When one note is played by two people, and one of them is out of tune, it can create a phenomenon known as beats. Beats occur due to the interference between two sound waves with slightly different frequencies.
Let's say one person is playing a note at a frequency of 440 Hz (for example, an A4 note), which is the correct pitch. The other person is playing the same note but is slightly out of tune, let's say at 442 Hz. The two sound waves will interact with each other when they reach your ears.
During the times when the two sound waves are in phase, meaning their crests and troughs align, they reinforce each other, resulting in a slightly louder sound. However, when the waves are out of phase, their crests and troughs misalign, causing them to partially cancel each other out, resulting in a softer sound.
This fluctuation between constructive and destructive interference creates a regular pattern of variations in the perceived loudness of the sound, which is what you perceive as beats. The rate at which the loudness fluctuates is equal to the difference in frequency between the two notes.
In the example above, the difference in frequency between the two notes is 2 Hz (442 Hz - 440 Hz), so you would perceive two beats per second. The speed of the beats is slower when the frequency difference is smaller and faster when the frequency difference is larger.
Beats can be a useful tool for tuning musical instruments. By listening to the beats produced when two notes are played together, you can adjust the tuning until the beats disappear, indicating that the two notes are in perfect harmony.
So, when you hear wave-like sounds when two people play the same note but one is out of tune, you are likely perceiving the beats resulting from the interference of the two sound waves with slightly different frequencies.