Does pushing 500 rms @ 4 ohms sound/make a sub perform differently than pushing 500 rms @ 2 ohms or 1 ohm?
No. 500 watts is 500 watts. If the driver into which you are putting the 500 watts has differing sensitivity numbers, or different specifications across the board, then they will sound or perform differently.
If you are using a DVC woofer, and you simply adjust the impedance from 2 ohms to 8 ohms (in the case of a DVC4 woofer) then there will be no difference in acoustic character.
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It all reminds me of something that Molière once said to Guy de Maupassant at a café in Vienna: "That's nice. You should write it down."
As far as your standard movement of the sub and power flowing into it, no there is no difference. There is a difference however in the THD% with most amps. I would say that atleast 90% of all amps that have THD% rated, are rated at a 4-ohm load and a THD% of <1%. When they go to lower ohm levels though, that number typically rises.
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Head Unit - Pioneer P80MP
Sub - 1 10" Soundstream T-5
Amps - 1 Kicker KX400.1, 1 Legacy crap 2 channel
Door Speakers - Audiobahn 600V Components
Box - Custom by Southern Sound in Statesboro, GA