I am going to try to explain better my question. Considering a simple control scheme made up of a forward-path amplification\$A\$ and a feedback-path frequency dependent attenuation \$\beta(j\omega)\$, assuming a "positive" feedback, such that the loop gain is \$T(j\omega)=A\beta(j\omega)\$ then an oscillation can be kept constant in time if certain conditions are met for a precise frequency value \$\omega_0\$, namely the Barkhausen criteria: \$T(j\omega_0)=1\$ or equivalently \$|T(j\omega_0)|=1\$ and \$\angle T(j\omega_0)=0\$. In this way, a sinusoidal signal follows the full loop path and comes back unchanged, without any amplitude or phase modifications.
Now, my questions is related to the phase condition: I have understood that the magnitude condition has to be satisfied in order not to have divergent or damped oscillations (no overall amplification or attenuation per each revolution of the loop),whereas why must the phase be \$0\$? I mean, if it is so, the signal is not delayed after each cycle but what happens if it is not so? Is the signal being shifted every cycle? What is then the resulting output waveform, something like a sliding sinusoid?
Since there are no signals being summed, like it happens in a classical feedback-system with an input, one cannot refer to constructive/destructive interference that would occur due to the phase shift between the two signals added; thus the phase shift of the loop gain \$\angle T(j\omega_0)\$ cannot bring the oscillation to grow up or drop if the magnitude is still \$1\$. Am I right? In other words, if the phase condition is not met, then no steady state oscillation can be produced, but anyway it is not possible to have oscillations that tend to increase or decrease.
EDITED SECTION: As additional doubt linked to the previous question, if the Barkhausen criteria are exactly met for a certain frequency, shouldn't one say something more also about the other frequencies, so that all of them are damped (band-pass behaviour)? In other words, how can one be sure that no other frequencies get amplified and enlarged? (even though, saying this I am not taking into account that at such frequencies the phase would not be 0 too, so I cannot figure out which is the effect of loop gain greater or equal than 1 but phase whatever so, the problem is degenerating again into the phase criterion).