There’s a growing body of research suggesting that binaural beats can reduce different forms of anxiety, from mild to chronic. One especially interesting study looked at the effects of binaural beats on anxiety among patients preparing to undergo surgery—a life circumstance that is pretty anxiety provoking for most anyone. Over a period of six months, patients spent 30 minutes on the day of their surgery listening to binaural beats. Compared to patients who listened to a soundtrack that did not include binaural beats—and patients who received no “beats” therapy at all—the binaural beat listeners experienced significantly greater reductions in their anxiety levels.
Meanwhile, the therapeutic benefits of listening to sound and music is a well-established principle upon which the practice of receptive music therapy is founded. The term 'receptive music therapy' denotes a process by which patients or participants listen to music with specific intent to therapeutically benefit; and is a term used by therapists to distinguish it from 'active music therapy' by which patients or participants engage in producing vocal or instrumental music.[37] Receptive music therapy is an effective adjunctive intervention suitable for treating a range of physical and mental conditions.[38]
I wouldn’t personally recommend listening to delta wave frequencies for depression, so I’m not sure who advised you to do that? People with depression usually have a higher ratio of theta and delta wave activity, so I would normally recommend listening to high alpha wave and low beta wave frequencies, to help balance things. I have some 10Hz alpha tracks for serotonin release, which you can try for free on my YouTube channel here: I also recommend you give these SMR (low beta wave) tracks a try: I hope that helps.
Generally speaking, the brain will usually entrain to the strongest stimulus which would be isochronic tones over binaural beats. So when you see people add binaural beats at a different frequency to the isochronic tones, that would not produce additional brainwave entrainment at another frequency. If they are both at the same frequency I haven’t seen any research to indicate whether that would be beneficial or not, but my belief is that it would weaken the potential for entrainment. When you look at the waveform of an isochronic tone there is a distinct empty space between each beat, making it very pronounced and effective. When you add binaural beats at the same frequency it looks like this: The depth of the waveform is now half as deep and less effective. This is before the binaural beats are formed inside your head, where the waveform is hard to determine and measure. From listening to that type of combination the beats sound much less pronounced, which has to make them much less effective in terms of a brainwave entrainment stimulus, compared to isochronic tones on their own.

Alpha Waves are electromagnetic oscillations in the frequency range of 8Ð12 Hz arising from synchronous and coherent (in phase / constructive) electrical activity of thalamic pacemaker cells in humans. They are also called Berger's wave in memory of the founder of EEG. They place the brain in states of relaxation times, non-arousal, meditation, hypnosis
A therapy that slows brainwave activity, helping to produce low-frequency waves, is likely to aid relaxation and sleep. But it’s not only lowering brainwave frequency that binaural beats may offer to sleep and relaxation. A small study (19 people) has found that exposure to binaural beats is associated with changes to three hormones important to sleep and well being:

But the notion of changing brain waves is a very appealing one, from a marketing stand point. People can visualize brain waves and we like synchrony. Also, in the computer age, we understand the notion of “programming.” We also have been prepped for the future by movies such as The Matrix, where people could master Kung Fu in minutes by simply “downloading” the knowledge. This gives the whole notion a superficial plausibility. But the science just isn’t there.

Are you someone who has turned to meditation or yoga as a way of relieving stress and improving your overall well-being? Yoga and meditation are time-proven methods, used for centuries, which restore mental, physical, and spiritual balance in people’s lives, and brainwave entrainment can be used in conjunction with these practices for even deeper levels of benefit. 

That is generally where the science ends and the pseudoscience begins. A number of companies and individuals have then extrapolated from the phenomenon of entrainment to claim that altering the brain waves changes the actual functioning of the brain. There is no theoretical or empirical basis for this, however. Entrainment is a temporary effect on the synchronization of neuronal firing – it does not improve or increase brain functioning, it does not change the hardwiring, nor does it cure any neurological disorder. There is no compelling evidence for any effect beyond the period of entrainment itself.
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: