How do diffraction and refraction affect wave propagation?

[hasan_researc]

Why do waves spread out after emerging from a narrow ($$\leq\lambda$$) opening?

Also, why do waves change direction at boundaries between regions where the wave speed differs?

I have tried to find answers to these questions by searching on the Net, but with no luck.:-(

 

[GRDixon]

Why do waves spread out after emerging from a narrow ($$\leq\lambda$$) opening?

Rather than re-invent the wheel here, I would refer you to Chapters 29 - 31 in "The Feynman Lectures on Physics," Vol. 1. Happy studying!

 

[hasan_researc]

Thanks for the reference!

By "re-inventing the wheel", do you mean working out the answer by myself?

Also, do you have any suggestions for my second question, please?

 

[johng23]

For why the direction of the wave changes at a boundary, I think it's most intuitive to think about conservation of momentum in the plane of the interface. Momentum is h/lamda times the cosine of the angle to the interface, it can't change in that plane since there is no force. The component of momentum has to stay the same as wavelength changes, so the angle changes.

Conservation of momentum perpendicular to the interface is what really throws me...

 

[GRDixon]

Thanks for the reference!

By "re-inventing the wheel", do you mean working out the answer by myself?

Also, do you have any suggestions for my second question, please?

In the "wheel" comment I meant that I would not re-type some of the material Feynman covers in the suggested reference. I have always found it easiest, regarding the change of beam direction at an interface, to consider a plane wave. Assuming the interface is also a plane, one expects the incident and transmitted waves both to be plane. Since one part of the incident wave reaches the interface before another part, and since the propagation speed is, say, lower in the new medium, the final, transmitted wave's Poynting vector must point in a different direction in the new medium.

 

[hasan_researc]

For why the direction of the wave changes at a boundary, I think it's most intuitive to think about conservation of momentum in the plane of the interface. Momentum is h/lamda times the cosine of the angle to the interface, it can't change in that plane since there is no force. The component of momentum has to stay the same as wavelength changes, so the angle changes.

Conservation of momentum perpendicular to the interface is what really throws me...

Thank you for your contribution! First, I have written below what I have understood from your post. Then, I have written about my own interpretation.

Let's consider electromagnetic radiation for the moment. We know that if light travels from a less dense to a denser medium, then the angle of refraction < angle of incidence. Therefore, the angle theta from the interface to the ray has increased after refraction.

This picture you have interpreted quantum-mechanically. A wave can behave as a particle with p = hk. In the direction parallel to the interface, p = hk cos(theta). k increases, so cos(theta) decreases, so theta increases. Therefore, observation agrees with the principle of momentum conservation applied parallel to the interface.

In the direction perpendicular to the interface, p = hk sin(theta). k increases, so sin(theta) decreases, so theta must decrease after refraction. But this is not what we observed in the second paragraph. So, there must be a loophole somewhere.

Either your analysis has a fallacy, or if that is not so, then the only logical conclusion is that there is a force acting on the photons perpendicular to the interface. This force arises in our discussion purely due to quantum-mechanical considerations, so the force cannot be described or found using classical physics. What might that force be?

I have used light as my example above. Now, how do we interpret quantum-mechanically the refraction of other waves, eg sound, water waves?

Can we not explain the origin of refraction using classical physics? Does the Huygen-Fresnel principle not explain refraction? I think we should try this approach as refraction must have been explained before the nineteenth century. I tried drawing secondary wavelets in a diagram of refraction to convince myself that the Huygen-Fresnel principle can indeed explain refraction, but the method is tedious and I've given up.

What are your thoughts?

 

[hasan_researc]

Assuming the interface is also a plane, one expects the incident and transmitted waves both to be plane.

If the interface is also a plane, why should we expect the transmitted wave to be plane?

Since one part of the incident wave reaches the interface before another part, and since the propagation speed is, say, lower in the new medium, the final, transmitted wave's Poynting vector must point in a different direction in the new medium.

I'm sorry I don't understand this very well.

 

[GRDixon]

If the interface is also a plane, why should we expect the transmitted wave to be plane?




I'm sorry I don't understand this very well.

If it weren't plane ... say if it was a spherical wave in the new medium ... what would determine where the center of curvature was, for example? Assuming the speed is single-valued in the new medium, the transmitted wave would have to be plane. Otherwise one part of the incident wave would, by inference, propagate at a different speed than another.