Boundary reflections

The decaying factor $\phi_r$ is given by the expression

$$\phi_r = \displaystyle{\prod\limits_{i=1}^{n_r}} R_i ,$$ (4.1)

where $n_r$ represents the total number of boundary reflections, and $R_i$ is the reflection coefficient at the $i$th reflection. The case with no reflections ($n_r = 0$) corresponds to $\phi_r = 1$. Generally speaking, boundaries can be one of four types:

• Absorvent: the wave energy is transmitted completely to the medium above the boundary, so $R$ = 0 and ray propagation is terminated at the boundary.
• Rigid: the wave energy is reflected completely on the boundary, with no phase change, so $R$ = 1.
• Vacuum: the wave energy is reflected completely on the boundary, with a phase change of $\pi$ radians, so $R$ = -1.
• Elastic: the wave energy is partially reflected, with $R$ being a complex value and $\left\vert R \right\vert < 1$.

The calculation of the reflection coefficient for an elastic medium (see Fig.4.1) is given by the following expression[10]:

$$R\left( \theta \right) = \frac{D\left( \theta \right) \cos\theta-1}{D\left( \theta \right) \cos\theta+1} ,$$ (4.2)

where

$$D\left( \theta \right) = A_1\left( A_2\frac{1-A_7}{ \sqrt{1-A_6^2} } + A_3\frac{A_7}{ \sqrt{1-A_5/2} } \right) ,$$

$$A_1 = \frac{\rho_2}{\rho_1} \hskip3mm , \hskip3mm A_2 = \fr... ... , \hskip3mm A_3 = \frac{\tilde{c}_{s2}}{c_{p1}} \hskip3mm ,$$

$$A_4 = A_3 \sin\theta \hskip3mm , \hskip3mm A_5 = 2 A_4^2 \h... ...n\theta \hskip3mm , \hskip3mm A_7 = 2A_5 - A_5^2 \hskip3mm ,$$

$$\tilde{c}_{p2} = c_{p2}\frac{1 - i\tilde{\alpha}_{cp}}{1 + \... ...- i\tilde{\alpha}_{cs}}{1 + \tilde{\alpha}_{cs}^2} \hskip5mm ,$$

$$\tilde{\alpha}_{cp} = \frac{ \alpha_{cp} }{40 \pi \log e} \h... ...alpha}_{cs} = \frac{ \alpha_{cs} }{40 \pi \log e} \hskip5mm ,$$

where the units of attenuation should be given in dB/$\lambda$.

Figure 4.1: Ray reflection on an elastic media.

In general the reflection coefficient is real when $\alpha_{cp} = \alpha_{cs} = 0$, and the angle of incidence $\theta$ is less than the critical angle $\theta_{cr}$, with $\theta_{cr}$ given by the expression

$$\theta_{cr} = \arcsin\left( \frac{ c_{p1} }{ c_{p2} } \right) .$$ (4.3)

Moreover, attenuation is negligible when $\theta < \theta_{cr}$, and for small $\theta$ the energy transfered to shear waves in the elastic medium is only a small fraction of the total energy transfered.