Hate to bring out the math, but here is the basic formula:
Power = RMS Current * RMS Voltage
Then if we substitute Ohm's Law
Current = Voltage / Load Resistance
and assume the amp can always supply enough current we get
Power = RMS Voltage * RMS Voltage / Load Resistance
The RMS Voltage in this case is the maximum time averaged voltage the amp can supply. It depends on two things - the maximum voltage the amp's output can swing and the shape of the waveform.
For a sine wave RMS Voltage = 0.707 * Max Voltage Swing
For a square wave RMS Voltage = Max Voltage Swing
So let's say we have a power amp who's output stage can swing plus or minus 40 Volts and we're driving an 8 Ohm speaker load. Sine waves are usually the standard waveform because they better approximate music content, so we'll use that formula. The power output of the amp is then:
Power = (0.707 * 40V)* (0.707 * 40V) / 8Ohms = 100 Watts
Now if we switch to a 4 Ohm speaker and assume the amp can still supply enough current we get:
Power = (0.707 * 40V)* (0.707 * 40V) / 4Ohms = 200 Watts
So the power output depends on the fixed maximum voltage swing the amp is capable of and the speaker load attached to it. BUT, not completely. In some case you may see specs like this:
Power Output:
100 Watts into 8 Ohms
150 Watts into 4 Ohms
You'd expect to see 200 watts for 4 Ohms, right? Well this spec illustrates an amp with limited current capacity. It can't reach it's maximum voltage swing with a 4 ohm load because it can't supply enough current to do it. So, why do some of those relatively low power amps in the hi-fi stores cost and weigh nearly as much as battle tanks? Because they have massive power supplies and arrays of output transistors providing huge current capacity such that their specs look very much like an ideal amplifier:
Power Output:
100 Watts into 8 Ohms
200 Watts into 4 Ohms
400 Watts into 2 Ohms
800 Watts into 1 Ohm
Oh, and that "peak to peak power" is kind of a bullshit rating calculated from a square waveform. In that case we could say: "Peak to Peak Power" = 40V *40V / 8Ohms = 200 Watts. But almost nobody plays music which looks anything like a pure square wave.
Hope this helps.
barefoot