Wave Optics (Interference due to reflection)

[cryora]

Homework Statement

A radio-wave transmitter and a receiver, separated by a distance d = 50.0m, are both h = 35.0m above the ground. The receiver can receive signals directly from the transmitter and indirectly from signals that reflect from the ground. Assume the ground is level between the transmitter and receiver and a 180 degree phase shift occurs upon reflection. Determine the longest wavelengths that interfere a) constructively and b) destructively.

Homework Equations

Since there is a 180 degree phase shift of the reflected signal, the conditions for constructive interference is:
δ = (m+1/2)λ
and conditions for destructive interference is:
δ = mλ
where δ is the path difference of the two waves, and m is an integer.

Unlike Young's Double slit, however, δ ≠ dsinθ, where d is the distance between two coherent light sources, because the extrapolated reflected ray does not make a 90 degree angle with the incoming ray before reflection.

The Attempt at a Solution

From the law of reflection, it can be deduced that the length of the path that the reflected ray takes from the source to the point of reflection is the same as the path it takes from the point of reflection to the receiver. Let's call this length l. Thus, we have an isosceles triangle where the sides are the paths of the direct ray and the ray that is reflected.

The point at which the ray is reflected is halfway (d/2) between the radio towers.

Knowing h, from the Pythagorean theorem, l = sqrt[h^2 + (d/2)^2]

Path difference δ = 2l - d
= 2sqrt[h^2 + (d/2)^2] - d

This is = to (m + 1/2)λ for constructive interference and mλ for destructive interference.

I have a feeling that I've already did something wrong at this point... but anyway the question asks for the greatest λ. Since h and d are constants, it seems the only thing I can vary is m. I could solve for λ and set m = 0 to maximize λ, but I get the wrong answer for constructive interference... and for destructive interference, it would mean λ → ∞.

So I'm quite confused about what I need to do. Any hints?

I just realized that m may not be variable but fixed, and unknown... This doesn't really improve things by much lol

 

[ehild]

Your work is correct, but you need to give physically possible solution. The wavelength of the radio wave must be positive. So find the m values which result in positive wavelengths and choose the longest ones.

ehild