Acoustic pressure outside the beam

The acoustic pressure outside the beam is calculated using geometric beams [4,5]

$$P(s,n) = \phi_r \frac {A(s)}{\sqrt{r(s)}} \phi(s,n) e^{ -i\left[ \omega \tau(s,n) - \phi_c \right] } ,$$ (12)

where $\tau(s,n)$ is calculated through linear interpolation, as described in [2],

$$\phi(s,n) = \left\{ \begin{array}{cl} \displaystyle { \fra... ...\leq W(s) \\ \\ 0 & \makebox{ else } \end{array}\right. ,$$ (13)

$W(s)$ represents the beam width, defined as

$$W(s) = \left\vert \frac {q(s) \Delta\theta }{ c_s \cos\theta (s)} \right\vert ,$$ (14)

$\Delta\theta$ is the angular separation between the rays and $n(s)$ is the normal distance from the receiver coordinates $(r,z)$ to the ray coordinates $\left[ r(s),z(s) \right]$. The acoustic pressure induced by $m$ rays is the superposition of the acoustic pressure induced by each ray:

$$P(r,z) = \displaystyle{\sum\limits_{i=1}^{m}} P_m(r,z) ,$$ (15)

where

$$P_m(r,z) = \phi_r^m \frac {A_m(s)}{\sqrt{r}} \phi_m(s,n) e^{-i\left[ \omega \tau_m(s,n) - \phi_c^m \right] } .$$ (16)