Think of your alternator and battery as being similar to the water main and toilet tank (it's early, good analogies come later in the day)
Your battery stores enough charge (even when the 'water' is off) to do things like power the starter motor to start the engine (flush). The drained battery is slowly and steadily replenished by the alternator (refill cycle). You can flush again at half full, but it's not as effective as waiting for it to fill completely.
You will notice when you start your car, that so much power is drained by the starter that your lights will dim, possibly to the point of seeming to go off. This is exactly what is going on with the sound system, albeit at a lesser draw but much higher rate of repetition.
Now measuring voltage (or current for that matter) is tricky because it's an average at best (whether the sampling period of a digital meter, or the mechanical hysteresis of an analogue needle). The only way to truly measure what's happening, would be to use a sampling oscilloscope, or some other kind of fast sample and record mechanism (at least twice your highest audio frequency...so about 40,000 times per second).
A larger battery is akin to a bigger tank, one that doesn't completely drain on the first flush, and so recovers more quickly (at a fixed rate of water input) for the next flush. As long as you don't exceed the maximum input flow on average, you can sustain flushing, with faster response times in general. Electrically speaking what you want is a bigger 'tank size' and battery capacities are measured in Ampere/hours, the more the merrier (and heavier)...so long as it still fits in your car.
Now the other problem with filling tanks is the length of the piping. The more piping, the more pressure (voltage in our analogy) drop you get at the use point. You can reduce this by using bigger pipe (fatter wire). You can also provide local pressure storage (imagine a big balloon on the water main right at the tank) which provides on demand water at full pressure...this is your capacitor. The more Farads of capacity, the bigger the 'balloon'. It's also a way to build up a reservoir of easily and quickly deliverable 'water' at the point of use, which is then replenished over time using a smaller pipe. This could be used to ensure that our toilet tank always refills in say 10s, rather than the 40s required by using the water pipe alone...which is exactly what we want to do for your audio power. Go ahead and install that cap as close to the amp as you can. It should come with a charge limiting resistor that goes in series between the battery and the cap, make sure that goes in the right place!
The last point not to neglect is the 'drain'...it needs to be at least as good as all the supply system...so if you can deliver 100 gallons per minute at 50 PSI...your drain better be able to handle the efflux
In our analogy the drain is the negative/ground system. It is particularly important to ensure your battery is very solidly grounded to the chassis, and that all your high power equipment also has a solid and robust ground. Keep in mind that steel isn't the best of conductors, and that even a few tens of milliohms of extra resistance can result in a voltage drop in the order of 1/2 volt or more at high power. For instance, assume 0.02 ohms extra resistance (versus copper #2
cable) to battery via steel chassis and chassis to battery strap. (This is probably conservative and it completely ignores how that resistance chances with frequency)...but at 40A, using Ohm's law...your voltage drop (or in this case rise) from battery -ve to amplifier -ve is V=IR, or in other words Vdrop = 40 A * 0.02 ohms or 0.8V
Hope I didn't lose you in all that, but if you tell me which model of battery you have and which one you are looking at I can help you understand if the new one is significantly increased in capacity.