Understanding Destructive Interference in Radio Waves

[jolly_math]

Homework Statement

A ship approaching harbor is transmitting at a wavelength of λ = 3.43 m from its antenna located h = 23 m above sea level. The receiving station antenna is located H = 160 m above sea level. What is the horizontal distance D between ship and receiving tower when radio contact is momentarily lost for the first time? Assume that the calm ocean reflects radio waves perfectly according to the law of reflection. (diagram below)

Relevant Equations

d sin θ = (m + 1/2) λ

The solution pretends that the ship is a two point source emitter, one h above the water, and one h below the water.
The one below the water is out of phase by half a wavelength.

I don't understand why then d sin θ = λ - wouldn't it be d sin θ = (1/2)λ since it is out of phase?

Thank you.

 

[TSny]

I don't understand why then d sin θ = λ - wouldn't it be d sin θ = (1/2)λ since it is out of phase?

The two point sources emit out of phase. That is, the waves start out from the two sources already with a (1/2)λ phase shift. The path difference (d sin θ) contributes additional phase shift where the waves arrive at the receiving station. If the path difference is (1/2)λ, what would be the total phase shift? Would that yield constructive or destructive interference?

 

[jolly_math]

The two point sources emit out of phase. That is, the waves start out from the two sources already with a (1/2)λ phase shift. The path difference (d sin θ) contributes additional phase shift where the waves arrive at the receiving station. If the path difference is (1/2)λ, what would be the total phase shift? Would that yield constructive or destructive interference?

I'm confused by how "the one below the water is out of phase by half a wavelength" is derived from the question. Also, why is the path difference (1/2)λ?

When the question states that "radio contact is momentarily lost for the first time", is it looking for constructive or destructive interference?

Thank you.

 

[TSny]

I'm confused by how "the one below the water is out of phase by half a wavelength" is derived from the question.

The underlying reason is that, when a radio wave reflects off of the surface of water, the radio wave undergoes a (1/2)λ phase shift. This is taken into account by taking the "phantom" underwater point source to be emitting waves that are (1/2)λ out of phase with the above water point source.

Did you discuss phase shifts due to reflection in class? There is a rule for deciding whether or not a light wave has a (1/2)λ phase shift due to reflection. It involves the indices of refraction of the two media where the reflection takes place. The same rule applies to radio waves.

[EDIT: In introductory physics courses, it is often stated that there will be a (1/2)λ phase shift at reflection if the ratio of the indices of refraction $n_2/n_1$ is greater than 1. Actually, there are exceptions to this that depend on the angle of incidence and the polarization of the waves. But, this is not usually covered in introductory classes. So, I guess that in this question you were just expected to assume that there will be a (1/2)λ phase shift due to reflection since $n_2/n_1$ is greater than 1 for an air-to-water interface.]

Also, why is the path difference (1/2)λ?

In the first post you asked, "I don't understand why then d sin θ = λ - wouldn't it be d sin θ = (1/2)λ since it is out of phase?"

I wanted you to see that if you take the path difference to be (1/2)λ, then the waves at the receiving station would not result in the receiving station losing contact with the ship. So, (1/2)λ path difference is not going to work.

When the question states that "radio contact is momentarily lost for the first time", is it looking for constructive or destructive interference?

The contact is lost because the waves from the two point sources "cancel out" at the receiving station. Would this be constructive or destructive interference?

 

[jolly_math]

The contact is lost because the waves from the two point sources "cancel out" at the receiving station. Would this be constructive or destructive interference?

This would be destructive interference. d sin θ = λ so that the overall phase difference is 1.5λ, which still ensures destructive interference - is this right?

 

[TSny]

This would be destructive interference. d sin θ = λ so that the overall phase difference is 1.5λ, which still ensures destructive interference - is this right?

Yes. Good.