Wave equation in a medium with smooth n(x) refractive index

[Killtech]

how is such an equation called?

Im looking for an simplest wave equation for a non-homogenous static medium with a smooth refractive index n(x). i am more interested into the case for acoustics, though i guess it will be quite the same for optics. i am failing to google the right thing, so i though i just ask people that can answer me right away. I know the eikonal equation is related, but i am looking for the equation of the actual wave.

[Markus Hanke]

It would probably be described by some variant of Cauchy’s equations of motion for inhomogenous media; see for example paragraph 3.1 in this article:

https://www.mdpi.com/2624-599X/3/4/45

[Killtech]

4 hours ago, Markus Hanke said:

It would probably be described by some variant of Cauchy’s equations of motion for inhomogenous media; see for example paragraph 3.1 in this article:

https://www.mdpi.com/2624-599X/3/4/45

Thanks. They suggest an equation with the simple form \(\partial_{t}^{2}\phi-c{}^{2}\partial_{x}^{2}\phi=0\) where \(\phi=Es(x)\) and \(c=c(x)=n(x)c_{0}\) i suppose, okay. That was my first guess, too. But i stopped there because simply replacing a constant \(c\) by a locally dependent one seemed a little too easy, specifically for a plane wave the the spatial dimensions can be easily treated independently.

Am i just blind and missing something or is this equation one approximation too many such that the smooth refractive index here won't produce any lensing effects?

[Killtech]

okay, nah, there is something wrong in the article. i don't see their factorization of the equation into 1st order PDEs to work, because \(c(x)\partial_{x}(c(x)\partial_{x})\neq c(x)^{2}\partial_{x}^{2}\) unless \(c(x)\) is constant. a term \((\nabla c)\nabla\) would creep into the wave equation.

[swansont]

16 hours ago, Killtech said:

Im looking for an simplest wave equation for a non-homogenous static medium with a smooth refractive index n(x)

 

Quote

Am i just blind and missing something or is this equation one approximation too many such that the smooth refractive index here won't produce any lensing effects?

What is x? If it’s the direction of propagation of a plane wave, why do you expect lensing? 

[joigus]

https://en.wikipedia.org/wiki/Eikonal_equation

[Killtech]

1 hour ago, swansont said:

 

What is x? If it’s the direction of propagation of a plane wave, why do you expect lensing? 

same as in the article x is meant to be the coordinates \(\boldsymbol{x}\) with 3 dimensions (i am struggling with using latex in the forums without an editor with better support). but they also mix it up and sometimes it just means the first component, specifically when they use \(\partial_x\).

hmm, does in that article \(c(x)\) vary only along one dimension?

1 hour ago, joigus said:

https://en.wikipedia.org/wiki/Eikonal_equation

as far as i understand it, the eikonal equation (the one in the wiki article) can be interpreted as the path a wavefront takes through the medium, but it is not the actual equation of the wave itself. so yes, it is very closely related to what i am looking for, but not exactly it.

[joigus]

32 minutes ago, Killtech said:

same as in the article x is meant to be the coordinates x with 3 dimensions (i am struggling with using latex in the forums without an editor with better support). but they also mix it up and sometimes it just means the first component, specifically when they use ∂x .

hmm, does in that article c(x) vary only along one dimension?

as far as i understand it, the eikonal equation (the one in the wiki article) can be interpreted as the path a wavefront takes through the medium, but it is not the actual equation of the wave itself. so yes, it is very closely related to what i am looking for, but not exactly it.

Sorry I didn't read carefully. You actually mentioned the eikonal equation.

What is it exactly that makes it non-wavy?

The problem with it is when the wave finds inhomogeneities of size the order of the wavelength, then it no longer is a good approximation. But otherwise it's quite wavy isn't it? I mean, if you solve for the amplitude and the direction of the 'rays' you're home free I suppose.

Edited November 11, 2023 by joigus
minor correction

[Killtech]

8 minutes ago, joigus said:

Sorry I didn't read carefully. You actually mentioned the eikonal equation.

What is it exactly that makes it non-wavy?

The problem with it is when the wave finds inhomogeneities of size the order of the wavelength, then it no longer is a good approximation. But otherwise it's quite wavy isn't it? I mean, if you solve for the amplitude and the direction of the 'rays' you're home free I suppose.

found what i was looking for here: https://wiki.seg.org/wiki/The_eikonal_equation

apart from the eikonal equation (4), there is also the corresponding wave equation (1) and it is indeed just the regular standard 2nd order wave PDE but with a non-constant \(c(x)\). (1) produces just the wavy solutions i was looking for.

[John Cuthber]

I'm pretty sure someone has the maths for things with a smooth change in refractive index.

https://en.wikipedia.org/wiki/Gradient-index_optics

[studiot]

The smooth change of refractive index condition is exactly what happens in marine acoustics.

(It is not the only thing though.)

 

In this context the eiconal equation is discussed chapter 2 page 5 et seq of the book most people refer back to by R J Urick

Sound Propagation in the Sea.

Originally written in 1979., but is now availble as a free pdf.

 

Alternatively the eiconal equation for optics and many other uses such as the propagation of discontinuities is discussed in Erich Zauderer's book

Partial Differential Equations of Applied Mathematics

 

Wiley 1989