If the head of the sail is moving in the same direction as the foot relative to the "ground" (which of course, it must) then assuming the true wind's direction is constant from top to bottom, both the wind's apparent and true direction must be the same from top to bottom.
Of course this doesn't take into account other effects like the vortexes and turbulence described above.
And this goes to remind all students of symbolic logic that you can prove anything from false premises.
The false premise, in this case, is that the true wind is the same from the peak of the sail to the foot.
It is not.
If you
really want to know why not, you open a whole sack-of-snakes:
-- physics of the air/sea boundary
-- laminar flow
-- wind/sea-state interaction
The list goes on.
I hope you paid close attention in those upper-division math courses on differential equations.
Bill Roosa is right.
This stuff goes far beyond the knowledge needed to trim your sail.
Don G's guidelines are what my HS German teacher called, "Simple rules for simple people."
They may be short on physics theory, but they work.
I certainly follow them when I sail.
But you may still want to get more into the science of all this.
If so, I suggest learning the vector geometry that defines True and Apparent wind.
That will help a lot in your understanding of what the wind that you see is doing.
Moreover, learning and practicing this vector geometry could lead you toward your learning about what the USN called 'maneuvering board solutions'.
Maneuvering board problems routinely deal with collision avoidance, a handy skill to have if you are sailing near a shipping lane.
[OK, on the submarine I was on, we were trying to get closer to the target, not avoid it.
But that was an application of the same geometry with a different objective.]
A final note:
I use the Weems & Plath 'VectorMaster' [enhanced] circular slide rule plotting tool for vector geometry at sea.
Is it necessary? Of course not.
Is it fun to play with? You bet, and sometimes it really helps.
Bob