Thus the square root itself will have a phase between as we require. In applications, would typically be chosen proportional to the maximum time on the data. Realistic modeling would have them damping with depth. We can expect a more concentrated pulse if we include the evanescent energy, even though it is small. There a stability problem will develop unless we begin from careful definitions as we are doing here.
Seismic theory includes an abstract mathematical concept known as branch-line integrals.
Such theory is most easily understood beginning from here.