Binaural beats work by playing a different tone in each ear, and in the process the brain creates a “phantom tone” which is the difference between the two tones (for example if you play a 300 hz tone in one ear and a 310 hz in the other ear the brain will produce a 10 Hz phantom tone.) If this phantom tone is 10 hz the brain will gradually be led to a dominant brainwave frequency of 10 hz (this has been measured by EEG devices). If the tone is 7 hz the brain will gradually be led to a 7 hz dominant frequency, and so on.

a concise P.S. I was in a horrible car accident when I was 18 and had to learn to read and write and walk and talk all over again. It was long ago and I’m pretty well over it, but such an injury has lifetime repercussions. The neurological damage sustained causes my left arm to shake uncontrollably anytime I use it in a situation requiring precise coordination. From simply listening to the demo, the shake [though not disappearing completely in the last 2 months] has lessened dramatically. I had been wondering for years what kind of neurological exercises I could undergo to help it…
Some studies have found that binaural beats can affect cognitive function positively or negatively, depending on the specific frequency that’s generated. For example, a study of long-term memory found that beta-frequency binaural beats improved memory, while theta-frequency binaural beats interfered with memory. This is something for scientists to continue to examine closely. For people who use binaural beats, it’s important to understand that different frequencies will produce different effects.

Binaural beats are considered auditory illusions. For a binaural beat to work, the two tones have to have frequencies less than 1000 Hz, and the difference between the two tones can’t be more than 30 Hz. The tones also have to be listened to separately, one through each ear. Binaural beats have been explored in music and are sometimes used to help tune instruments, such as pianos and organs. More recently, they have been connected to potential health benefits.

Did you ever wonder why someone who "has everything" could be depressed and unhappy and why someone who has very little can be so happy? Of course, we can answer that material wealth, beauty, situation, good environments aren't everything. (Sure are nice though :+)) But that explanation ends up seeming like a rationalization and not an explanation. Consider the idea that the answer could reside in our brainwaves. Certain brain frequencies promote a feeling of well being and certain frequencies promote a feeling of depression. The state of our brainwaves also influences our tendency to be overly reactive or detached.
A binaureal beat is created by playing a different tone in each ear, and the interference pattern between the slightly differing frequencies creates the illusion of a beat. It's intended to be heard through headphones, so there's no cross-channel bleed across both ears. Listen to this, I'll play a simple binaural beat, and I'll slide the pan control back and forth from one ear to the other. You can see that there isn't actually any beat, it's just an acoustic illusion:
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