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Prior RMS velocity

Substituting the theoretical interval velocity $v(\tau)$ from equation (3.40) into the definition of RMS velocity $V(\tau )$ (equation (3.25)) yields:
$\displaystyle \tau  V^2(\tau)$ $\textstyle =$ $\displaystyle \int_{0}^{\tau} v^2(\tau')  d \tau'$ (47)
  $\textstyle =$ $\displaystyle v_0^2  \frac {e^{\alpha \tau} - 1} {\alpha} .$ (48)

Thus the desired expression for RMS velocity as a function of traveltime depth is:
V(\tau) \eq v_0 \
\frac{e^{\alpha \tau} - 1 }{\alpha \tau}
\end{displaymath} (49)

For small values of $\alpha \tau$, this can be approximated as
V(\tau) \quad\approx \quad v_0 \sqrt{1 + \alpha \tau / 2} .
\end{displaymath} (50)