I am fortunate to be working with Deepak Chopra, M.D., and Dr. Rudy Tanzi, co-authors of the bestselling book Super Brain, on a technology called Brain Wave Entrainment. Deepak is very well known, but Rudy is an amazingly interesting person as well. He is the Kennedy Professor of Neurology at Harvard Medical School and Vice-Chair of Neurology at Mass General Hospital. Rudy co-discovered three of the four original Alzheimer's genes and today runs the Alzheimer's Genome Project. He also plays the keyboards, including, at times, for Aerosmith. He is kind of a real life "Buckaroo Banzai."
The effects are strongest while you are listening to the tones because your brainwaves are synchronized and tuned into the frequency range you desire at that time.  After you've stopped listening the effects can still linger for a while afterwards.  The timescale will vary from person to person and be affected by what you do after you've stopped listening.
One of the first pieces of research into using binaural beats as a sleeping aid was In a paper called “Tests of the Sleep Induction Technique” published in 1975 by Dr Arthur Hastings. He noted the effects of subjects who listened to a tape containing binaural beats starting from beta slowly descending to delta. Using an EEG machine he was able to monitor the subject’s brainwaves as they listened to the tape, becoming increasingly relaxed before falling asleep. He concluded; “patterns in the various stages suggested that the tape was influencing the subject’s state”.
Binaural beats were first discovered in 1839. The effects of using audio instead of light for brainwave entrainment were noted in 1959. Oster’s research paper in 1973 brought attention to the use of binaural beats in a medical setting and this inspired a range of studies in the 1980s indicating very positive findings. The effects of brainwave entrainment using binaural beats has become of a hot topic of research ever since but despite their reported successes, they have never quite shook off the sometimes negative connotations of being an “alternative medicine”.
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OK – so some of this is pretty common marketing hype. Consumers have become almost numb to such hyperbole. But still we see some of the common – almost ubiquitous – elements of pseudoscientific scams. The company claims that the device is “effort-free”, that the results are “rapid and long-lasting,” that it is the “most effective” method of its kind, and that it is useful for a broad range of applications (hey, why limit the market). I am used to such hype about dishwashing detergent, but find it intolerable when applied to a pseudoscientific device with medical applications.

Beta brainwaves are further divided into three bands; Lo-Beta (Beta1, 12-15Hz) can be thought of as a 'fast idle', or musing. Beta (Beta2, 15-22Hz) is high engagement or actively figuring something out. Hi-Beta (Beta3, 22-38Hz) is highly complex thought, integrating new experiences, high anxiety, or excitement. Continual high frequency processing is not a very efficient way to run the brain, as it takes a tremendous amount of energy. 
Now for the good part: depending on the frequency, binaural beats can be used to manipulate what kinds of waves your brain is sending out to promote sleep, deep thinking, etc. The lowest frequencies, from 0.1-4 Hz, are the best for deep sleep, pain relief, and reduction of stress hormones like cortisol. Additionally, these frequencies aid in the release of DHEA and melatonin, two of the sleep hormones. The next lowest, from 4-8 Hz, promotes REM sleep (when vivid dreaming and memory development occurs), deep relaxation, and is great for meditation. The higher frequencies, from 14 up to 100 Hz, have benefits like increasing focus, awareness and cognitive functioning for activities like information processing and problem-solving.
Since it’s humble beginnings, the science of brainwave entrainment has evolved to be one of the most potent and powerful ways to unlock the full potential of the human mind. Our audios have been created by brainwave entrainment engineer, Ashton Aiden, and his years of research, expertise, personal experience, and creativity. We are confident that the audios we offer on this site are of the best quality you will ever find, anywhere.
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.

To get a full answer you should really get in touch with the owner of the app, because it does really depend on how their track is constructed and how they intend it to work. I have one 50-minute sleep track which takes you down into delta (http://www.mindamend.com/shop/sleeping-and-dreaming/sleep-induction-isochronic-tones/) and that is designed to just get you to sleep, so you just let play until the end. If you played that track on repeat it might make you jump and wake you up, as the track begins at a higher frequency. I have another 8-hour sleep track which is meant to be played all throughout the night (http://www.mindamend.com/shop/sleeping-and-dreaming/deep-sleep-8-hour-sleep-cycle/). So it does depend on the individual track you are using.
Listening to Theta Meditation will guide your mind into relaxation that is so deep that most people are unable to stay awake. You won't go into deep sleep but stay in a place of not being asleep and not being awake either. Some people experience dreams. Others don't remember anything. Some people are able to have normal thoughts but after the meditation they seem as if the thoughts were part of a dream.
To fall asleep we need to go from our normal waking state, to a relaxed theta, and then finally all the way to delta. Binaural beats are a scientifically proven way to speed up this process, so you get to the delta state quicker and as a result fall asleep faster. For more about the science of binaural beats, have a read of the article Fall Asleep Fast With Binaural Beats.
In order to understand brainwave entrainment you first have to understand something about brain waves and electroencephalograms (EEG). Neurons are cells that communicate with each other through electrical conduction. When a neuron fires it creates a small electrical and magnetic field. These fields are far too small to measure by placing electrodes on the scalp, or even on the surface of the brain. The only reason we can detect the electrical fields of the brain is because many neurons are firing together – in synchrony. All brain waves that are measured by an EEG, therefore, represent a large group of neurons firing together.
“Binaural beats are not very noticeable because the modulation depth (the difference between loud and quiet) is 3 db, a two-to-one ratio. (Isochronic tones and mono beats easily have 50 db difference between loud and quiet, which is a 100,00-to-1 ratio). This means that binaural beats are unlikely to produce an significant entrainment because they don’t activate the thalamus.”

This is something which will come to you with practice. For instance you may feel an energy around and inside you like you have shifted dimensions but are still there. Instead of thinking this is God, the Zone or whatever, think of this as your soul, you essence, your expanded pure light being. In this way you will begin to train yourself to find your identity in your expanded being and your internal identity. This is what we want - to let go of the idea that we are a limited ego.
There is also evidence to suggest meditation plays a protective role against the natural reduction in grey matter volume associated with aging. One study found evidence that Zen meditators experienced a slower age related decline rate for cerebral gray matter volume in the putamen which plays a role in learning, cognitive flexibility and attentional processing [14] This could suggest a better attentiveness in aging meditators versus non-meditators.
Well, except for one reason: The power of suggestion. If I give you a music track and tell you that it will cure your headache, you're more likely to report that it cured your headache than you are to say "Well it didn't effect my headache, but it made my short-term memory better." An interesting experiment would be to buy a binaural track claimed to induce drunkenness, for example; play it for five friends without telling them the claim, and then ask how it made each of them feel. Give them multiple choices to select from. Chances are they're going to respond all over the map. If you have a friend who is a believer in binaural beats, I suggest going ahead and setting up this little test.