Thanks for the explanation and especially for the TL;DR versionI am not an engineer, but I might be able to help a little.
It is a good analogy, it is called the water flow analogy. It is used all the time, to visualize electricity by replacing it with water.
If you think of a water pump as the power source (or amplifier). The pressure (psi) of the water, is the same as Volts in electricity. The flow of the water or liters per min, is the same as current in electricity. And the diameter of the pipe, is the same as resistance (Ohms) in electricity.
You can solve for the power output of electricity by multiplying Volts by Amps to get watts. And you can solve for power output from a water system by multiplying psi by lpm to get, also watts. Here the thing. I hear it asked all the time here, but in a different form. People want to know "what is the impedance at the output?". I believe really what they are asking is what are the amperes at output, lol. And that is a difficult question to answer because it always is changing. BUT there is a trick, Ohms law (who else? lol). It says that the current (amperes) is equal to the voltage divided by the resistance (Ohms). In more cases than not (for me anyway) it is easier to figure out the amperes by knowing the resistance, than figuring out the resistance by knowing the amperes. In either case you will need to know the voltage to solve. And the figure for resistance, I am pretty sure, would be just the resistance (Ohms) from the headphones ( I say this because how else are you going to quantify resistance? Add it all up, Headphone resistance plus transformer resistance? I don't think so. I think there is just one resistance figure and it is the highest one.). Anyway, Divide the output voltage by headphone ohms and that will give you the output amperes. I may be wrong but I believe that the output Ohms by nature always match the actual Ohms of headphones.
Your analogy was slightly off in the concept of a constant 500mW, (and also calling it a volume of water. The volume of water is represented by liters per min, not watts). But the idea that a hose is delivering defined and constant water from a pump is more or less correct, but how that water turns into power is changed very much by what happens after the pump. Mostly by the size of the pipe, and in your example the introduction of a finger (as it does change the size of the pipe). Which is a very good comparison to an impedance switch that can be flipped on or off independently of what the original pipe size had been. BUT, it is important to remember what that finger does, lol. It may be an impedance, but what is it doing? It is raising psi (and so does the pipe),,, or voltage... SO while it may be called impedance, because it also raises pressure it is also voltage. Your example shows how power output is never constant (if it was you would just hear one loud tone in your ear). As you put your finger over the hose, you greatly increase psi or volts, the lpm may go down a bit, but the overall power output goes way up, and can knock over the heavy object or turn a water wheel. Here is the thing, you can do it other way around too. A hose with enough liters per min going through it with very low psi or resistance, could still knock that same heavy object down. It is just generally you don't have extra lpm (amperes) to throw at stuff, but introducing psi (volts) through only resistance is usually easier. But anyway the point here is that you will always have a change going on in power output based on these factors. Which is all after output, don't forget the music itself creates variations in output. Silent is nearly no watts while loud is many. I believe what creates confusion is the power ratings manufacturers share with us. But these are maximum power ratings. They are based on the factors they know about their amps. Namely how many volts it can produce...
Look at the water pump like the tubes and transformers. They are all going to have certain ratings, and you can't get something from nothing. So in the water example By knowing you have a 1HP pump (for example), you can then figure out what flow of water you can produce, then by using a pipe, you can direct that flow, producing power. Same with an amp. The tubes will be capable of a certain output. In most headphone amps these tubes are not being used to their full potential. Which means they have some headroom, well hey what better way to use some of that headroom? Release it when changing impedance modes. I think what has happened with the MHA200 is that it's tubes don't give it this headroom. As the impedance increases on this one, the voltage is being sacrificed and goes down. Where an amp with more room to play within the tubes, will let the volts increase through all the impedance changes, building higher power ratings with each mode change, due to increased resistance.
I don't know how the manufactures actually make their impedance circuits work, so the last part may or may not be correct. I look at the impedance switch like a faucet or hose spigot/finger, and then the impedance from headphones as a straw. If I set the water to either a trickle, low, medium, fast, or gushing; I will find I can more easy direct the water into different size straws. The slow trickle into a swizzle stick, low flow into a small straw and so on. Overall the actual numbers will be determined by all these factors, and constantly changing within a range. And as far as how the amplifiers are actually working, lol, I am just making an educated guess.
So this is not in any way meaning to say the MHA200 is a bad amp, lol. In fact many may say it is the way an amplifier should be built. I wouldn't disagree, although I would like to see that power rating around 1-2 watts if it is going to be constant through impedance modes. But I also have never listened to this amp. I can say Mcintosh amplifiers are built on a rock solid reputation, that much is for sure.
The water-flow analogy compares Volts to psi (pressure), Amps to lpm (flow), and Ohms (resistance) to pipe size. Volts X Amps = Watts , psi X lpm = Watts
Watts are rarely ever constant, always changing. In fact watts are probably more constant in a water system than they are in a sound system.
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New McIntosh MHA200 Headphone amp
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