The most important Mitzva of Rosh haShana is listening to the Shofar. The Shofar is an animal horn which was modified removing its inside material (keratin) and opening a mouthpiece in its upper narrow ending. The most common horn used to make a Shofar is a ram's horn. When blown, the Shofar emits a deep and loud sound. The sound is produced by releasing air through its mouthpiece.
If you try to blow a Shofar do not place it inside your mouth in between your lips the way you would blow-up a balloon. Place it on or against your lips, usually on one side of your mouth (most people use their right side). Then you should blow air with your lips tightly closed. Buzzing or emitting the sound of the letter 'P' while vibrating your lips as you release the air. Don't inflate your cheeks and don't force your lungs. Breath as normally as possible. The most difficult part is adjusting the placement of the Shofar on your lips, avoiding air to escape from any other point of your mouth.
The Shofar normally produces one sound. In Rosh haShana we play three different voices using the same sound. The three voices have theoretically the same length but are divided differently. The first voice Teqi'a is a plain uninterrupted sound. Then we have Shebarim a voice divided into three smaller unit. And finallyTeru'a consisting of nine very short units.
The typical combination of sounds in Rosh haShana is the following:
teqi'a/ shebarim-teru'a/ teqi'a.
teqi'a/ shebarim/ teqi'a
teqi'a / teru'a/ teqi'a.
This formula repeats itself several times until we reach more that 100 sounds.
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Great tutorial on the technique you need to blow the Shofar.