Light Waves and Colors Legacy Problem #24 Guided Solution

This is certainly a difficult question and probably what makes it most difficult is picturing the situation. So before I begin to discuss the physics of this problem, I'd like to call your attention to the link at the bottom of the page that goes back to the Physics Classroom Tutorial to a page titled Other Applications of Two-Point Source Interference. There you'll see situations such as this discussed and heavily illustrated in a step-by-step fashion and it might be worth a read if you're having difficulty with this type of problem. Now Mr. H is trying to pick a home out in Flagstaff in the retirement community and his favorite station out there is of course K-Fizz. And he would wish to receive signals from K-Fizz at his home. The problem is that just beyond his community is large cliffs which are reflecting radio waves that come from the source back towards all the homes in the neighborhood. Now if you can kind of picture this, there's a transmitting station many kilometers away from his home and it's transmitting these 1420 kHz frequency signals. And these signals come to his antenna and are received by his antenna but they're also passing his antenna reflecting off these steep cliffs back towards his home and being picked up by the antenna there as well. So there's actually two sources of waves, one being direct from the source, the antenna that transmits the signal and the other being the reflected waves coming off the cliffs from behind his home. And they could interfere destructively at just the right distances. And the condition for destructive interference for two sources of waves is that if the difference in distance traveled, we call that the path difference, from the sources to the receiver is equal to a half or three halves or five halves or some half number of wavelengths then we get destructive interference in equation form. That's put path difference, pd, equal m times lambda. Lambda is the wavelength and m is a half number, the smallest of which is one half. So here in this question what we're asked is to determine the six nearest distances from the cliffs that will result in the destructive interference. So those six differences correspond to different m values, a half, three halves, five halves, seven halves, all the way up to eleven halves. So what I need to do is calculate the wavelength and then find the path differences. Once I do that I have an additional step. These differences in distance traveled, these path differences, are equal to twice the distance from Mr. H's home to the cliffs. Because the signal has to go past the antenna of Mr. H's home to the cliffs and back. That's a 2d distance, d down, d back. So once I get path difference, I'm going to have to divide them by two in order to get all my answers. So that's the strategy. The mathematics is relatively simple. As long as I recognize that the 1420 kilohertz is a frequency, then I can use it to calculate the wavelength. These are radio waves, which is a form of light waves, which travel at 2.998 times 10 to the 8th meters per second. So what I'm going to do is use my wave equation and rearrange it to wavelength equal the speed of light divided by the frequency. I'm going to go 2.998 times 10 to the 8th meters per second divided by the frequency in hertz, which is 1.420 times 10 to the 6th. That gives me a wavelength of 211.1268 meters. I'm going to take this 211 value and I'm going to multiply it by half and multiply it by 3 halves and by 5 halves and by 7 halves, etc. I'm going to get a list of possible path differences. And then I'm going to take those values and I'm going to say that they're equal to 2d. And I'm going to divide each one of them by 2. And that gives me the answers you see listed here. It gives me 53 meters, 158 meters, 264, 369, 475, and 581 meters. And that's all there is to this problem.