Understanding Minimum Path Difference for Waves Below the Centerline

[kent davidge]

Homework Statement

Homework Equations

The Attempt at a Solution

(Sorry my poor English). I just don't understand why should the minimum path difference be 0.75 wavelengths for angles below the centerline instead 0.25 wavelenghts as it is for angles above the centerline.

 

[TSny]

For angles below the centerline, is the path length from A to the detector greater or less than the path length from B?

 

[kent davidge]

It's greater, but how would it change the expression for interference?

 

[TSny]

Yes, the the path from A is greater when the angle is below the center line.

The wave from A has a "head start" compared to the wave from B due to the vibration of the speaker at A being 1/4 period ahead of the vibration at B. When the waves arrive at the detector, does the extra distance for A tend to make the phase of the wave from A even farther ahead of B or not as much ahead?

You might consider the case where the extra distance for A is 1/4 wavelength. A sketch will help.

 

[kent davidge]

(Sorry my poor English). Okay, but instead adding some value, it seems we have to subtracting one so that the right side of the equation equals to 0.75 wavelength

X_a - (1/2)λ - X_b = λ/4
X_a - X_b = 3λ/4

 

[TSny]

(Sorry my poor English). Okay, but instead adding some value, it seems we have to subtracting one so that the right side of the equation equals to 0.75 wavelength

X_a - (1/2)λ - X_b = λ/4
X_a - X_b = 3λ/4

I'm sorry, but I'm not quite following what you did here.

The extra distance that the wave from A travels will shift the phase of A in a direction that makes the phase of A "behind" the phase of B. But the wave from A started out 1/4 λ ahead of the wave from B. So, if the extra path distance that A has to travel is 1/4 λ, the waves from A and B will end up in phase at the detector.

You need to find the path difference in order for A and B to end up out of phase at the detector.

 

[kent davidge]

oh thank you TSny I understand it now. For any point above the centerline the distance from A will be always smaler than the distance from B. If waves from B start 1/4λ behind of waves from A, and if the distance from B is 1/4λ greater than distance from A, we add 1/4λ to find the position of the wave from B, which will be 1/4λ + 1/4λ = 1/2λ out of phase with waves from A. Analogous situation occurs from points below the centerline. Am I right?

 

[TSny]

oh thank you TSny I understand it now. For any point above the centerline the distance from A will be always smaler than the distance from B. If waves from B start 1/4λ behind of waves from A, and if the distance from B is 1/4λ greater than distance from A, we add 1/4λ to find the position of the wave from B, which will be 1/4λ + 1/4λ = 1/2λ out of phase with waves from A. Analogous situation occurs from points below the centerline. Am I right?

Yes. That's right.