Understanding the constraint on distance between microphones in microphone array for Direction of Arrival estimation The distance between microphones in a two-microphone array for direction of arrival estimation, should be at least less than half of the wavelength of the signal that is being detected by the microphone, when the signal is periodic.

I am having trouble understanding this constraint on the distance between microphones. As far as I remember, the path difference between the sound arriving at one microphone and the other shouldn't be a multiple of half the wavelength, because if $x = \frac{\lambda}{2}$, the phase difference between the two signals for a continuous periodic signal would be zero, and thus we wouldn't be able to measure the time difference between the sound arriving at the two micropones.

However, I am unable to understand how a less than constraint appeared, and on the distance between the microphones and not on the path difference of the sound arriving at the two microphones.
Say the microphones $A$ and $B$ are omnidirectional and exactly half a wavelength apart. If the sound field propagates in the direction of the vector $A\to B$, then a chosen part of the waveform will arrive at $...,$ $A,$ $B,$ $A,$ $B,$ $...$ at intervals that equal half the period of the waveform. If the sound field propagates in the direction of the vector $B\to A$, the chosen part of the waveform will arrive at $...,$ $B,$ $A,$ $B,$ $A,$ $...$ at identical intervals. You won't be able to tell the difference between the two cases from the arrival times because for a periodic waveform there is no "first arrival". This ambiguity is resolved by the constraint you mention, as you then know that you can safely choose for comparison arrival times at $A$ and $B$ that are separated by at most some known time less than half the period of the waveform.