Theoretical background

Ray tracing requires the solution of the ray equations to determine the ray coordinates. Amplitude and acoustic pressure requires the solution of the dynamic ray equations, which are described in detail in [1].

For a system with cylindrical symmetry the ray equations can be written as [2]

$$\begin{array}{cccccc} \displaystyle { \frac{dr}{ds} } & = & ... ...\frac {1}{c^2} \frac{\partial{c}}{\partial{z}} } , \end{array}$$ (1)

where $r(s)$ and $z(s)$ represent the ray coordinates in cylindrical coordinates and $s$ is the arclenght along the ray; the pair $c(s)\left[ \xi(s),\zeta(s) \right]$ represents the tangent versor along the ray. Initial conditions for $r(s)$, $z(s)$, $\xi(s)$ and $\zeta(s)$ are

$$r(0) = r_s \hskip3mm , \hskip3mm z(0) = z_s \hskip3mm , \h... ... , \hskip3mm \zeta(0) = \frac {\sin\theta_s}{c_s} \hskip3mm ,$$

where $\theta_s$ represents the launching angle, $(r_s,z_s)$ is the source position, and $c_s$ is the sound speed at the source position. The coordinates are sufficient to obtain the ray travel time:

$$\tau = \displaystyle{\int\limits_{\Gamma}^{}} \frac {ds}{c(s)} \ ,$$ (2)

which is calculated along the curve $\left[ r(s),z(s) \right]$.