My understanding of Huygens' principle is, that it describes the way how a wave moves: Instead of moving straightforward, it propagates in all directions via producing secondary wavelets. 'Secondary wavelets' and 'wave's propagation' are synonyms.
But I realized that many others assume that waves move straight, and Huygens' principle doesn't say anything about the original wave, just that in addition to the original wave that moves straight, there are also secondary wavelets which are spherical. And these wavelets just continue straight, because Huygens' principle only applies to waves, not wavelets. I wonder if it could be the correct interpretation, because it seems to void most of Huygens' principle. For example, if refraction occurs due to Huygens' principle, refracted waves shouldn't be able to refract again, because they are composed of those wavelets, since the original wave just continues straight. So it's not a 'real' wave, and Huygens' principle doesn't apply to it.
Now to my main question (with my understanding of Huygens' principle): Let's start with the most simple case: propagation of waves in free space. Suppose we have a planar wavefront of troughs followed by a wavefront of crests of the same waves (½ wavelength behind). After one revolution, we have a lot of semicircular secondary wavefronts superimposed on each other all along the initial wavefront. (We should really have full spherical wavefronts, but for now let's ignore backward waves, and focus on 2 dimensions only, considering only the forward semicircles. It's enough complicated as is.) The result is a thick wavefront starting from the location where the wavefront initially was, and ending one wavelength ahead. After another revolution, we have a wavefront twice as thick. The starting point never changes (as long as we ignore backward waves), but the endpoint does, so the wavefront is just growing. The same is true for the wave crests (and everything in between), just their start and end points are a half wavelength behind. So behind the very first part of the waves - where the troughs have no crests to compete with - we should have destructive interference in every wave.
Here is how it should look after one revolution: Whatever the answer should be, one needs to be careful that only straightforward waves should survive. And I'm not concerned specifically about backward waves (and I have no idea why people assumed so), but about every possible direction.
I think that the answer may be that although there are troughs and crests all over, they have different densities, which causes that some parts should survive. But I'm waiting for others to confirm it before diving in the details.
When it comes to refraction, the problem with direction gets worse. If every point of a wavefront produces waves in every direction with and without refraction, then how can waves change direction? What exactly is different after refraction than before?
The question gets even more complicated, when we have refraction and diffraction simultaneously. In that case the side waves don't get canceled (this is the cause for diffraction), so how can refraction have an effect?
Some commenters told me that Huygens' principle is not accurate. It's just an approximation. I wonder if they're right, because I didn't see such a statement anywhere. If they're right, then I'm not interested in understanding Huygens' principle, because I'm interested in the real facts of wave propagation.
One of them told me that the exact truth is Maxwell's equations. But I couldn't find an interpretation of Maxwell's equations which predicts wave propagation, at least not a well explained one.
This is really a fundamental question. If you know of an article or inexpensive ebook etc. that explains this in a way that all my questions will be answered, please give me a link. (in addition to, or without, your answer.) (It's not easy to find. I searched a lot before posting this question.)