Why does transmission probability decrease, increase, then decrease again? We did a quantum tunneling lab online. We used a Java program to model the electron wave function and show what happens when there is a step potential (U is less than E). Our value for the transmission probability had a pattern:
Barrier Width (nm)   T   Barrier Width (nm)    T
1.0                  0.97      6.0            0.82
2.0                  0.91      7.0            0.88
3.0                  0.84      8.0            0.95
4.0                  0.80      9.0            1.00
5.0                  0.80      10.0           0.99

It seems to decrease, increase, then decrease again? Why is this? 
 A: The transmission probability depends on the relation between the barrier width and the de Broglie's wavelength of the electron (within the barrier). The waves reflected from the front and the end of the barrier interfere constructively or destructively depending on this relation. The description of a similar phenomenon for light can be found, e.g., at https://en.wikipedia.org/wiki/Thin-film_interference#Phase_interaction (where they discuss reflection, rather than transmission). 
A: Firstly, the transmission is always between zero and one, and there are some sweet spots where it is one exactly so in between two sweet spots it has to decrease then increase again. But that's a bit uninformative and uneducational. And you asked about why, so let's look at the why of tunneling and quantum dynamics in general.
You start with a wave packet that is normalizable and comes in towards the barrier from one side, evolve it in time until you have two separate packets one on each side of the barrier each heading away from the barrier, then you get the transmission probability as the ratio of the L2 norm of the transmitted part over the L2 norm of the incoming packet.
Of course a wave packet doesn't have a fixed energy so the whole $E<U$ argument loses some force or relevancy. But you can see a true why to a realistic situation of tunneling when you look at a realistic situation. A sharp barrier is unrealistic so we will loom at a situation where the potential rises/falls over a small region.
If the barrier increases over a region you have a wave packet that was naturally spreading even before it arrived at the beginning of the barriers. To see why they spread you can note that from the time derivative of the probability density $\rho=|\Psi|^2$ you can get a current $\vec J$  such that $\partial \rho /\partial t =- \vec \nabla \cdot \vec J,$ just like in electromagnetism. And the current $\vec J$ is like a velocity and in facts it reacts to the classical potential at a location right there just like a classical potential affects the classical velocity right there. Except there are other things that affect the current too. In fact, all quantum effects happen because there is an additional term that affects $\vec J$ as if the the total potential was the classical potential $V$ plus an additional potential $Q$ also affecting it equally to how the classical potentials affects it. So given a wave,
$\Psi,$ it acts like there is an additional potential $Q=-\frac{\hbar^2\vec \nabla^2 |\Psi|}{2m|\Psi|}.$
So now you can see that if your incoming wave has a middle region where $\frac{\vec \nabla^2 |\Psi|}{|\Psi|}$ is pretty constant, then there is little deviation from classical dynamics. But near the edges of the wave packet that result, $\frac{\vec \nabla^2 |\Psi|}{|\Psi|},$ can't be constant and in fact is just perfect to make it spread. The current $\vec J$ will change (differently than they would if there was just a classical potential) near the edges of the wave packet as if there was a packet shape dependent potential driving regions with lower $\frac{\vec \nabla^2 |\Psi|}{|\Psi|}$ to regions with larger $\frac{\vec \nabla^2 |\Psi|}{|\Psi|}.$
Wave packets spread because even when there is no classical potential the current has to change as if there is an additional potential $Q.$  As you get to the classical barrier where the classical potential goes up, the current would classically go down (particles slow as they reach a barrier) and indeed it does decrease but when the leading edge decreases and the next part hasn't yet that makes the density go up there (see the continuity equation) and this causes the potential $Q$ to change and this actually pushes that leading edge and it's current through the barrier faster than it classically would go through the barrier (of course the spreading that happened before the packet reached the barrier was also making it move faster than the packet as a whole and faster than it classically would). But the fact that the classical speed was lower on the leading edge can cause a traffic pile up behind which can slow down those behind thus making them pile up like a traffic slow down, each slower moving one creates a bit more density behind it which can slow down the things behind it.
This can happen all throughout the rise in the classical potential, once through there it is just spreading from the $Q$ in the region where the classical potential is level until you get to the place where the classical potential drops back down which makes the current speed up and as a bit approaches that drop the part that hits it first gets pulled away faster than it was before and thus $|\Psi|$ gets smaller behind it and this again changes $Q$ and leaving the wave behind it makes it get pulled back but that just means less pushing than it normally got from the $Q$ spreading it had at the top level portion of the classical potential.
So when you approach a classical potential rise the leading edge slows down less than it normally would, that's tunneling. This is the why for everything quantum (well there is spin, and particle creation and such) and in  particular this is why any part of a packet flows the way it does. But this nice picture of why something happens has become a bit local, so we can't see the effect of the barrier width very easily. But we can do it.
So imagine an incredibly wide wavepacket similar to a pure sinusoidal wave like $\Psi=e^{i(kx-\omega t)}$ modulated by an envelope that is a standard normal with an astoundingly huge standard deviation. Then near the middle it is very very flat. And it thus takes a long time to travel (we fixed $k$ then made the standard deviation large compared to that). In particular we can make it large compared to the barrier width. And every part is very flat. If you want near perfect transmission you can think of a setup like that in a circuit element with a steady current, the charge density piles up imbalanced charges until the current flows steady through the circuit element. Similarly, we can get that near perfect transmission if you can build up (because of the slow down) to make a $Q$ before, in, and after the barrier so that current flows in at the same rate it net travels through the barrier and then out.
To get that during the time the middle part of the packet is going through the barrier the region anywhere close to near the barrier should look similar to the state that has a non normalizable wave coming and matched up at the boundaries. That state should have equal sized amplitudes in the asymptotic reflection and transmission (because we have a near monochromatic waveform).
And to do that you need a particular width to line them up just like for fixed energies in a well you needed certain width of a well. If your width isn't that perfect width you can still get a near steady state by having unequal amplitudes that just means you had to get buildup on the incoming side if the barrier before you had a steady current throughout.
This is what affects the total transmission. If the steady state in the middle requires a bigger buildup on the incoming side that pile up propagates all the way through and eventually starts to eat away at the spreading on the tail of the incoming packet
A bit like a slinky the formerly tail end of the incoming wave eventually becomes a leading edge of a reflected wave thus not all of the wave goes through the barrier.
You basically have a transmission probability that is just how different the current coming out is compared to the average net total current of the incoming packet and you can find that by how much the wave needs to buildup on the incoming side before the quantum potential $Q$ is enough to overcome the barrier and even out the current (if it is a wide packet). A buildup within the finite barrier isn't a big deal as that is finite and transitory, but if the steady state requires a build up on the incoming side that propagates to a buildup throughout the entire incoming side that starts to gives the whole incoming side less current, and this eventually results in the incoming side having at least part turn around and go back.
Some analysis is easier to flat potentials, but not everything generalizes when a special case has two things be equal that in general are not equal. But this is a good why for tunneling, and for any dynamical effect in quantum mechanics.
