Stationary phase condition: searching for an understanding

Having noted that Isaac Newton made a lot of progress during self-isolation, I've been back to the large Quantum Mechanics text book I was working through last year. Luckily, I kept a diary so one year later I have been able to pick up the threads. I got stuck on the stationary phase condition for deriving the wave packet needed to represent a particle in QM.

 It's the integral above that was bothering me. p stands for momentum and by the De Broglie equation, p=h/lambda. But since lambda is inversely proportional to frequency, then p is another way of choosing different frequencies. Instead of sine or cos, the wave is represented in its more obscure complex exponential form. Integration means adding up. The clue is superposition. So all we are doing is putting waves of different frequencies one on top of the other.

I drew what we'd do for superposition of different frequencies with a L6 class. Three waves are shown, all with different wavelengths and thus different momenta. Notice that they all coincide constructively at x=0 but don't elsewhere. The green curve is only just out with the red curve and just those two added would still give a reasonable tall wave. But the blue waves interferes destructively in a lot of places. So this is my take on the stationary phase condition - in one place and across a narrow range of frequencies, you get constructive interference. Since we are adding an infinite number of frequencies, elsewhere we get total; destructive interference. Hence overall we get a wave packet - a narrowly defined space with waves. It does bother me that it would seem that constructive interference at x = 0 would give infinite amplitude. I have drawn all 3 with the same amplitude but the equation implies that amplitude depends on momentum. So if most momenta have a very small amplitude, perhaps in the limit you get a defined amplitude.