The attenuation of a wave through a medium can be modeled by the Beer-Lambert Law using an attenuation coefficient. If $I$ is the intensity, and $I_r$ is a reference intensity, then what is the physical difference between modelling attenuation using the two functions below?
I am looking for a conceptual understanding of the difference between $I_1$ and $I_2$. What does the integral in $\beta_1(\tau,\omega)$ imply?
In the equations below, $\tau$ is the time, $\omega$ is angular frequency, and $Q(\tau)$ is a real-valued function of time $\tau$.
I have seen both $\beta_1$ and $\beta_2$ functions used in the context of the attenuation of seismic waves through layered media, and I am wondering if one model is perhaps better than the other.
${I_1} = {I_r}{\beta _1}(\tau ,\omega )$
${I_2} = {I_r}{\beta _2}(\tau ,\omega )$
${\beta _1}(\tau ,\omega ) = \exp \left[- {\int\limits_0^\tau {\frac{\omega }{{2Q(\tau ')}}d\tau '} } \right]$
${\beta _2}(\tau ,\omega ) = \exp \left[- {\frac{{\omega \tau }}{{2Q(\tau )}}} \right]$
A reference for these two equations is Y. Wang, Seismic Inverse Q Filtering (link). See for example, pg. 121.
