Vibrations and Waves Legacy Problem #15 Guided Solution

This is a very difficult problem. Like all difficult problems, you'll have to practice the habits of an effective problem solver. Reading the problem carefully and visualizing the situation, recording the things that you know and the things that you're looking for, and thinking real hard about physics concepts and formulas in order to get from the known information to the unknown information. Here we read about two youngsters are out on the piers at a harbor as waves come in and they're being rocked up and down in alternating fashions such that when one of the students is up, the other student is at a low position on his pier. The waves are coming in, rocking these piers up and down one complete cycle every 6.6 seconds. I read that and I write down one cycle per 6.6 seconds. The piers are positioned a distance of 24 meters apart. When one's up, the other's down, and there's exactly one wave crest between them when this is true. We wish to calculate the wavelength, frequency, and speed. I'm going to begin by dealing with that one cycle in 6.6 seconds. That's information to help you find the period and the frequency. One of the things I'm looking for in this problem is the frequency. So I have to know the distinction between the two. The distinction is this. The frequency is the number of things that happen per time, and the period is the time per number of things which happen. The thing that has happened is just one up and down cycle, and that happens in 6.6 seconds. So if I go cycles over seconds, I'm calculating frequency. That's what I wish to find here. So I go one cycle divided by 6.6 seconds, and end up getting 0.151515 repeating. That would be in units of cycles per seconds, also known as hertz. I record that to two significant digits, 0.15 hertz. Now I've got to deal with this 24 meter separation distance between the two pairs, and the idea that when one's up, the other's down, and there's a crest between them. When I read that information, one of the first things I do is I sketch out a pattern of a wave. Your math teachers would call it a sine wave pattern. I just sketch that out, a collection of crests, followed by trough, followed by crests, etc. And I put a dot at the top of a crest. That's when one student is up. The other student's at a trough, but it's not at the next trough on my diagram. It's at another trough past the next crest, because there's a crest between them, as the verbal statement reads. So I use a little diagram to help me out here. I put a dot at the crest, and I go to the next crest, and then to the next trough, and I put a dot. Now the distance between these two dots, horizontally on my wave pattern, is 24 meters. And it's equivalent, you should count it, it's equivalent to 1.5 wavelengths. Because going from crest to the next crest is one wave, and then going to the next trough is another half of a wave. So I write 24 meters equal 1.5 wavelengths. And I can solve this for wavelength by dividing each side by 1.5. End up getting 16 meters. Now, that's the wavelength. I've got the frequency calculated. Now I can find the speed, as v, speed, equals f times wavelength. Where the f is the 0.151515, and the wavelength is 16 exactly. When I do my math, I end up getting 2.424242 repeating. I can round that to two significant digits, such that it's 2.4 meters per second.