How do non planar waves reflect?

[chrom68]

I have some questions regarding the reflective properties of sound waves...

If a spherically propagating wave encounters a plane, the reflection is similar to a light ray, shoots off at the critical angle it approached at, with a new 'image' centre. If the spherical waves encounter another object say a convex object like another sphere (of a smaller radius) what happens? Do they return back along the vector direction from the smaller sphere to the point source? (puzzling to picture i know). What wave parameters would change? Are there set equations governing this behaviour?

 

[HallsofIvy]

Waves reflect at individual points. If you have a spherical wave reflecting from a spherical object, it reflects like this: From each point on the spherical object, draw a line to the center of the spherical wave. Also draw a tangent plane to the object at that point. The ray reflects off the tangent plane making equal angles.

 

[chrom68]

I'm struggling to picture the reflected rays, because shouldn't they reflect off the spherical object as the spherical waves they approached at? with the difference being that the reflected spherical waves now have a new (imaginary) point source. If this is correct shouldn't the centre of the spherical object be this new point source?

 

[chrom68]

Additionally, do the waves reflect off the object sphere as rays or as waves?

 

[bobthenormal]

I'm not sure if this is correct, but I was reading your question and picturing this reflection, and I think I have an easy way to explain what (I think) would happen... the reflection would actually be really symmetric!

Say you have a point source, S, and it emits a spherical wavefront. That wavefront propagates as concentric spheres... say that at radius R, the wavefront is tangent at a single point P, to a completely reflecting sphere of radius r. So, you can kind of picture a large sphere (the wave) centered at S, with a tangent small sphere a distance R+r from S.

Now, imagine the mirror image of the wave front across the tangent plane, enlarged. So now you have the same large wavefront, but instead of being centered at S, it is centered at a distance 2(R+r) from S, and is also tangent to the small sphere at point P (the small sphere would be inside of this sphere, tangent at point P).

That new sphere is what the reflection would look like, except that it would only be a part of it (it would look like a convex disk moving toward you).

There is one problem with this explanation that I can think of... and that is I'm not really sure the "reflected sphere's" radius is really R+2r (which is what I put above).

To answer your other questions... the wavefront can be thought of as being made up of lots of rays, but since tracing a line along the front of them makes a nice spherical shape (in this case), why not call it a wave?

--Bob

 

[chrom68]

After making a quick sketch i agree that there is a convex disc moving back towards the source of the spherical waves, however its still unclear where the apparent source of the ones reflected back come from. I believe its the centre of the smaller sphere object, Bob thinks 2R+r. Its tricky because the incoming spherical wave will reflect in layers off the sphere object. Any insights?

 

[chrom68]

If each wave consists of particles, are the directions of their reflection in someway related?