How an AP with several omnidirectional antennas can steer its signal towards one client, and what that does to everyone else.
With a mouse you can also drag either client. Not to scale: one grid square is one wavelength (λ), about 12 cm at 2.4 GHz or 6 cm at 5 GHz.
Every antenna on the AP is still omnidirectional: on its own it sends the same signal in every direction. Beamforming sends the same signal from several antennas at once, each slightly earlier or later than the others. The AP picks those timing offsets (phase shifts) so the waves from all antennas arrive at client A in step and add up. Everywhere else they arrive out of step by different amounts, so they add up in some places and cancel in others.
The AP splits the same total power across its antennas. At the target the waves add in step, so the signal there is stronger than one omnidirectional antenna would give by up to 10 × log₁₀(number of antennas): +3 dB with 2 antennas, +4.8 dB with 3 and +6 dB with 4. The power isn't created, it's moved: other directions get less, and some get almost nothing.
Client B sits somewhere in the same pattern. Depending on where it is, it may get more signal than before, about the same, or much less if it lands in a null where the waves cancel. Move B around the AP to see the peaks and nulls, and use the buttons to jump to them. Sending to several clients at once with deliberate nulls on the others is the idea behind MU-MIMO.
An AP doesn't know where clients are on a map. With explicit beamforming (802.11ac onwards) it sends a short sounding frame, the client measures how the signal from each AP antenna arrived and sends that measurement back, and the AP works out the timing offsets from it. The result is the same as here: the waves add up at that client.
The wavelength is drawn much larger than real life so the waves are visible, and the map is a flat, two-dimensional slice. The clients are only a few wavelengths from the AP; real clients are usually many metres away, where the pattern is the same shape but much larger.
This is free space. Indoors, reflections from walls and furniture reach the client by several paths, and real beamforming uses the measured channel to cope with that, separately for each subcarrier. The principle, adjusting timing so the signals add up at the client, is the same.
The colours show how much stronger or weaker the signal is than a single omnidirectional antenna with the same total power, at the same spot. The normal weakening with distance is left out; the Free Space Path Loss tool covers that. For more on multiple antennas, see MIMO and Spatial Streams and OFDMA vs. MU-MIMO.