Minimum amount of hops for radiowave to travel from Amsterdam to Tokyo

[Shukie]

Homework Statement

I don't know if 'hops' is the correct English term, but that's the term my assignment uses in Dutch. The assignment is about the reflection of radiowaves by the f1-layer in the atmosphere and a hop is defined as one such reflection. The question is, what is the minimum amount of hops needed for a radiowave to travel from Amsterdam to Tokyo? We have to use Mathematica to solve it.

Homework Equations

First of all we have to calculate the spherical distance from Amsterdam to Tokyo. I calculated this to be 9293.71 km. Then we have to calculate the angle between Amsterdam and Tokyo, as seen from the centre of the earth, in radians, which I found to be 2.3541.

The Attempt at a Solution

I'm pretty much at a loss now, how do I continue?

 

[mgb_phys]

Assuming the radio waves travel in straight lines and bounce off the surface (or are relayed) and the ionosphere - you are trying to draw a polygon that fits between these two circles.

 

[Shukie]

Ok, I think I see where you are going, but not quite yet. The best way for me to send that radiowave would be to send it completely horizontally (so at an angle of 0), right? I drew a picture in paint:

http://home.orange.nl/~kuip3148/radiowave.JPG

The f1-layer has an altitude of 300 km. All I would have to do is figure out how far the radiowave would have traveled after one hop and then I could simply divide the distance Amsterdam-Tokyo by the distance traveled after one hop to get the number of hops, is that correct? If it is, is there an easy way to calculate that?

 

[mgb_phys]

Almost - except the wave should hit the Earth at a tangent to get the least number of hops