I can't seem to wrap my head around how a bunch of organic transistors (neurons) could possibly produce a wave? What exactly is a brain wave? If you graph a brain wave what exactly would your X and Y axis represent? For example a sound wave is graphed with X axis being time and Y axis being pressure. If someone told me to recreate a brain wave and for this purpose gave me a bunch of neurons with complete and exact control of which neuron fires when how would I possibly create said wave?
In the general case of EEG, the x-axis is time, and the y-axis is the voltage potential measured by an electrode placed on the scalp.
When others talk about "brain waves" they typically refer to oscillations visible in EEG electrode signals that have more power in a certain frequency range. For example, "Alpha brain waves" usually refer to EEG signals with maximum power in the range of 7-13 Hz. "Gamma waves" have frequencies >30 Hz.
The dominant theory of the origin of the waves is that they are the result of large populations of individual neurons firing action potentials in a synchronized fashion.
So, if you had the ability to make, say, a 1,000 neurons near the surface of the scalp fire at the same time, a scalp electrode placed above those neurons should register a significant voltage potential deviation around the time of the trigger.
The EEG (or Local Field Potential - LFP) is the result of currents flowing in and out of every neuron in the brain. As this current flows, a dipole is created which creates an electric field - this is what is measured at the electrode. There are details in this excellent review.
It is true that neurons tend to fire synchronously during epochs of oscillations but that is only part of the story. The voltage measured by the electrode also includes sub-threshold activity and it must be noted that the contribution of a current is scaled by 1/(r^2) where r is the distance to the electrode. This means that EEG usually only measures activity from neurons close to the electrode. Activity of subcortical structures (hippocampus, amygdala) get attenuated because they are too far away. That is why you have to actually stick an electrode into the brain to measure from these structures.