I see home sound systems rated at 300w, etc. but a 50w guitar amp is massively louder.
I assume that the first one is some useless marketing rating, but how are the two figures calculated to be so different?
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Audio power is actually rather complicated and hence some "easy to measure" specifications have been made up by the industry, none of which is particularly useful for a real world application.
The worst offender is the so called "Peak Music Power Output" or PMPO. This is not defined by any standard and doesn't relate in any reproducible way to a real world measurement. It's basically "made up" for marketing purposes.
Other power ratings are RMS power, long term thermal power, peak power, etc. applications. RMS power is often defined as the power of a sine wave close to clipping (1% THD) into a nominal restive load. Peak power is twice that. Turns out music isn't sine waves and speakers are definitely NOT a resistive load so that's dubious specification. Most amplifiers are not capable of delivering their rated RMS power for an extended period of time, because they never have to deliver that amount of power in a real world application.
An actual audio amplifier has certain set of constraints
If you combine this with a complicated input waveform and a highly reactive and partially non-linear load you can see a fairly complicated picture emerging.
The most "relevant" spec is "long term thermal power" which is the power level that the amp can support for an infinite amount of time without overheating. Since music is actually not a sine wave and has a significantly higher crest factor a good design target would be around 1/3 of the RMS power or even less if you primarily drive high frequency devices.
A guitar amp is much louder primarily because of the driver, not the amplifier itself. Guitar speakers have VERY high efficiency in the high mid-band which makes them perceptually really, really loud even at low power input. Real HIFI speakers are designed for flat frequency response and "cleanliness" which makes them a lot less efficient.
It is not about the wattage of the amplifier, which simply rates how much power it can give out (or consume), it tells nothing how the watts are actually being used.
For instance, your guitar amplifier might have a sensitive 4-ohm speaker that pushes out 90dB at 1W measured at 1m, and your home system might have a insensitive 8-ohm speaker that pushes out 80dB at 1W measured at 1m. Therefore you need 10W, or 10 times more power, to reach same 90dB volume level. For comparison, 80dB is approximately the sound level of one lawnmower.
The guitar amp may also be rated to allow for higher distortion than your Hi-Fi set for the same wattage.
Also, guitar amps typically effects like distortion, overdrive, compression etc which can make the audio sound louder. It's basically gives a perception of being louder, much like a sound of someone shouting but played back at low volume level, than someone whispering at high volume level, even though a decibel meter indicates the levels are equal.