What are brain waves, exactly?

brain waves don't propagate like transverse (light, seismic S waves, etc) or longitudinal (sound, seismic P waves, etc) waves do. They're rhythmic oscillations produced by activity of the brain's neurons. The brain's frequencies are categorized into five bands: delta, theta, alpha, beta, and gamma. These waves aren't unique to a single individual, everybody has them and our brains fluctuate between all of them depending on the state we are in.

These waves are measured using a device called an electroencephalogram (EEG), it's non invasive so it'd be safe to assume that all Ricks have some sort of EEG/detection device enhancement that allows them to view others' brain waves, hence how they can recognise other Ricks. This is more of a fictional concept, since you can't detect brain's across cities let alone dimensions, but this is Rick and Morty.

Despite humans having the same five of the frequency bands aforementioned, this doesn't mean everybody has identical brain activity. That said, each individual has their own unique patterns, hence what Rick likely meant when stating that every Rick has distinct and traceable brain waves.

"See, when a Rick is with a Morty, the 'genius' waves get cancelled out by the, uh… 'Morty' waves."

Rick then draws a simple diagram of two waves out of phase. This is called destructive interference, when one wave's peak aligns with the other wave's trough as shown in the drawing. Destructive interference essentially causes the waves to 'subtract' from one another, and perfect cancellation, as Rick might be implying, requires both waves to be perfectly out of phase and to have matching amplitudes (max height of a wave) and have matching frequencies (how many waves pass per second). This is where many problems with this whole explanation comes in.

Brain waves don't behave like the simple waves that Rick drew, brain waves are incredibly complicated and perfect cancellation in this case, would be extremely unlikely. I'm going to assume that Rick oversimplified his explanation so Morty can understand the gist of it. I'm going to re-interpret 'cancel' as 'unrecognizable by the detector', hence why he also used the word camouflage.

1. 'Complementary' waves, like he said, don't really work here. Waves being in destructive interferences has nothing to do with the type of waves they are, they can essentially be achieved by any two types of waves, even if they give out different outcomes. If we take Rick's words literally, and that their waves truly cancel out perfectly, this would imply they have the same brain waves (refer back to how perfect cancellation requires two waves to have the same features). This means that Morty wouldn't be as dumb as Rick is smart, contrary, Morty would have to share characteristics of 'Rick's genius'. Their waves would just have to be perfectly out of phase for Rick's assumption to work.

2. Wave interferences are not permanent. They are interactions that occur when waves overlap. One moment, Rick and Morty's waves might be destructive, but in another, they could be constructive (when a wave's peak aligns with another wave's peak, this causes a new, bigger wave).

Considering this, we will have to assume: Rick was just being an ass hole when he implies that Morty was dumb/that these waves are due to intelligence and that the explanation is more complicated than that, that a Rick and Morty's brain waves somehow have a very unique interaction which is why another person wouldn't be able to fill this role and that, somehow, Ricks and their Mortys' waves are in a constant destructive pattern. Again, I'm assuming all of these inaccuracies are due to simplification so that Morty could understand in that scene.

So, how could it really work?

1. This is a bit harder to back up since as you see in 2, this might be achievable using other companions, and I think that's the case. Rick might not actually need a Morty specifically for this brain camouflage to work. It's mentioned by Mr. Nimbus that he had another sidekick called Kyle, and not to forget, the whole two crows situation. It also suggests further that Rick was being unserious when he implied their wave cancellation is due to their IQ difference, since the crows are not necessarily stupid… Though I don't know how their animal IQ might compare to Morty. You get the gist of it.

2. Rick might have some sort of implant or device that mimics destructive interference. Rather than actually manipulating Morty or Rick's brain waves to constantly stay out of phase (which would be neural manipulation, this would primarily affect the person's thoughts, emotions, behaviours, etc, which isn't enough), this device could detect the signals produced by their brains and modify how it might be perceived externally. To another Rick's hypothetical 'EEG' (quotes since they don't necessarily work like that) device, the waves would present as destructive, rendering them unrecognisable.