Light Waves and Colors Legacy Problem #23 Guided Solution

Light waves interfere, and they can interfere destructively whenever a crest, a so-called crest from one source, interferes or meets up with a trough from another source. Here are radio stations broadcasting signals at 582 kilohertz from two different antennas. And at the Robinson Hall, they actually interfere destructively to give static on the dial. Now we're given the distance from the nearest antenna to the Robinson Hall. It's 13.78 kilometers. What we wish to know is what's the distance from the furthest antenna to the Robinson Hall that results in this destructive interference. In order to do that, we need to understand the concept that as long as the difference in distance traveled of the wave from one antenna to the house compared to the other antenna to the house is a half number of wavelengths, then you'll get destructive interference. After all, if the difference in distance traveled is a half number of wavelengths, a crest would be meeting up with a trough. Now we know, therefore, that the difference in distance between the 13.78 kilometers and the other distance, what we're looking for, is going to be one half of a wavelength. So if I can just find the wavelength, I can take a half of it and I can add it to 13.78 kilometers. So how do I get the wavelength? Well, I recognize that that 582 kilohertz, that that's a frequency. Expressed in the usual hertz unit, that would be 5.82 times 10 to the 5th hertz. And for light waves which travel at 2.998 times 10 to the 8th meters per second, I can find the wavelength using the wave equation. Wavelength equal the speed of light divided by the frequency. So I take the 2.998 times 10 to the 8th meters per second and I divide it by the 5.82 times 10 to the 5th hertz, and I get a wavelength that comes out to be 515.1203 meters. Now if I take half that value, I'm getting the minimum path difference required in order to cause this destructive interference. Minimum half of the 515 value comes out to be 258, 257.56 to be exact. Now I need to take that 258 and I need to add it on to the 13.78 kilometers. That's a meter added on to kilometers, doesn't work too well. So I need to convert the 13.78 to units of meters. That's going to be 1.378 times 10 to the 4th meters. Adding 257.56 to it gives me 14,037.56. I can round that to four significant digits and convert it back to kilometers. It becomes 14.04 kilometers.