Destructive Interference problem

[jhoffma4]

The problem states:

Waves broadcast by a 1152 kHz radio sta-
tion arrive at a home receiver by two paths.
One is a direct path, and the second is from
re°ection o® an airplane directly above the re-
ceiver. The airplane is approximately 137 m
above the receiver, and the direct distance
from station to home is 18:7 km.
What is the height (within 1% error) of the
airplane if destructive interference occurrs?
Assume no phase change on reflection.
Answer in units of m.

First of all I am kind of confused as to where the receiver is...on the house or the radio station. I was thinking it should be on the house, but 137 meters seems very low for an airplane to be flying above a house.

What I know/tried:
I know that since it's destructive interference, it is out of phase, so we use ∆L=(lamda/2). and we can find L1 and L2 with the values given, but I have no idea how to include frequency...maybe using v=lamda/f? I tried applying the small angle approximation (where sinө~tanө), but the trigonometry is really confusing and I wasn't having much success. Could someone help?

 

[tiny-tim]

Hi jhoffma4!

First of all I am kind of confused as to where the receiver is...on the house or the radio station. I was thinking it should be on the house, but 137 meters seems very low for an airplane to be flying above a house.

It doesn't matter, does it?

… but I have no idea how to include frequency...maybe using v=lamda/f? I tried applying the small angle approximation (where sinө~tanө), but the trigonometry is really confusing and I wasn't having much success. Could someone help?

Yes, v=lamda/f should do fine.

And just use Pythagoras.

 

[Moetasim]

I understand the problem and the concept of destructive interference. In this scenario, the waves from the radio station are arriving at the receiver through two different paths - one direct and one reflected from an airplane flying overhead. The height of the airplane is unknown and we are trying to find it using the fact that destructive interference is occurring.

To solve this problem, we can use the formula for destructive interference, which is ΔL = λ/2. In this case, we have two different lengths - one for the direct path and one for the reflected path. We can calculate these lengths using the given values of 18.7 km for the direct distance and 137 m for the height of the airplane.

We also know that the frequency of the radio waves is 1152 kHz, which we can convert to meters by using the formula v = λf. This gives us a wavelength of approximately 260 m.

Now, we can set up the following equation:

ΔL = L1 - L2 = λ/2

Substituting in the values we have, we get:

18.7 km + 137 m - L2 = 260 m/2

Solving for L2, we get:

L2 = 18.7 km + 137 m - 130 m = 18.7 km + 7 m = 18.707 km

Since we know that the height of the airplane is 137 m, we can now use the Pythagorean theorem to find the height (h) of the airplane above the receiver:

h^2 = (18.707 km)^2 - (18.7 km)^2

h = √((18.707 km)^2 - (18.7 km)^2) = √(0.007 km^2) = 0.083 km = 83 m

Therefore, the height of the airplane is approximately 83 meters above the receiver. This is within 1% error of the given value of 137 m, which shows that our calculations are accurate.

In summary, by using the formula for destructive interference and considering the wavelength and frequency of the radio waves, we were able to solve for the height of the airplane in this scenario.