One of the first parts of breathing basics is learning to breath through your nose as well as through your mouth. If you can breathe equally well through your mouth or your nose you can choose which method you use at any give time.
In general, inhaling through the mouth tends to be frowned upon in yoga circles, the main reason seeming to be that if you breathe through your nose then your nose helps to filter the air your breathe. This doesn't happen when you inhale via your mouth.
Another argument is that your nasal cavity has prana receptors that help to draw prana, or energy, into your body when you inhale through through your nose. According to this logic, when you inhale through your mouth prana can't enter your body.
I would suggest that on occasions when you need to draw large amounts of air into your lungs quickly (or on occasions when your nose is clogged up) then breathe through your mouth. You can also use mouth breathing to practice controlling the passage through which you draw your air into your body. You can use your lips and even your tongue as a regulator. When you've experienced either of these methods of breath regulation you may then find it easier to use your throat to regulate your breath when breathing through your nose.
If you are mouth breathing you can shape your lips to control the size of the hole you breath through. If you tighten your lips and make a small hole to breathe through you'll find you have to do more work both to inhale and exhale. If you enlage the hole that your lips make then you'll find it easier to breath with less effort.
One exercise you can play with is to make the hole bigger while inhaling and then smaller while exhaling. See if you can make your exhales twice as long as your inhales. Try and keep the muscular effort the same, so don't work harder on your exhales, simply narrow the diameter of your lips so that it takes you longer to exhale.
This may be challenging to begin with. You can work towards doubling your exhales by first making your exhales the same length of time as your inhales. Then see if you can make your exhales a quarter longer than your inhales. So if you inhale for 8 seconds, exhale for 10. Then add another quarter once you are comfortable and inhale for 8 and exhale for 12. Then do three quarters, inhaling for 8 and exhaling for 14. Finally see if you can make your exhales twice as long for example by inhaling for 8 seconds and exhaling for 16.
Making your exhales twice as long as you inhales is said to have a calming effect. However if you struggle to keep the breathing ratio (1:2) then you are probably negating any calming effect. You can try to reduce the time that you inhale and exhale so that the ratio is the same but your breaht cycle is shorter. For example you can try a four second inhale and an eight second exhale (4:8) and gradually increase the time of both.
Another benefit of being able to double your exhales is that you gain control of your ability to regulate your breath.
One other method you can play with for breath regulation is to use your tongue. You can press your tongue up to your top teeth. Breathe betweent the top of your tongue and the roof of your mouth.
To increase the breathing space try to curve your tongue more moving the center of your tongue away from the roof of your mouth so that your tongue makes a trough shape from front to back.
To narrow the passage press your tongue up towards the roof of your mouth.
Another similar method is called sittali breathing where you stick your tongue out between your lips and make a tubular shape with it. This time you create the passage between your tongue and your upper lip. The passage way created with this method is generally quite small. However one advantage is that you may find that it has a cooling effect. The air actually feels cool as it passes over the surface of your tongue.
With the possibility of breathing in and out of your nose or in and out of your mouth there are four basic possible combinations of nose and mouth breathing:
To work towards nose breathing, you can start by inhaling through your nose and exhaling through your mouth. Do a few breaths like this at a time then if you need to, go back to mouth breathing to rest, then try again.
To make nose breathing easier, you might try using your cheek muscles to flare your nostrils. Prior to this align your jaw with your head. Slightly open your mouth to do this (you can keep your lips closed or only slightly open) and align the center of your jaw (from left to right) with the center of the roof of your mouth.
I'd also suggest that you touch the tip of your tongue to the roof of your mouth and keep it there.
With your jaw aligned, smile, and spread the sides of your nose. The affect is like using one of those bandages that athletes use to keep their nostrils open without actually requiring the bandage.
With your cheeks spread, practice nose breathing until you can comfortably inhale through your nose and exhale through your mouth. Then practice both inhaling and exhaling through your nose.
Try to keep your nostrils open.
While breathing, whether through nose or mouth, another method for regulating your breath is by controlling the diameter of your throat. You can learn to control the diameter of your throat just at or below where your swallow reflex happens.
By narrowing the throat diameter you restrict the passageway through which you draw air into your lungs and this can happen whether mouth breathing or nose breathing. If you then try to draw the same amount of air in as when your throat is open, your respiratory muscles have to do more work. This can result in two things.
While breathing through your nose, one of the tricks or games you can play is trying to fill or draw the air through the nasal cavities, the spaces behind your cheeks and above your pallet.
If there are indeed prana receptors in your nasal cavities, then this method is perhaps one way of getting the air to those prana receptors so that you can then draw prana in.
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