Can Snell's Law Be Proven Mathematically for Sound and Light Refraction?

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Discussion Overview

The discussion revolves around the mathematical proof of Snell's Law in the context of sound and light refraction, particularly focusing on the scenario where waves strike a surface perpendicularly. Participants explore the implications of this condition on the behavior of waves and the application of Snell's Law.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that when light or sound waves strike a surface perpendicularly, they do not refract according to Snell's Law.
  • One participant questions how to mathematically prove the absence of refraction in this scenario using Snell's Law.
  • Another participant explains that the sine of 90 degrees is 1, leading to the conclusion that no refraction occurs when applying Snell's Law.
  • A different participant clarifies that when light falls perpendicularly, the angle is 0 degrees, leading to the application of Snell's Law as n1sin(θ1) = n2sin(θ2), resulting in θ2 being 0 and thus no refraction.
  • One participant elaborates on the wave-vector component's behavior at normal incidence, stating that continuity requires no deflection of the wave, which aligns with the implications of Snell's Law.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of Snell's Law in the context of perpendicular incidence. While some agree that no refraction occurs, the discussion remains unresolved regarding the mathematical proof and implications of this scenario.

Contextual Notes

There are unresolved assumptions regarding the definitions of refraction and the mathematical steps involved in applying Snell's Law in this context.

Misr
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when light rays fall perpendicularly on a separating surface between two media it doesn't refract according to Snell's law
and it is the same with sound waves

how to prove something like that mathematically??
Thanks in advance
 
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Misr said:
when light rays fall perpendicularly on a separating surface between two media it doesn't refract according to Snell's law
and it is the same with sound waves

how to prove something like that mathematically??
Thanks in advance

You mean using something other than Snell's law, which mathematically says that there is no refraction?
 
No I mean using Snell's Law
which mathematically says that there is no refraction?
I want to know how
 
Well the sin 90 degrees, or perpendicularity, is 1. So...multiplying any number by zero will give the same number, ergo no refraction.
 
Well the sin 90 degrees, or perpendicularity, is 1. So...multiplying any number by zero will give the same number, ergo no refraction.
I don't understand what are you trying to say but when light falls perpendicular to the surface means that the angle is 0 not 90
so on appling snells law
n1sin(θ1) = n2sin(θ2)
where sin0=0
therefore n2sin(θ2)=0
and since n2 can't be zero of course
then θ2=0 so no refraction takes place
right?
 
Misr said:
I don't understand what are you trying to say but when light falls perpendicular to the surface means that the angle is 0 not 90
so on appling snells law
n1sin(θ1) = n2sin(θ2)
where sin0=0
therefore n2sin(θ2)=0
and since n2 can't be zero of course
then θ2=0 so no refraction takes place
right?

Yes. "No refraction" means the two angles are the same, and 0=0, ergo there is no refraction.
 
Oh ok thanks very much
 
More fundamentally, the wave-vector component parallel to the interface must be constant across the interface to ensure continuity between EM fields in both media. Snell's law is derived from this condition.

When a wave strikes a boundary at normal incidence, there is no wave-vector component parallel to the interface (by definition). Due to continuity, there can be no parallel wave-vector component in either medium, and so the wave is not deflected in any way.

This is what Snells law says in a nutshell.

Claude.
 

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