Physics Question (young's double slit)

Question

Radio waves from a star, of wavelength 250 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 25 degrees above the horizon. Find the height of the cliff. (assume no phase change on reflection')

since it's destructive interference,

i use ........d*sin25=250/2

and got d=296m

i figure that you're suppose to get the y (distance between the first wave hitting the telescope and the second wave hitting the water) and then using that to get the height but i don't really know so if anyone can help me it be appreciated

Answer 1

The distances the two waves travel from the star differ by half a wavelength.

So draw the picture and compare the non-matching parts of the paths.  There's a ray coming straight to the telescope, and another reflecting off the ocean and then going to the telescope.  The rays are parallel coming from the star (since the star is, for the purposes of this question, an infinite distance away), so when the reflected ray hits the water you can draw the perpendicular and it will also be perpendicular to the unreflected ray.  Everything beyond that perpendicular (off toward the star, that is) can be ignored, since it's the same for both rays.  

Let's call the base of the cliff B, the point at which the ray is reflected R, the location of the telescope T, the star S, a distant point on the water off toward the horizon H, and the intersection of the perpendicular with the unreflected ray P.  .  

The difference in distances the two rays travel is RT - PT, so one equation is RT - PT = 250/2.  (*)

There are two triangles on the diagram, BRT and PRT.  The distance RT traveled by the reflected ray is the hypotenuse of both triangles, so we can get an equation by saying that its length as the hypotenuse of  one is the same as its length as the hypotenuse of the other, after we work through what all the angles are.  

The angle SRH of the star above the horizon, given to be 25 degrees.  By basic geometric optics, the angle BRT between the horizontal and the reflected ray is also 25.  Angle SRP was chosen to be 90 degrees.  Those plus TRH add up to 180, so SRH is 180 - 90 - 25 - 25 = 40.

From triangle BRT, RT = BT/sin(25). (**)

From triangle PRT, PT = RT*sin(40).

Then from the latter equation, RT - PT = RT - RT*sin(40).  But RT - PT = 125, from the interference relation (*) above, so RT - RT*sin(40) = 125.  Solve that for RT, set equal to the value of RT from the equation (**) from triangle BRT, and solve for the number BT that you needed to find.