This is a poster in a hallway here in the Texas A&M physics building:

Sort of an odd question, but an interesting one. Sound waves carry energy, and if that energy we being absorbed by your coffee because you were yelling at it, how long would you have to do so before it was piping hot?

But let’s put that question aside for a minute and talk about sound levels. Most of the time you’ll hear sound intensity quoted in *decibels*. Decibels confuse a lot of people because it’s a logarithmic scale rather than a linear one. Logarithmic scales are useful for measuring quantities that can span a very wide range of values, because logarithms turn multiplication into addition. Here’s what I mean by that: each increase of 10 decibels corresponds to the level of sound increasing by 10 times. So if you’re having a quiet conversation at a level of 30 dB, a 40 dB sound carries 10 times the intensity of the 30 dB sound. A 50 dB sound carries 10 times the intensity of the 40 dB sound and 100 times the intensity of the 30 dB sound. A 60 dB sound is 1000 times the intensity of a 30 dB sound, and so on.

A jet aircraft taking off might output sound of some 150 dB close to its engine – how much more intense is that sound than the quiet conversation? Well, 150 dB – 30 dB = 120 dB. And 120 decibels represents 12 increases by a factor of 10, so the jet is 10^12 = one trillion times more intense than the casual conversation.

If you’re on top of things you may have noticed that the 0 dB level is pretty arbitrary in this kind of scale. 0 dB doesn’t represent absolute quiet, instead it’s 1/10 as intense as a 10 dB sound, or 1/100 as intense as a 20 dB sound, or 10 times more intense than a -10 dB sound, or 100 times more intense than a -20 dB sound, and so on. No sound at all would be represented by -infinity dB. A little counterintuitive, but it’s the price we pay for a logarithmic scale.

The 0 dB level corresponds to an arbitrarily set reference intensity of 10^-12 watts per square meter. A person shouting might reach 100 dB, which is 10^10 times more intense than the 0 dB reference level. That means the sound intensity emitted by a shouting person is about (10^10)*(10^-12 W/m^2) = 0.01 W/m^2. Which is not very much. If we generously assume the coffee cup has a cross-sectional area of 100 cm^2 and absorbs all the sound energy incident on it, it will absorb something like 0.0001 watts.

If I’ve done the math right this works out to something along the lines of 12 hours per gram per degree. There are probably worse ways to heat you morning coffee, but I think it’s also safe to say that there are better ways too.

Here, by the way, is the coffee-heating essay the poster is promoting. (I avoided looking it up before writing this) Their presentation and numbers are a bit different than mine, but the overall answer of “it takes a very long time” is the same.