Eikonal equation with n(y)

  • Thread starter avisha03
  • Start date
In summary: Hi avisha03!Haven't you come across cosh and sinh? If not, see http://en.wikipedia.org/wiki/Sinh#Standard_Integrals"(See also standard integrals in the PF Library)There are tables in which you can look up, for example, cosh(x) and cosh-1(x).The integral result is logarithmic, isn't it?Yes, but, as you've noticed, it's difficult to invert in that form (ie ln(x + √(x2 + a2))) :wink:I do know cosh & sinh,
  • #1
avisha03
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0

Homework Statement



A duck is walking in a flat desert (yes, the proff. is creative...). According to atmospheric conditions the Refractive index is n(y)=1.0006-0.0003y. y in KM.
An observer is on the ground, 26.2 KM far. Where will he see the duck? Show the gragh of the ray path.


Homework Equations


d(ne)/ds= grad n
e- the ray path direction

The Attempt at a Solution


I used the eikonal equation, and some matematical manipulations to get the following equation:
dy/dx= sqrt((a-by)^2/c^2-1)
(a=1.0006, b=0.0003, c=1.0006*cos(theta_0))
In order to try and find y as a function of x, I tried to solve the latter eq. by seperating it, and using the variable replacment: m=a-by
I got:
ln(m+sqrt(m^2-c^2))=exp(-bx/c)
now I don't know wheather my way till now is correct, and how to display y as y(x).
Thanks for your help!
 
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  • #2
Welcome to PF!

Hi avisha03! Welcome to PF! :smile:

(have a square-root: √and a theta: θ and try using the X2 tag just above the Reply box :wink:)
avisha03 said:
… I used the eikonal equation, and some matematical manipulations to get the following equation:
dy/dx= sqrt((a-by)^2/c^2-1)
(a=1.0006, b=0.0003, c=1.0006*cos(theta_0))
In order to try and find y as a function of x, I tried to solve the latter eq. by seperating it, and using the variable replacment: m=a-by
I got:
ln(m+sqrt(m^2-c^2))=exp(-bx/c)
now I don't know wheather my way till now is correct, and how to display y as y(x).
Thanks for your help!

You mean dy/dx= √((a-by)/c)2 - 1) ?

(I haven't checked it, but assuming that it's correct …)

Easier to put n = (a-by)/c, dn = -bdy/c

dn/√(n2 - 1) = -(b/c)dx

and then use a hyperbolic trig substitution. :wink:
 
  • #3
Thanks for your help & advices. :)
What do you mean by hyperbolic trig substitution?
The integral result is logarithmic, isn't it?
 
  • #4
avisha03 said:
Thanks for your help & advices. :)
What do you mean by hyperbolic trig substitution?

Hi avisha03! :smile:

Haven't you come across cosh and sinh? if not, see http://en.wikipedia.org/wiki/Sinh#Standard_Integrals"

(See also standard integrals in the PF Library)

There are tables in which you can look up, for example, cosh(x) and cosh-1(x).
The integral result is logarithmic, isn't it?

Yes, but, as you've noticed, it's difficult to invert in that form (ie ln(x + √(x2 + a2))) :wink:
 
Last edited by a moderator:
  • #5
I do know cosh & sinh, but since I'm not a native english speaker- I wanted to make sure. Thanks alot, anyway.
 

What is the Eikonal equation with n(y)?

The Eikonal equation with n(y) is a partial differential equation that describes the propagation of waves in a medium with varying refractive index, represented by the variable n. It is used in optics, seismology, and other fields to model the behavior of waves.

What is the physical interpretation of n(y) in the Eikonal equation?

n(y) represents the refractive index of the medium at a given point in space, y. This parameter affects the velocity of waves in the medium, and can be used to describe the behavior of light, sound, or other types of waves.

What are the main applications of the Eikonal equation with n(y)?

The Eikonal equation with n(y) has a wide range of applications in various fields, such as optics, seismology, geophysics, and medical imaging. It can be used to model the propagation of light in lenses and other optical devices, seismic waves in the Earth's crust, and ultrasound waves in medical imaging.

How is the Eikonal equation with n(y) solved?

The Eikonal equation with n(y) is a non-linear partial differential equation, and its solution can be obtained using various numerical and analytical methods. Some commonly used techniques include the finite difference method, the finite element method, and the method of characteristics.

What are the limitations of the Eikonal equation with n(y)?

The Eikonal equation with n(y) is a simplified model that does not take into account certain physical phenomena, such as diffraction and scattering. It also assumes that the refractive index of the medium is known and constant, which may not always be the case in real-world scenarios. Additionally, the equation only describes the behavior of waves in a homogeneous medium and cannot be applied to inhomogeneous or anisotropic media.

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