An alternator does not "take over". ALL the power for your car at all times comes from the battery. BUT under all normal circumstances when the engine is running, the alternator is also charging the battery at the same time that the rest of the system is drawing power. By design, the minimum charging rate at idle is more than enough to make up for the current draw from the battery for the rest of the car, so the battery does not get drained. However, if you were to add to your system a new power consumer with significant current draw when the engine is just idling, then the alternator output may NOT exceed the total current drain, and your battery will be supplying more power than the alternator can make up.
You installed a speaker that uses at max 120 W, or 10 A at 12 VDC. It is not clear whether the entire speaker and amp system uses 10 A max, or whether the speaker itself uses up to 10 A and the amp may be consuming more than that. Whichever, when the speaker actually IS potting out a loud bass sound and thus using nearly its max current rating, that may be in the range that an idling alternator cannot supply. Don't forget, the alternator is designed to be able to produce more current at higher rpm, but at idle rpm its output is just enough to offset the car's "normal" draw at idle. In fact, when the alternator is forced (by high use of the subwoofer for loud bass tomes) to produce the max current it can at idle speed, that represents an increased load on the engine turning the alternator, which is why you note the engine rpm's are down and it feels like it is shuddering and might even stall.
A capacitor will do NOTHING for this.A good one can supply a surge of current for less than a second so a single bass drum "thump" can be delivered correctly, but then it needs to be recharged before the next "thump" needs extra power. If the power supply to the line cannot deliver that right away, it does not solve your problem.
You could ask an auto tech to check whether the engine idle speed is actually set correctly. This is not something normally adjusted by the end user, but there is a spec for what it should be and how to adjust.Commonly it is spec'd when the car is in Drive and idling but stopped, just like at an intersection red light. BUT that's with what the car designers called "normal" electrical load at idle. If it is already too low, just setting it back up to spec might help. Alternatively, you might ask if the tech can make the adjustment to spec WITH the new subwoofer operating and using significant power with lots of bass music. Running the engine at a slightly higher idle speed would increase the alternator output under the new increased-load version of "idle".
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