Attenuation

As shown in the previous chapters the general solution for the acoustic pressure can be written as

$$P(\mbox{$\mathbf{r}$},\omega ) = A e^{-i\omega \tau}$$

with

$$A = \frac {1}{4\pi}\sqrt{ \frac {c(s)}{c(0)} \frac {\cos\theta(0)}{J} }$$

for the classical solution, and

$$A = \frac {1}{4\pi}\sqrt{ \frac {c(s)}{c(0)} \frac {\cos\thet... ... \exp\left[ -\frac{1}{2}i\omega \frac {p(s)}{q(s)} n^2 \right]$$

for the Gaussian beam approximation. Either expression does not take into account the dissipation of energy due to material absorption, and due to the transfer of energy every time the acoustic wave bounces on a boundary. Such dissipation is frequency-dependent (and more relevant as frequency increases), and implies using a corrected amplitude $a$, which corresponds to the original amplitude, multiplied by two decaying factors $\phi_r$ and $\phi_V$:

$$a = A \times \phi_r \times \phi_V$$

where $\phi_r$ and $\phi_V$ stand, respectively, for the decay due to boundary reflections, and due to volume absorption. Both factors are described in the following sections.