A speaker relies upon an electromagnetic coil, referred to as a voice coil, to oscillate and vibrate a cone or dome, which is attached to the coil. If the cone or dome becomes disattached, partially or fully from the coil, the speaker is blown. This often occurs when playing music at volumes that are too high for the amplifier/speaker to handle.
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Listen to the speaker. A blown speaker will sound distinctly distorted and your audio will be noticeably off.
Discern if there's no sound coming from the speaker or simply a sonic fuzz. If there is no sound, first check the connections to make sure everything is hooked up properly. A speaker that is completely blown will emit no sound or be extremely distorted.
Feel the woofer. Woofers move noticeably when producing low frequencies and often pump air out that you can feel. Pull the grill off the speaker and feel if the woofer is moving. If the speaker is blown it will not be moving.
Use a multimeter. Turn the stereo off. Connect a multimeter to the speaker terminals and read the impedance. A blown speaker will read infinite impedance, whereas a speaker that is not blown should read "1."