Interference from reflection off water across a lake

[Richard L]

#89 chapter 35
A microwave transmitter at height a above the water of a wide lake transmits wavelength lambda to a receiver across the lake height x above the water. Reflected wave interferes with waves arriving directly.
Assumptions: width of lake D is much greater than a or x, and lambda is greater than or equal to a

Write an expression for distance x for which the signal is a maximum.

I can write an expression for the path length difference using Pythagoras. I know to apply the formula: path difference = (m+0.5)lambda because the reflection will give a phase shift of 0.5 lambda.

I think that I can assume that the direct path is equal to D because of the first assumption.
My equation is a mess and difficult to solve for x without making use of another assumption.
I don't know what terms I can omit based on the assumptions allowing me to omit terms and simplify the equation

The answer is supposed to be x=(m+0.5)Dlambda/2a

 

[Simon Bridge]

I don't know what terms I can omit based on the assumptions allowing me to omit terms and simplify the equation.

... without seeing your equation, we cannot tell either.
The clue in the problem statement is the bit about D >> x,a ... this means that any terms in x/D will be very small ... you will usually be expected to make a binomial approximation someplace, or a small-angle approximation where there is trig. So look for the little angles.

Also check you have used all the relations you know... for instance, there are three triangles in your setup - two are right-angle triangles. The right angle triangles have both trig and pythagoras going for them.