Eikonal

The title of this article is ambiguous. For the secret service cooperation of the same name, see Operation Eikonal .

The eikonal ( ancient Greek εἰκών eikon = image, image ) is the path of a light beam between the starting point and the end point in geometric optics ; meanwhile the term mostly describes the Bruns-Eikonal .

Bruns-Eikonal

The Bruns-Eikonal or Brunssche Eikonal is a function which, according to Fermat's principle, describes the shortest path between two points separated by optical media . It was published by the German mathematician Heinrich Bruns in 1895 and used in ray optics. The name Eikonal comes from Bruns, but the process was already known to William Rowan Hamilton , who called it a characteristic function ( Hamilton-Jacobi equation ) and used it in optics and mechanics.

The Bruns-Eikonal is used for acoustic waves and other wave phenomena , e.g. B. in seismology to calculate the propagation of seismic waves .

Derivation

In the following, the eikonal equation is to be derived as a high frequency approximation of the acoustic wave equation . In quantum mechanics , a similar method is used, the semi classical WKB approximation .

So we start from the acoustic wave equation with the pressure , the position vector , the position-dependent propagation speed and constant density ${\displaystyle p}$ ${\displaystyle {\vec {x}}}$ ${\displaystyle c=c({\vec {x}})}$

${\displaystyle \nabla ^{2}p-{\frac {1}{c^{2}}}{\frac {\partial ^{2}p}{\partial t^{2}}}=0}$

We are looking for a time-harmonic high-frequency approach for which a frequency and time-independent amplitude and the transit time function can be assumed. She has the shape ${\displaystyle P({\vec {x}})}$ ${\displaystyle \phi ({\vec {x}})}$

${\displaystyle p({\vec {x}},t)=P({\vec {x}})e^{\mathrm {i} \omega (t-\phi ({\vec {x}}))}}$

First the time derivatives of the wave equation are calculated:

${\displaystyle {\frac {\partial p}{\partial t}}=\mathrm {i} \omega P({\vec {x}})e^{i\omega (t-\phi ({\vec {x}}))};\quad {\frac {\partial ^{2}p}{\partial t^{2}}}=-\omega ^{2}P({\vec {x}})e^{i\omega (t-\phi ({\vec {x}}))}}$

Now the local derivatives follow:

${\displaystyle \nabla p=\nabla Pe^{\mathrm {i} \omega (t-\phi ({\vec {x}}))}-\mathrm {i} \omega Pe^{\mathrm {i} \omega (t-\phi ({\vec {x}}))}\nabla \phi =(\nabla P-\mathrm {i} \omega P\nabla \phi )e^{\mathrm {i} \omega (t-\phi ({\vec {x}}))}}$

Because of the vector identity : ${\displaystyle \nabla \cdot \left(a({\vec {x}}){\vec {b}}({\vec {x}})\right)=\nabla a({\vec {x}})\cdot {\vec {b}}({\vec {x}})\ +a({\vec {x}})\nabla \cdot {\vec {b}}({\vec {x}})}$

${\displaystyle \nabla ^{2}p=\nabla \cdot \nabla p}$

${\displaystyle =\nabla \cdot (\nabla P-\mathrm {i} \omega P\nabla \phi )e^{\mathrm {i} \omega (t-\phi ({\vec {x}}))}+(\nabla P-\mathrm {i} \omega P\nabla \phi )\cdot \nabla e^{\mathrm {i} \omega (t-\phi ({\vec {x}}))}}$

${\displaystyle =\left(\nabla ^{2}P-\mathrm {i} \omega \nabla P\cdot \nabla \phi -\mathrm {i} \omega P\nabla ^{2}\phi -\mathrm {i} \omega (\nabla P-\mathrm {i} \omega P\nabla \phi )\cdot \nabla \phi \right)e^{\mathrm {i} \omega (t-\phi ({\vec {x}}))}}$

${\displaystyle =\left(\nabla ^{2}P-2\mathrm {i} \omega \nabla P\cdot \nabla \phi -\mathrm {i} \omega P\nabla ^{2}\phi -\omega ^{2}P(\nabla \phi )^{2}\right)e^{\mathrm {i} \omega (t-\phi ({\vec {x}}))}}$

The two derivatives inserted into the wave equation result after division by ${\displaystyle e^{\mathrm {i} \omega (t-\phi ({\vec {x}}))}}$

${\displaystyle -\omega ^{2}P\left((\nabla \phi )^{2}-{\frac {1}{c^{2}}}\right)-\mathrm {i} \omega \left(2\nabla P\cdot \nabla \phi +P\nabla ^{2}\phi \right)+\nabla ^{2}P=0.}$

Division by then leads to ${\displaystyle -\omega ^{2}P}$

${\displaystyle \left((\nabla \phi )^{2}-{\frac {1}{c^{2}}}\right)+{\frac {\mathrm {i} }{\omega P}}\left(2\nabla P\cdot \nabla \phi +P\nabla ^{2}\phi \right)-{\frac {1}{\omega ^{2}P}}\left(\nabla ^{2}P\right)=0.}$

Since the real and imaginary parts of the equation must be equal to zero independently of one another, it follows:

${\displaystyle \left((\nabla \phi )^{2}-{\frac {1}{c^{2}}}\right)-{\frac {1}{\omega ^{2}P}}\left(\nabla ^{2}P\right)=0}$

The approximation assumes that the amplitude is only weakly dependent on location, that is , it is limited. Since neither the transit time nor the amplitude are frequency-dependent at the same time , the second term for very high frequencies is small compared to the first term and the equation is simplified to: ${\displaystyle P}$ ${\displaystyle \nabla ^{2}P}$ ${\displaystyle \phi }$ ${\displaystyle P}$

${\displaystyle \left(\nabla \phi \right)^{2}={\frac {1}{c^{2}}}}$

The solution of the eikonal equation assigns the time of flight of the wave to each point in space . Lines with the same transit time can accordingly be interpreted as wave fronts . ${\displaystyle \phi ({\vec {x}})}$