Vibrations and Waves Legacy Problem #23 Guided Solution

Here's a question which demands more than just being able to plug and chug into some chosen physics formula. This question involves a need to understand a concept in physics, a big concept in physics, and that's the concept that the speed of a wave within an elastic cord depends only upon the properties of the elastic cord, and not upon the frequency and wavelength. And so if you change anything about the frequency and or the wavelength of a wave, you'll do nothing to change the speed of the wave. Here what we have is a 1.6 meter long wave, a meter long rope, and waves are traveling through it at 2.4 meters per second. The frequency is 1.5 hertz. So right away I know the V value, and I know the F value, and the first question they ask is what would be the new wavelength and the new speed if they double the frequency of vibration of the cord? That is, if instead of shaking it with a frequency of 1.5 hertz, what if they shook it at a frequency of 3 hertz? What would be the new speed and the new wavelength of this wave? Now there's a couple of ways to approach the question. The first one might be to just simply calculate the wavelength of the wave when the frequency is 1.5 hertz. You need the wave equation to do that. You'd have to go V equal F times lambda, where lambda is the wavelength, maybe rearrange it so that lambda is equal to V over F, and then divide the 2.4 meters per second speed by the 1.5 hertz frequency, and you end up getting 1.6 meters as the wavelength of this wave happens to be the same as the length of the rope, telling us that what we must have here is a second harmonic vibration. Now what would happen if we doubled the frequency? Would it still be 1.6 meters, or would it change one way or another? Well, the first thing you have to understand is that the speed isn't going to change. It's still going to be 2.4 meters per second, and you're going to have a frequency of 3.0 hertz because you've doubled it from the 1.5 hertz. So now I can calculate the new wavelength by going wavelength equal V over F, 2.4 divided by 3.0 hertz, and that gives me a wavelength of 0.8 meters. Now I calculated that last wavelength using the wave equation, but I could have just as easily taken the 1.6 meters and reasoned that if you double the frequency, you'd have to half that 1.6 meters to get half the wavelength. That's the only way that that speed can come out to be the same, 2.4 meters per second. So our answers here are the new wavelength is 0.8 meters, and the new speed is the same as the old one, 2.4 meters per second.