Then let me teach it to you in an intuitively obvious way by using some imagery. Imagine that you ride on a crest of the wave in a special vehicle. In front of you there is a meter with a single needle that looks like a clock but has divisions in degrees. As you ride along with the wave, the needle turns at a constant rate.
Rule: Every time the needle makes one complete revolution, the vehicle has advanced the distance of one wavelength and the phase of the wave, expressed as a sine or cosine, has advanced by ##2\pi.##
Now imagine two observers, one on each wave, in two separate vehicles. They start together with their wave-o-meters synchronized, go their separate ways and eventually meet again and compare the readings on their meters.
- If the two needles point at the same value, you have constructive interference
- If the two needles point in exactly antiparallel directions (not necessarily 0 and 180 degrees), you have destructive interference
- If the needles point in none-of-the-above directions, you have in between interference.
In this problem, when the wave is reflected off the metal plate the needle attached to that wave gets bumped ahead by 180 degrees and which means that the path is bumped by half a wavelength.
Does this help?