Making sense of cavity in a 3 ropes system

I fail to understand why cavity resonance happens. Let’s say for the sake of example we have 3 ropes, the first and the last half-infinite and the middle one is of some length $L$. Now, we generally get that for the limit that the middle finite rope is very light compared to the ropes it is connected to, and if the length of the rope is a natural number times half the wave length of the waves there, if we send a wave from $-infty$, when it gets to the middle rope we will get in this limit infinite amplitude (constructive superposition of a wave going right and a wave going left in this rope, and nothing gets out because the outer ropes are very very dense). The math told that (in case I’m not mistaken), but why does it happen physically? Why would the amplitude of the trapped waves get to infinity (theoretically…)?

Another weird observation, is, if we say the wave trapped inside if of this form:

$$psi(x,t) = Ae^{i(frac{pi}{L}x-omega t)}+Be^{i(-frac{pi}{L}x-omega t)}$$

We get that the total energy is $$int_{0}^{L} rho_E dx=0$$
Where $rho_E$ is the energy density of the wave.
I’m quite confused at this point, why do all these things happen? Have I done some mathematical mistake?