Suppose you have a device that extracts energy from ocean breakers in direct proportion to their intensity. If the device produces 10.0 kilowatts of power on a day when the breakers are 1.20 meters high, how much will it produce when they are 0.600 meters high?
We’re told in the problem statement that 10.0 kilowatts of power are produced when breakers are 1.20 meters high. We can write that relationship as an equation. The power produced 𝑃 when the breakers are 1.20 meters high is equal to 10.0 kilowatts. We want to solve for the power produced when the waves are half as tall, 0.600 meters.
The problem tells us that energy is extracted from the waves in direct proportion to their intensity. The intensity of a wave and its amplitude are connected via a relationship: intensity 𝐼 is directly proportional to the amplitude of a wave squared.
That means we can write that the ratio of the power produced when the waves are 0.600 meters high to the power produced when the waves are 1.20 meters high, as the statement tells us, the intensity of the wave when its amplitude is 0.600 meters to the intensity of the wave when its amplitude is 1.20 meters and that this also equals, by our intensity relationship, the amplitude of the smaller wave squared divided by the amplitude of the larger wave squared.
We can remove our middle ratio, and if we multiply both sides of the equation by the power produced by a wave of 1.20-meter amplitude, then that term cancels out of the left side of our equation. And we can plug in 10.0 kilowatts for that power produced by the 1.20-meter tall wave.
When we multiply these numbers through, we find a result of 2.50 kilowatts. That’s how much power is produced by a wave 0.600 meters high. Notice it’s one-fourth as much as the power produced by a wave with twice the amplitude. That’s because of the relationship between intensity and amplitude squared.