Where does sound energy go?

Coldplay Speed of Sound

Today in class, a student brought up an interesting question: If sound waves carry energy, where does the energy go when the sound is no longer audible? Since sound waves consist of moving particles, it seems reasonable to assume that the particle movement will become less ordered over time and therefore the sound energy is eventually converted to heat.

So can you warm up a room by yelling? Hm. So far our reasoning seems to say yes. I decided to try a back-of-the-envelope calculation, looking for the upper limit of how much you might warm something up with sound.

Let’s try a loud sound. According to internet sources, the intensity of sound in the front row of a rock concert is 0.1-1 W/m2. Take the upper limit: 1 W/m2.

What shall we warm up with sound? Keep it simple: a cup of water. Suppose all of the rock concert sound energy incident on the top surface of the water is absorbed by the water, not reflected. If the top surface of the water has an area about 10 cm by 10 cm (I imagined the cup has a square opening in order to approximate the area), that means the sound is incident upon an area of 0.1 m x 0.1 m or 0.01 m2.

1 W/m2 incident upon a 0.01 m2 surface means 0.01 W being absorbed by the water, or 0.01 J/s.

We’ll assume all of that energy is converted to heat. After all, what other form of energy could it reasonably become? To get a better feel for the numbers, we’ll figure out how much the temperature of the water will increase.

Heat energy is related to temperature by Q = mcΔT. Q is heat energy, m is the mass of the water, c is the specific heat of water, and ΔT is the temperature change in degrees celsius.

Imagine the cup contains about 100 mL of water. With a density of 1 g/mL, that’s 100 g, or 0.1 kg, of water. The specific heat of water is about 4000 J/kg degC.

If we just look at a time interval of one second:
Q = mcΔT
0.01 J = (0.1 kg)(4000 J/kg degC)(ΔT)
Result: ΔT = 0.25 x 10-4 degC each second.

Er. That’s not very much. At this rate, how long would it take to get the temperature of the water to rise by just 1 degC? Well, 0.25 x 10-4 degC each second is equivalent to 4 x 104 seconds per degC, or about 10 hours per degC.

End result: If you sit in the front row of a rock concert holding a cup of water, it would take at least 10 hours for the temperature of the water to increase by just 1 degree celsius due to the sound waves hitting it.

I’m sure you can imagine all sorts of other effects that I’ve neglected, or maybe better ways of going about the calculation. If so, let me know.

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