Solving the Problem of Plane Height for Destructive Interference

[atse1900]

Waves broadcast by a 1500-kHz radio station arrive at a home reciever by two paths. One is a direct path, and the second is by reflection off an airplane directly above the home receiver. The airplane is approximately 100 m above the home receiver, and the direct distance from the station to the home is 20 km. What is the exact height of the airplane if destructive interference is occurring? (Assume that no phase change occurs on rfelective from the plane.)

I have no idea how would you solve this problem. Can someone help me out please? Thanks. =].

 

[Astronuc]

To have destructive interfence at some point, the waves must be 180° out of phase, i.e. the maximum of one coincides with the minimum of the other. So the difference between path lengths must include one-half of the wave length, but not necessarily exactly one-half of a wavelength.

 

[kyle8921]

I can provide some information to help solve this problem. Destructive interference occurs when two waves of the same frequency and amplitude meet and cancel each other out. In this case, the waves from the radio station are reaching the home receiver by two paths - one direct and one reflected off the airplane. The key to solving this problem is understanding the concept of path length difference.

The path length difference is the difference in distance traveled by the two waves from the source (radio station) to the receiver (home). In this case, the direct path has a distance of 20 km, while the reflected path has a distance of 20 km + 2*100 m = 20.2 km. This means that the path length difference is 0.2 km.

To achieve destructive interference, the path length difference must be equal to half of the wavelength of the waves. The wavelength of a 1500-kHz radio wave is 200 m (speed of light/frequency). Therefore, half of the wavelength is 100 m.

Now, we can set up an equation to solve for the height of the airplane:

0.2 km = 100 m * n, where n is the number of half wavelengths in the path length difference.

Solving for n, we get n = 0.002. This means that the path length difference is equal to 0.002 wavelengths. Since the path length difference is equal to the difference in height between the direct and reflected paths, we can say that the airplane is 0.002 wavelengths or 0.4 m above the home receiver.

Therefore, the exact height of the airplane is 100 m + 0.4 m = 100.4 m.

I hope this helps in solving the problem. It is important to understand the concept of path length difference and how it relates to interference in order to solve this type of problem.