So, in summary, binaural beats certainly do not work the way the sellers claim, but there's no reason to think they're any less effective than any other music track you might listen to that effects you in a way you like. If they make you sleepy (like they all do for me), use them to go to sleep. If they relax you or get you amped, use them for that. But don't expect them to be any more effective than regular music. If someone you know claims that they are, put them to the test, and bust the myth.
Both brainwave entrainment and neurofeedback deal with brainwaves, but the similarity stops there. Entrainment pushes your whole brain into a pre-determined state, while neurofeedback teaches you how to move specific parts of your brain on your own. It is the differeence between forcing the brain into a given position, and skills building so you can move it there yourself. 
Earlier evidence out of UCLA suggested that meditating for years thickens the brain (in a good way) and strengthens the connections between brain cells. Now a further report by UCLA researchers suggests yet another benefit. Long-term meditators have larger amounts of gyrification ("folding" of the cortex, which may allow the brain to process information faster) than people who do not meditate. A direct correlation was found between the amount of gyrification and the number of meditation years, possibly providing further proof of the brain's neuroplasticity, or ability to adapt to environmental changes.
However, those with ASD are sometimes also diagnosed with ADHD. Although they are different conditions, some of the symptoms can be intertwined, which it appears may be the case with your grandson. So I would start by trying out my tracks for ADHD. You can try those out for free on my YouTube channel and I’ve put them together in a playlist here: https://www.youtube.com/watch?v=RKao2oZZPnw&list=PLveg0IEcZWN5iVmMduR9UjLzX_YgCetds. The corresponding MP3 versions for those tracks can be found on my site here: https://www.mindamend.com/intelligence-and-brain-power/. I hope that helps.

Deeper Meditation • Near The Stage Of Sleep • Vivid, Dreamlike Imagery • Creative Visualization • Feel More Open & Connected • Advanced Problem Solving • Super Creativity • Insight • Intuition • Inspiration • Deeper Subconscious To Super-Conscious Mind • Trance-like • GABA • Immune System • Serotonin • Endorphins • Acetylcholine • Lower Cortisol • Deeply Relaxed • Sleep Better • Emotional Intelligence
If you search the Internet for "binaural beats" you'll quickly find there's a whole industry built on the idea that listening to binaural beats can produce all kinds of desired effects in your brain. It can alter your mood, help you follow a diet or stop smoking, get you pumped up for a competition, calm you down, put you to sleep, enhance your memory, act as an aphrodisiac, cure headaches, and even balance your chakra. Binaural-Beats.com offers a $30 CD that they call the world's first "digital drug". They claim it can get you drunk without the side effects. I-Doser.com offers a range of music tracks that they say simulates a variety of actual pharmaceuticals, such as Demerol, Oxycontin, and Vicodin. Suffice it to say that no matter what superpower you're looking for, someone on the Internet sells a binaural beat audio file claimed to provide it.

Binaural beats are created when two tones are played simultaneously, one in each ear. As long as the difference between the two frequencies is less than 30 hertz, the brain perceives the difference in the tones as a “beat” which, if listened to with enough focus, the brain will begin to follow (this is known as the “frequency following response”). By changing the difference between the two beats, the brain will change its wavelength and enter into different “states” of being.

Deeper Meditation • Near The Stage Of Sleep • Vivid, Dreamlike Imagery • Creative Visualization • Feel More Open & Connected • Advanced Problem Solving • Super Creativity • Insight • Intuition • Inspiration • Deeper Subconscious To Super-Conscious Mind • Trance-like • GABA • Immune System • Serotonin • Endorphins • Acetylcholine • Lower Cortisol • Deeply Relaxed • Sleep Better • Emotional Intelligence
In order to take full advantage of binaural beats to tackle insomnia, you should first buy a high-quality recording. Although it’s true that any two tones that have wavelengths that are seven hertz or less fewer apart will create a binaural beat, there are other factors to take into consideration. Many people find the tones that create binaural beats unpleasant, so a good recording will also have a “pink noise” to help mask the sound (falling rain is one popular such noise).; this noise will also help you focus on the binaural beat instead of letting your mind wander.
The binaural-beat appears to be associated with an electroencephalographic (EEG) frequency-following response in the brain (3). Many studies have demonstrated the presence of a frequency-following response to auditory stimuli, recorded at the vertex of the human brain (top of the head). This EEG activity was termed "frequency-following response" because its period corresponds to the fundamental frequency of the stimulus (Smith, Marsh, & Brown, 1975). Binaural-beat stimulation appears to encourage access to altered states of consciousness.
Business executive and radio producer Robert Monroe started experimenting with brainwave entrainment and has a series of powerful out of body experiences using it. In 1971 he published his cult classic “Journeys Out of the Body” sharing his experiences. He later created one of the first audio entrainment companies called Hemi Sync, alongside the Monroe Institute of Applied Sciences.
The Alpha state is an intensely pleasurable and relaxed state of consciousness, essential for stress reduction. Alpha is associated with deep physical and mental relaxation, but not quite meditation. Alpha waves instill a sense of mild euphoria, calm your body, reduce Cortisol levels, strengthen your immune system and counteract the negative effects of stress. Alpha waves also increase the production of Melatonin to greatly improve the quality of your sleep.

Recently developed entrainment software has been designed to correct specific imbalances in hemispheric activity associated with undesirable mind-states. For example, people suffering with depression often have more activity in the right hemisphere, and specially designed brainwave music decreases this activity while increasing activity in the left hemisphere, reducing depression.
Entrainment is a term originally derived from complex systems theory, and denotes the way that two or more independent, autonomous oscillators with differing rhythms or frequencies, when situated in a context and at a proximity where they can interact for long enough, influence each other mutually, to a degree dependent on coupling force, such that they adjust until both oscillate with the same frequency. Examples include the mechanical entrainment or cyclic synchronization of two electric clothes dryers placed in close proximity, and the biological entrainment evident in the synchronized illumination of fireflies.[13]

In 1973, biophysicist Dr. Gerald Oster published a famous article in Scientific American titled “Auditory Beats in the Brain”, which found that when two pure tones of varying frequencies were combined, a third rhythmic beat was created which he called binaural or monaural beats. According to Oster, monaural beats occur when two tones are combined and sent through a loudspeaker, while binaural beats occur when stereo headphones are used to deliver each tone separately to each ear. Oster concluded that monaural beats were a more effective form of brainwave entrainment.
Binaural beats can easily be heard at the low frequencies (< 30 Hz) that are characteristic of the EEG spectrum (Oster, 1973). This perceptual phenomenon of binaural beating and the objective measurement of the frequency-following response (Hink, Kodera, Yamada, Kaga, & Suzuki, 1980) suggest conditions which facilitate entrainment of brain waves and altered states of consciousness. There have been numerous anecdotal reports and a growing number of research efforts reporting changes in consciousness associated with binaural-beats. "The subjective effect of listening to binaural beats may be relaxing or stimulating, depending on the frequency of the binaural-beat stimulation" (Owens & Atwater, 1995). Binaural beats in the delta (1 to 4 Hz) and theta (4 to 8 Hz) ranges have been associated with reports of relaxed, meditative, and creative states (Hiew, 1995), and used as an aid to falling asleep. Binaural beats in the alpha frequencies (8 to 12 Hz) have increased alpha brain waves (Foster, 1990) and binaural beats in the beta frequencies (typically 16 to 24 Hz) have been associated with reports of increased concentration or alertness (Monroe, 1985) and improved memory (Kennerly, 1994).
When signals of two different frequencies are presented, one to each ear, the brain detects phase differences between these signals. "Under natural circumstances a detected phase difference would provide directional information. The brain processes this anomalous information differently when these phase differences are heard with stereo headphones or speakers. A perceptual integration of the two signals takes place, producing the sensation of a third "beat" frequency. The difference between the signals waxes and wanes as the two different input frequencies mesh in and out of phase. As a result of these constantly increasing and decreasing differences, an amplitude-modulated standing wave -the binaural beat- is heard. The binaural beat is perceived as a fluctuating rhythm at the frequency of the difference between the two auditory inputs. Evidence suggests that the binaural beats are generated in the brainstem's superior olivary nucleus, the first site of contra-lateral integration in the auditory system (Oster, 1973). Studies also suggest that the frequency-following response originates from the inferior colliculus (Smith, Marsh, & Brown, 1975)" (Owens & Atwater, 1995). This activity is conducted to the cortex where it can be recorded by scalp electrodes.  
The review by Cahn also notes findings describing a heightened emotional state of meditators. A more complex study, conducted in 2008 by Lutz et al., focused on emotional response during meditation.[9] This investigation involved the creation of a “compassion meditation” state by novice and experienced meditators and testing the meditators response to emotionally charged sounds. fMRI results indicated heightened activity in the cingulate cortex but also in the amygdala, temporo-parietal junction, and right posterior superior temporal sulcus in response to the emotional sounds. The authors of this study believe this indicates greater sensitivity to emotional expression and positive emotion due to the neural circuitry activated.[9]
A 2008 study at Hofstra University played two different binaural beats and a control sound (a babbling brook) to patients with high blood pressure. There was no difference between the groups. In one small study from Japan that was published in the Journal of Neurophysiology in 2006, they played various binaural beats to nine subjects, and observed the resulting EEGs. They found great variability in the results. Their conclusion was that listening to binaural beats can produce activity on the human cerebral cortex, however the cause was more likely a conscious auditory reaction and was not correlated to the frequency of the binaural beat. However, a 2005 study published in Clinical Neurophysiology found that they were able to induce a desired frequency in the EEG matching the phantom beat frequency encoded in a binaural beat, however this was with a single subject and was neither blinded nor controlled.
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