in (49) depicts Bessel-type characteristics, dampened by the exponentially depleting correlation of \(\mathrm{\gamma}_{\rm{p}}\) [in (49)].