Vibrations and Waves Legacy Problem #16 Guided Solution

This is a very difficult problem. Like all difficult problems, I'll have to follow the habits of an effective problem solver. I'll have to read the problem carefully, visualizing the physical situation, identifying the things that I know, things that I'm looking for, and then think really hard about concepts and formulas in physics in order to get from the known information to the unknown information. I read of Nick and Kara are out on the waters at Lake Bluebird, and they're spaced 1.8 meters apart. And a boat zooms by, and here comes these ripples bobbing the rafts of Nick and Kara up and down. And it says it makes four up and down cycles in 8.4 seconds. And I know that's important, so I write it down. Four cycles per 8.4 seconds. When Nick's raft is at a high point, Kara's raft is at a low point. There's no crest between them. So I begin to visualize this situation of a wave pattern traveling through the water, a sine wave pattern is what your math teacher would call it. And I draw a pattern on my sheet of paper. I put a dot at a crest. I go to Nick's trough, low point, put a dot there. So when Nick's at the high point, a crest on the pattern, Kara's at the low point, there's no crest between them. And that distance between them, we're told, is 1.8 meters. I'm looking to calculate the wavelength, frequency, and speed of these ripples. I'm going to first deal with this four up and down cycles in 8.4 seconds. That's information relating to the period and the frequency, so I have to know the distinction. The period is simply the time per number of cycles. The frequency, on the other hand, is the number of cycles per time. Cycles per second, for instance, that would be units of hertz. So I'm going to take the four cycles and divide it by the 8.4 seconds. That would give me my frequency in cycles per second, also known as hertz. It comes out to be 0.47619 hertz, and I can round that to two significant digits. Now I'm going to deal with this 1.8 meter separation along the water. When one's up, the other's down. I'm going to deal with the diagram part of the situation. So what I recognize is that the distance from the high point across to the low point where the car is at, the trough, is not a full wavelength. It's actually one half of a wavelength. So I write this math statement down. 1.8 meters equal one half times wavelength. And I'm going to solve for wavelength using that equation. I multiply both sides by two, which cancels the fraction of a half on the right, and it gives me wavelength equal 3.6 meters. Now, the last thing to calculate is the speed of these waves, and I can do that using the wave equation. Speed, or v, is equal to f times lambda. I know f is 0.47619 hertz, and I just calculated wavelength, or lambda, is 3.6 meters. I multiply the two. I get my v, or my speed, and it comes out to be 1.7143 meters per second, and I can round that to two significant digits, 1.7 meters per second.