I'm confused about speaker ratings and amplifier ratings and how they interact.

Some pages say your speakers should be rated higher than your amp. Other pages say that your amp should be rated higher than your speakers. Both of these cannot be true.

Let's start with why we're concerned about this at all. I think we all understand that a higher rated amp is going to be able to produce a louder sound than a lower rated one (assuming their ratings are equivalently determined). And a higher-rated speaker probably performs the same way. So in the best of all worlds, we'd all get amps and speakers with massively high ratings and be done with it. But economics kicks in.

I know Doug will pipe up and say that higher power ratings means more headroom, which is likely to mean a cleaner output, which I'm all happy to agree with, but that doesn't help me if I'm trying to reduce cost, and therefore reducing power rating.

And what we're interested in mostly with this line of questioning is preventing damage to this equipment. Once we determine how to mix numbers we can approach quality.

So, what causes damage?

The obvious one is clipped waveforms, which have much more power than unclipped waveforms. What causes clipping? Assuming I don't go over 0dB on the empeg, the source isn't clipping, so it must happen somewhere in the amp or speakers. But it's unlikely that your amp (unless it's really cheap) is causing clipping on its own, I think, so it must be some interaction of the speakers and amp (since the speaker itself is obviously not able to cause it).

Some would argue that an underpowered amp causes this, as the high-powered speakers try to draw more power than the amp could handle, like hooking up a bunch of space heaters to one power outlet in your home. This could be a bad analogy because the amp varies voltage while your house does not (not to mention the whole AC/DC thing).

Others would argue that an overpowered amp causes it, like a power surge hitting your delicate computers.

Maybe both of those are true. One page argues that if your speakers are rated way higher than your amp, then even if the signal is clipped that it won't be able to produce enough power to actually damage them. I suppose now is where we get into quality, as that begs the question that even if the speakers won't be damaged, they won't sound good, and is it the high power rating of the speaker contributing to the clipping?

What other causes are there for speaker damage? Overdriving, causing mechanical damage to the coil, surround or cones? What causes this? It would seem that that would be caused by the speaker trying to generate sound louder than it is able to. That would seem to be caused by an overpowered amp. (Obviously the amp would have to be "told" to produce that much volume by turning up the volume on the source.)

Here's another question: what causes a change in volume? Is it voltage? Variation in voltage is what describes the waveform, right? And speakers are designated to be of specific impedances. Is there a set voltage maximum for audio systems or is it that 4V specifies the max deflection of the speaker cone in one speaker, but only halfway in another? Or is current a variable, too? But the waveform voltage doesn't specify volume directly. It's the degree of variation in the waveform that's volume. That is a sinus waveform that travels between 3V and 4V wouldn't be very loud, but one that travels between -4V and 4V would be, despite that its additive value would be zero. Of course, that -4V to 4V sinus wave at 200Hz isn't going to seem as loud as a -4V to 4V sinus wave at 5000Hz. I see now that audio signals are actually AC, aren't they? And you get into power factors with all of this, and not even linear ones, which makes all of this numbering even more complicated than I thought.

Okay, now I'm in too deep, and I just want someone to tell me if I need speakers more powerful than my amp, or speakers less powerful, or that they should be the same, or that I should just go stick my head in a Gramophone just to keep my brain from turning to goo with all this confusing information.

And please show your work. There's more than enough conjecture without any basis. Hell, there's a lot of inconclusiveness pretending to be an answer. To be honest, I don't really care that much about the math, but there are obviously a lot of people out there just repeating what they've heard, so I need some basis for a conclusion now. (That is, if everyone said "your speakers need to be rated at 1.414 times your amps rating", I wouldn't really care about any basis, but as it is, that 1.414 can be replaced by anything from one-tenth to ten and it seems that someone will agree with it.)