Wave interferance and determining an unknown height

[chris097]

Homework Statement

Radio waves from a star, of wavelength 231 m, reach a radio telescope by two separate paths. One is a direct path to the receiver, which is situated on the edge of a cliff by the ocean. The second is by reflection off the water. The first minimum of destructive interference occurs when the star is 23.7° above the horizon. Calculate the height of the cliff. (Assume no phase change on reflection.)


The attempt at a solution


Since its destructive interference, the difference between the two lengths is half a wavelength (=115.5 m).
So let's say L-D=115.5 m.
D = Lsin(90-23.7) = Lsin(66.3)
L - (0.916)L = 115.5 m
0.0843L = 115.5 m
L = 1370 m

Height = Lsin(23.7)
= 550.7 m


Any idea what I'm doing wrong? Thanks :)

 

[Redbelly98]

The attempt at a solution

Since its destructive interference, the difference between the two lengths is half a wavelength (=115.5 m).
So let's say L-D=115.5 m.
D = Lsin(90-23.7) = Lsin(66.3)

I don't think 23.7 is the angle to use here. Did you draw a figure to show what is going on?

The rest of it looks okay:

L - (0.916)L = 115.5 m
0.0843L = 115.5 m
L = 1370 m

Height = Lsin(23.7)
= 550.7 m


Any idea what I'm doing wrong? Thanks :)

 

[kerimek]

I would first like to commend you for your attempt at solving this problem using wave interference principles. It shows that you have a good understanding of the concept. However, there are a few things that could be improved upon in your solution.

Firstly, the equation D = Lsin(90-23.7) is incorrect. The correct equation should be D = Lsin(180-23.7), since the reflected wave travels an additional distance of L before reaching the receiver. This would give us D = Lsin(156.3).

Secondly, the value of L that you have calculated (1370 m) seems to be incorrect. Using the correct equation above, we can calculate L as follows:

L = D/sin(156.3)
= 115.5 m / sin(156.3)
= 1372.7 m

This value of L should be used in the final calculation for the height of the cliff.

Therefore, the correct height of the cliff would be:

Height = Lsin(23.7)
= (1372.7 m) * sin(23.7)
= 552.2 m

In summary, it seems that the incorrect equation for D and a minor calculation error for L led to the incorrect height calculated in your solution. But overall, your approach using wave interference principles is correct. Keep up the good work!