Gaussian beams and dynamic ray equations

Under the Gaussian approximation the acoustic pressure along a ray beam is defined as [2,3]

$$P(s,n) = A \sqrt{ \frac {c(s)}{ r q(s) } } \times \exp\left[... ...a \left( \tau(s) + \frac {p(s)}{2q(s)}n^2 \right) \right] ,$$ (4)

where $n$ represents the normal distance from the central ray and $A$ is an arbitrary constant, which can be fixed using an expansion of a point source into beams. After substitution into the Helmholtz equation one can obtain the dynamic ray equations [1,4,5]

$$\frac{dq}{ds} = c(s) p(s) \hskip5mm , \hskip5mm \frac{dp}{ds} = -\frac {c_{nn}}{c^2(s)} q(s) ,$$ (5)

where [2]

$$c_{nn} = c_{rr} N_r^2 - 2 c_{rz} N_r N_z + c_{zz} N_z^2 ,$$ (6)

$$c_{rr} = \frac{\partial{^2c}}{\partial{r^2}} \hskip5mm , \hs... ...5mm c_{rz} = \frac{\partial{^2c}}{\partial{r \partial z}} ,$$

and

$$N_r = c(s) \zeta(s) \hskip5mm , \hskip5mm N_z = -c(s) \xi(s) .$$ (7)

Initial conditions for $p(s)$ and $q(s)$ can be

$$q(0) = 0 \hskip5mm\makebox{ and }\hskip5mm p(0) = 1 .$$

Thus, the acoustic pressure along the ray can be calculated as

$$P(s) = \phi_r \frac {A(s)}{\sqrt{r(s)}} e^{ -i\left[ \omega \tau(s) - \phi_c \right] } ,$$ (8)

where the amplitude is given by [5]

$$A(s) = \left\vert \frac { c(s) \cos\theta_s }{ q(s) } \right\vert ^{1/2} .$$ (9)

The factor $A(s)/\sqrt{r(s)}$ represents the ray amplitude due to cylindrical spreading; it should be multiplied by the decaying factor $\phi_r$ in order to account for the loss of energy due to boundary reflections; the factor $\phi_c$ represents the shift caused by the formation of caustics. Both $\phi_r$ and $\phi_c$ are described in the following subsection.