I'm learning cagniard de-hoop method(de Hoop, A. T., A modification of Cagniard’s method for solving seismic pulse problems, Appl. Sci. Res. B 8, 349-356 (1960). ZBL0100.44208.), but I don't quit understand the derivation of some formulas in it.
In Eq.(2.12)
$U(x,y;s)=\frac{F(s)}{2\pi i}\int^{i\infty}_{-i\infty}exp[-s(px+\eta|y|)]\frac{1}{2\eta}dp\tag{2.12}$
in which
$\eta=(1/v^2-p^2)^{1/2} (Re\ge 0)\tag{2.13}$
p is a complex variable in the p-plane, the only singularities of the integrand in Eq(2.12) are branch points at $p=1/v$ and $p=-1/v$. In view of subsequent deformations of the path of integration we take $Re (\eta)\ge 0$ everywhere in p-plane. This implies that branch cuts are introduced along $Im(p)=0$, $1/v<|Re(p)|<\infty$
The next step towards the solution of the transient problem is to perform the integration in the p-plane along such a path that the right-hand side of (2.12) can be recognized as the Laplace transform of a certain function of time. The analysis which follows will show that the path has to be selected such that
$px+\eta|y|=\tau\tag{2.14}$
which $\tau$ is real and positive
I konw the certain function of time is real and positive, but why is $px+\eta|y|=\tau$ real and positive? In other words, why is $px+\eta|y|$ the certain function of time.