So to summarize their claim, they're saying that entrainment means that a binaural beat will cause your brain's electroencephalogram to match the pattern of the phantom beat. Well, if it did, entrainment certainly doesn't apply and would not be part of the equation, so we can scratch that off the list. But it doesn't make the claimed observation wrong. We do know that certain electroencephalogram waveforms are often associated with certain kinds of activity. For example, physical activity or REM sleep often produces an electroencephalogram with a sine wave of between 4 and 8 Hz, which we term a theta pattern. Waking relaxation with eyes closed often produces a pattern from 8 to 12 Hz, which is called an alpha pattern. There are only a few characterized patterns, and pretty general descriptions of what kinds of activities go with them. The claim made by the binaural beat sellers depends on much more granular and specific matches. For example, the claim that a binaural beat with a frequency of X produces the same effect in your brain as Vicodin is wholly implausible. Such claims presume that we know the exact frequency of the electroencephalogram in each of these desired conditions, and the fact is that brain waves don't work that way. It is wholly and absolutely implausible to say that desired brain condition X will occur if we get your EEG to read exactly X Hz.
All techniques, whether mantra, focus on the breath, contemplation of a koan, or any other, are simply the vehicles we use to take us into (hopefully) a meditative state. Put another way, the purpose of repeating a mantra is not to get proficient at repeating a mantra. The purpose is to take us to a meditative state where the mantra slips away and is no longer necessary.
Entrainment is like priming the pump of meditation. It doesn’t “cause” meditation any more than pouring that little bit of water into the pump initially “causes” water to appear in the well. The water was already there. The conditions for the pump to do its job just needed to be set up. What entrainment does is to create the conditions and the internal environment typically associated with meditation. You could get to that point without entrainment, certainly, but the entrainment allows you to get there quicker, more consistently, and to remain there for longer periods of time than you would ordinarily be capable of doing on your own.

Binaural beats require two separate tones from two sources that are combined inside the listener’s brain to form the target tone. The lower frequency sound is called the carrier tone, and it is combined with a higher frequency sound known as the offset tone. Because of this, binaural beats must be listened to with stereo headphones or the effect is lost. Binaural beats create a hypnotic effect, but they are not the most effective tool for brainwave entrainment, and binaural beats are often ineffective for people with hearing loss.
In the 1980s, a researcher in Japan, Tsuyoshi Inouye described how light stimulation creates synchronization of brain hemispheres. Since then, other researchers have detailed the positive effects of hemispheric synchronization including a 1984 study by researcher Dr. Gene W. Brockopp stating that hemispheric synchronization resulted in improved intellectual functioning as well as improvements in long-term memory, and these effects are cumulative over time.
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When we are mentally active various groups of neurons will be firing and the EEG will look like a jumble of different waves at different frequencies. When we are in a relaxed state, however, our brains settle into a steady rhythm – when fully awake this is the alpha rhythm, which as a frequency of 8-12 hz and other recognizable features. When drowsy our brainwaves slow to the theta range, 6-7hz, and when in deep sleep into the delta range, 4-5hz.
Dealing with sudden stress—a phone call with bad news, a last-minute assignment from your boss, or an argument with your spouse—triggers a cascade of physical and mental symptoms that can be hard to stop. The first thing to do? Pay attention to your breathing pattern, and make an effort to start taking slow, deep breaths as stress relievers. “Breathing can change how we feel because emotions and breathing are closely connected,” says Emma Seppala, PhD, author of The Happiness Track and a Stanford University psychologist who’s done research on yogic breathing. “A revealing research study showed that different emotional states are associated with distinct respiration patterns.” In a follow-up study, participants actually started to feel the emotions that corresponded to an assigned breathing pattern. “This finding is revolutionary: We can change how we feel using our breath!” Dr. Seppala says. Deep breathing gets more oxygen to your brain and may lower levels of the stress hormone cortisol.
Brainwave entrainment music can be used almost anywhere and anytime, making this mood and self-improvement method versatile and flexible enough to do at work, while traveling, or at other times during the day. When used in the workplace during short rest periods, brainwave entrainment techniques can enhance concentration, communication, and work productivity.
Sleep issues are becoming much more of a common problem. I think it's great to look at alternative ways to help with insomnia and other sleep related issues, especially when they don't involve the need to take medication. Something proved to be effective is 'Sleepstation' a UK based organisation who use CBT techniques to cure insomnia. Their online sleep therapy course is brilliant and very convenient as it's delivered online. Their website is definitely worth a look if you're having sleep issues

You are already far beyond the need for using an entrainment product as a meditation aid. Those who sincerely believe this to be the case would not have come searching for a meditation site such as Project Meditation to help them learn and practice meditation and brainwave entrainment. They would probably not even bother with using an entrainment aid; hence this entire discussion would be irrelevant to them. The fact that you are here, reading this, seeking to benefit from incorporating meditation and brainwave entrainment to aid in your existing meditation practice suggests you are not one of these folks whose development is so advanced that entrainment is of no use to you.
Slightly higher-frequency entrainment can lead to hyper suggestive states of consciousness. Still higher-frequency EEG states are associated with alert and focused mental activity needed for the optimal performance of many tasks. Perceived reality changes depending on the state of consciousness of the perceiver (Tart, 1975). Some states of consciousness provide limited views of reality, while others provide an expanded awareness of reality. For the most part, states of consciousness change in response to the ever-changing internal environment and surrounding stimulation. For example, states of consciousness are subject to influences like drugs and circadian and ultradian rhythms (Rossi, 1986; Shannahoff-Khalsa, 1991; Webb & Dube, 1981). Specific states of consciousness can also be learned as adaptive behaviors to demanding circumstances (Green and Green, 1986).  
The advice “take a deep breath” may seem like a cliché, but it holds true when it comes to stress. For centuries, Buddhist monks have been conscious of deliberate breathing during meditation. For an easy three- to five-minute exercise, sit up in your chair with your feet flat on the floor and hands on top of your knees. Breathe in and out slowly and deeply, concentrating on your lungs as they expand fully in your chest. While shallow breathing causes stress, deep breathing oxygenates your blood, helps center your body, and clears your mind.

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The mechanism for this is that when your eyes or ears are exposed to a particular frequency of pulses or beats, the thalamus first distributes this information to the entire brain, including the visual and cerebral cortex where neural activity begins to synchronize to the incoming frequency, producing hemispheric synchronization and a balance of brainwave activity across the brain.
Because the mind and body are a single system, changing our brainwaves and spending more time in harmonious, relaxed, and restorative mind-states also affects our physical health. Physical health then reinforces our mental-state, and a feedback loop of either positive or negative processes becomes established. Research studies have shown beneficial effects of using brainwave entrainment for treating migraine headaches, premenstrual syndrome, and for managing physical pain.

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:

Brainwave entrainment is a colloquialism for such 'neural entrainment', which is a term used to denote the way in which the aggregate frequency of oscillations produced by the synchronous electrical activity in ensembles of cortical neurons can adjust to synchronize with the periodic vibration of an external stimuli, such as a sustained acoustic frequency perceived as pitch, a regularly repeating pattern of intermittent sounds, perceived as rhythm, or of a regularly rhythmically intermittent flashing light.

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