Understanding the constraint on distance between microphones in microphone array for Direction of Arrival estimation
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.