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Explaining the error of my waves

2010 April 12
Explaining the error of my waves

Several months back I set off to answer a question from reader Kristine Olstrom on why longer boats are generally faster than shorter ones. I dove into some physics skills that I apparently had forgotten long ago and, well, didn't answer the question correctly. My error was caught by astute reader Robert Rosner of the University of Chicago and Stanford University. Rosner, in addition to being associated with these esteemed universities and several national labs, does research in the behavior of deep water waves. While I am a little embarrassed by my error, I am glad to see that my column was of interest and that Rosner took the time to point out my error and quite a bit more time to further explain a correct answer.

I began my explanation with the fact that boats with longer waterlines in general will have a faster theoretical hull speed. The first part of the reasoning behind this is that displacement hulls behave a lot like waves of equal length. I then went on the try to describe why longer waves are faster and treaded off into some pretty bad physics. My first claim was that wave speed was equal to wave frequency times wavelength and that wave frequency was constant, and this was just not correct. Had it been accurate, wave (boat) speed would have a linear relationship to wave length, it would not involve the square root that we all know and love.

Rosner explained that water waves are gravity waves, waves in which gravity is the restoring force. If a wave is caused by a passing boat or a strong wind, it is gravity that settles the water. He went on to explain that behavior of a water wave can be approximated by the behavior of a simple pendulum, another system where gravity is the restoring force. The frequency of a pendulum is described as (g/l)^(1/2), where g is the acceleration of gravity and l is the length of the pendulum. Doing a little mathematical gymnastics, we yield a formula that says the frequency of a wave varies as the inverse square root of the wave length. Following this, longer waves have a smaller frequency (longer period) and vice versa, and as Rosner mentioned to me, this is easily observed on the water.

I struggled to understand the physical argument that related wave speed to wave length as an inverse square root, but using a pendulum as a model for the behavior of a wave makes a lot of sense. Rosner explained that "distorting the surface of the water is like pulling a pendulum off center and letting go, in the sense that the water surface bounces back and forth for exactly the same reason a displaced pendulum returns to center (and overshoots)-gravity does the pulling."

In an attempt to get my head around the inverse square root relationship, Rosner explained that only two quantities could possibly matter, the length of the pendulum and the strength of gravity. The mass of the pendulum doesn't matter, as Galileo figured out by dropping two balls of different weights from the Leaning Tower of Pisa and found that they arrived at the ground at precisely the same time. The inverse square relationship just develops mathematically to relate these two physical quantities.

I think I now have all the understanding of this topic that I need. To seal the deal, Rosner paraphrased one of my favorite physicists, Richard Feynman, "sometimes the explanation of a physical law reduces simply to the fundamental equation that describes that law and no 'intuitive' explanation will do better than that."