Velocity gradients

Inserting the Snell wavefield expression into the scalar wave equation, we discover that our definition of a Snell wave does not satisfy the scalar wave equation. The discrepancy arises only in the presence of velocity gradients. In other words, if there is a shallow constant velocity v₁ and a deep constant velocity v₂, the equation is satisfied everywhere except where v₁ changes to v₂. Solutions to the scalar wave equation must show amplitude changes across an interface, because of transmission coefficients. Our definition of a Snell wave is a wave of constant amplitude with depth. The paraxial wave equation could be modified to incorporate a transmission coefficient effect. The reason it rarely is modified may be the same reason that density gradients are often ignored. They add clutter to equations while their contribution to better results--namely, more correct amplitudes and possible tiny phase shifts--has marginal utility. Indeed, if they are included, then other deeper questions should also be included, such as the question, why use the acoustic equation instead of various other forms of scalar elastic equations?

Even if the paraxial wave equation were modified to incorporate a transmission coefficient effect, its solution would still fail to satisfy the scalar wave equation because of the absence of the reflected wave. But that is just fine, because it is the paraxial equation, with its reflection-free lenses, that is desired for data processing.

EXERCISES:

1. Devise a mathematical expression for a plane wave that is an impulse function of time with a propagation angle of 15$^\circ$from the vertical z-axis in the plus z direction. Express the result in the domain of
2. [] (a)$\ \ (t,x,z)$
3. [] (b)$\ \ ( \omega ,x,z)$
4. [] (c)$\ \ ( \omega , k_x , z)$
5. [] (d)$\ \ ( \omega ,p,z)$
6. Find an amplitude function A(z) which, when multiplied by f in equation (47), yields an approximate solution to the scalar wave equation for stratified media v(z).