What happens if you mix brain waves?
Everyone has a mix of ongoing oscillations, of brain waves, at any given point. Some frequencies of waves are more or less likely to co-occur with other types of waves.

In general, brain activity follows roughly a power law distribution - that is, there is more power at lower frequencies and that power decreases in a predictable way as you go to higher frequencies.

When researchers talk about brain waves at a particular frequency, they are talking about a deviation from power law at that frequency - if they say "gamma waves" what they really mean is "there is more energy at gamma frequencies than we would expect if things were random."

Some of these deviations are predictable with changes in brain state - for example, low frequency (delta) activity is greater during sleep, and higher frequency (gamma) activity is higher during cognition.

It should also be understood that these terms really come from a method of data analysis such as Fourier analysis that breaks a signal down into component frequencies. It isn't necessary that a signal is actually produced by combinations of sine waves to break that signal down into component sine waves and describe it that way.