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metacogitans

How to Use Metric Tensors to Describe How Waves Interact with Surface Area?

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Posted (edited)

I'm trying to figure out how to describe the geometrical coordinates of a wave contacting and reflecting off the surface area of a spherical object/particle; specifically waves traveling at the speed of light.

 

My goal is to be able to explain the torsion in a region of surface area over time making contact with a wave reflecting off it, considering how the velocity of the object/particle changes as well, and how the geometry of the reflecting wave changes too.

 

I have a good picture in my head of how it all comes together geometrically, but I don't know how to go about writing it down with tensor geometry on paper.

 

I'm still very new to using tensor calculus; I would be very excited to be able finally to actually write out the math of a concept I'm thinking about instead of hitting a roadblock at tensor calculus every time and not being able to do anything after that - so, teachers are welcome =)

 

I know calculus up to about what a second year student would know, and understand the FToC, how differentiation and integration are inversely related, etc.. I even know 3-coordinate volume integrals and planar derivatives, as long as its not too complex of a problem. So, it seems to me like learning tensor geometry and calculus in physics is right where I should be in looking for what to tackle next.

Edited by metacogitans

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Posted (edited)

I'm trying to figure out how to describe the geometrical coordinates of a wave contacting and reflecting off the surface area of a spherical object/particle; specifically waves traveling at the speed of light.

 

My goal is to be able to explain the torsion in a region of surface area over time making contact with a wave reflecting off it.

 

I have a good picture in my head of how it all comes together geometrically, but I don't know how to go about writing it down with tensor geometry on paper.

 

I'm still very new to using tensor calculus; I would be very excited to be able finally to actually write out the math of a concept I'm thinking about instead of hitting a roadblock at tensor calculus every time and not being able to do anything with it after that - so, teachers are welcome =)

 

I know calculus up to about what a second year student would know, and understand the FToC, how differentiation and integration are inversely related, etc.. I even know 3-coordinate volume integrals and planar derivatives, as long as its not too complex of a problem. So, it seems to me like learning tensor geometry and calculus in physics is right where I should be in looking for what to tackle next.

 

The very first thing you need is to decide what sort of wave you mean and how to describe it.

 

By what sort I don't mean sound or light or gravity, I mean its geometric distribution in space, in short its 'shape'.

Is it 1D, 2D or 3D?

Is it plane spherical or what?

Clearly (to you I hope) it is a travelling wave since you say it is reflected.

 

Edit

 

A good clear source for the transition from simpler mathematics to tensor methods in this subject is

 

Modern Optics

 

By Robert Guenther

 

pub Wiley.

Edited by studiot

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Posted (edited)

 

The very first thing you need is to decide what sort of wave you mean and how to describe it.

 

By what sort I don't mean sound or light or gravity, I mean its geometric distribution in space, in short its 'shape'.

Is it 1D, 2D or 3D?

Is it plane spherical or what?

Clearly (to you I hope) it is a travelling wave since you say it is reflected.

I was initially thinking of the wave as an infinitely thin sphere propagating out in all directions from its center, and following the inverse square law having a diminishing intensity with distance.

 

I was trying to figure out if perhaps the degree of curvature of the wave (being 'flatter' the further the wave propagates), proportional to the curvature of the spherical object/particle, would determine its intensity.

 

As for the type of wave, a force-carrying wave in its simplest form (if there is such a thing), traveling at the speed of light. Or it could just be considered light, or some other simple electromagnetic wave that transfers inertia.

Edited by metacogitans

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I was initially thinking of the wave as an infinitely thin sphere propagating out in all directions from its center, and following the inverse square law having a diminishing intensity with distance.

 

 

 

This doesn't describe a wave it describes either a wavefront or a soliton pulse.

 

 

 

 

I was trying to figure out if perhaps the degree of curvature of the wave (being 'flatter' the further the wave propagates), proportional to the curvature of the spherical object/particle, would determine its intensity.

 

 

For many purposes spherical waves can be approximated by plane waves.

This was part of the reason for my original response.

Many useful simplifications have been developed over the years.

 

 

 

 

As for the type of wave, a force-carrying wave in its simplest form (if there is such a thing), traveling at the speed of light. Or it could just be considered light, or some other simple electromagnetic wave that transfers inertia.

 

I specifically said I was not initially interested in the nature of the wave from this point of view.

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People model the radiation pressure effects on satellites. You might use the time-honored tradition of cribbing from work folks have already done.

 

However, I'm not sure if there would be any effect on a spherical object, unless there are gradients in the intensity.

 

Also not sure that tensors would need to be involved. You should just be able to apply conservation of momentum, and from that deduce torque, from the reflection angle dependence for any element on the half-sphere (integrated over the surface)

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