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''geometrical-optics approximation, ray approximation''
 
''geometrical-optics approximation, ray approximation''
  
 
A series of the form
 
A series of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044200/g0442001.png" /></td> </tr></table>
+
$$
 +
\sum _ {s = 0 } ^  \infty 
  
that formally satisfies the equation describing a wave phenomenon (or a system of equations when the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044200/g0442002.png" /> are vectors).
+
\frac{u _ {s} ( x, y, z) }{(- i \omega )  ^ {s} }
 +
 
 +
e ^ {i \omega \tau ( x, y, z) - i \omega t }
 +
$$
 +
 
 +
that formally satisfies the equation describing a wave phenomenon (or a system of equations when the $  u _ {s} $
 +
are vectors).
  
 
In order to solve wave propagation problems in the high-frequency regime (cf. [[Diffraction, mathematical theory of|Diffraction, mathematical theory of]]) the so-called [[Ray method|ray method]] for the construction of geometric approximations has been developed [[#References|[1]]], [[#References|[2]]]. Presumably, the resulting series is an asymptotic expansion of the solutions sought wherever the terms of the geometric approximation have no singular points. This hypothesis can be proved in special cases. There also exists a non-stationary analogue of the geometric approximation.
 
In order to solve wave propagation problems in the high-frequency regime (cf. [[Diffraction, mathematical theory of|Diffraction, mathematical theory of]]) the so-called [[Ray method|ray method]] for the construction of geometric approximations has been developed [[#References|[1]]], [[#References|[2]]]. Presumably, the resulting series is an asymptotic expansion of the solutions sought wherever the terms of the geometric approximation have no singular points. This hypothesis can be proved in special cases. There also exists a non-stationary analogue of the geometric approximation.
  
The construction of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044200/g0442003.png" /> is based on considering the field of rays, i.e. the extremals of a functional (cf. [[Fermat principle|Fermat principle]]):
+
The construction of the functions $  u _ {s} $
 +
is based on considering the field of rays, i.e. the extremals of a functional (cf. [[Fermat principle|Fermat principle]]):
 +
 
 +
$$
 +
\int\limits
 +
\frac{ds }{c ( x, y, z) }
 +
,
 +
$$
 +
 
 +
where  $  c( x, y, z) $
 +
is the wave speed in the isotropic physical medium under consideration and  $  ds $
 +
is the element of arc length. Let some pair of parameters  $  \alpha , \beta $
 +
characterize the ray, let the parameter  $  \tau $
 +
characterize the points on the ray, and let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044200/g0442004.png" /></td> </tr></table>
+
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044200/g0442005.png" /> is the wave speed in the isotropic physical medium under consideration and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044200/g0442006.png" /> is the element of arc length. Let some pair of parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044200/g0442007.png" /> characterize the ray, let the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044200/g0442008.png" /> characterize the points on the ray, and let
+
\frac{d \tau }{ds }
 +
  = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044200/g0442009.png" /></td> </tr></table>
+
\frac{1}{c( x, y, z) }
 +
.
 +
$$
  
The parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044200/g04420010.png" /> may be used as curvilinear coordinates. The transition from these coordinates to the orthogonal Cartesian coordinate system is given by the formula
+
The parameters $  \alpha , \beta , \tau $
 +
may be used as curvilinear coordinates. The transition from these coordinates to the orthogonal Cartesian coordinate system is given by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044200/g04420011.png" /></td> </tr></table>
+
$$
 +
\mathbf r ( \alpha , \beta , \tau )  = \mathbf r ( x, y, z).
 +
$$
  
The surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044200/g04420012.png" /> are orthogonal to the rays. At the points where the field of rays has no singularities, the magnitude
+
The surfaces $  \tau = \textrm{ const } $
 +
are orthogonal to the rays. At the points where the field of rays has no singularities, the magnitude
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044200/g04420013.png" /></td> </tr></table>
+
$$
 +
= \left | \left [
  
which is known as the geometric spreading or divergence, is non-zero. The magnitude <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044200/g04420014.png" /> forms part of recurrence relations interconnecting the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044200/g04420015.png" />, and plays a fundamental role in all constructions of geometric approximations.
+
\frac{\partial  \mathbf r }{\partial  \alpha }
 +
,\
  
====References====
+
\frac{\partial \mathbf r }{\partial \beta }
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> F.G. Friedlander,  "Sound pulses" , Cambridge Univ. Press (1958)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.M. Babich,  V.S. Buldyrev,  "Asymptotic methods in the diffraction of short waves" , Moscow  (1972)  (In Russian)  (Translation forthcoming: Springer)</TD></TR></table>
 
  
 +
\right ] \right | ,
 +
$$
  
 +
which is known as the geometric spreading or divergence, is non-zero. The magnitude  $  J $
 +
forms part of recurrence relations interconnecting the functions  $  u _ {s} $,
 +
and plays a fundamental role in all constructions of geometric approximations.
 +
 +
====References====
 +
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> F.G. Friedlander, "Sound pulses" , Cambridge Univ. Press (1958) {{MR|0097233}} {{ZBL|0079.41001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> V.M. Babich, V.S. Buldyrev, "Asymptotic methods in the diffraction of short waves" , Moscow (1972) (In Russian) (Translation forthcoming: Springer)</TD></TR></table>
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Kline,   I.W. Kay,   "Electromagnetic theory and geometrical optics" , Interscience (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> L.B. Felsen,   N. Marcuvitz,   "Radiation and scattering of waves" , Prentice-Hall (1973) pp. Sect. 1.7</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Kline, I.W. Kay, "Electromagnetic theory and geometrical optics" , Interscience (1965) {{MR|0180094}} {{ZBL|0123.23602}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> L.B. Felsen, N. Marcuvitz, "Radiation and scattering of waves" , Prentice-Hall (1973) pp. Sect. 1.7 {{MR|0471626}} {{ZBL|}} </TD></TR></table>

Latest revision as of 19:41, 5 June 2020


geometrical-optics approximation, ray approximation

A series of the form

$$ \sum _ {s = 0 } ^ \infty \frac{u _ {s} ( x, y, z) }{(- i \omega ) ^ {s} } e ^ {i \omega \tau ( x, y, z) - i \omega t } $$

that formally satisfies the equation describing a wave phenomenon (or a system of equations when the $ u _ {s} $ are vectors).

In order to solve wave propagation problems in the high-frequency regime (cf. Diffraction, mathematical theory of) the so-called ray method for the construction of geometric approximations has been developed [1], [2]. Presumably, the resulting series is an asymptotic expansion of the solutions sought wherever the terms of the geometric approximation have no singular points. This hypothesis can be proved in special cases. There also exists a non-stationary analogue of the geometric approximation.

The construction of the functions $ u _ {s} $ is based on considering the field of rays, i.e. the extremals of a functional (cf. Fermat principle):

$$ \int\limits \frac{ds }{c ( x, y, z) } , $$

where $ c( x, y, z) $ is the wave speed in the isotropic physical medium under consideration and $ ds $ is the element of arc length. Let some pair of parameters $ \alpha , \beta $ characterize the ray, let the parameter $ \tau $ characterize the points on the ray, and let

$$ \frac{d \tau }{ds } = \ \frac{1}{c( x, y, z) } . $$

The parameters $ \alpha , \beta , \tau $ may be used as curvilinear coordinates. The transition from these coordinates to the orthogonal Cartesian coordinate system is given by the formula

$$ \mathbf r ( \alpha , \beta , \tau ) = \mathbf r ( x, y, z). $$

The surfaces $ \tau = \textrm{ const } $ are orthogonal to the rays. At the points where the field of rays has no singularities, the magnitude

$$ J = \left | \left [ \frac{\partial \mathbf r }{\partial \alpha } ,\ \frac{\partial \mathbf r }{\partial \beta } \right ] \right | , $$

which is known as the geometric spreading or divergence, is non-zero. The magnitude $ J $ forms part of recurrence relations interconnecting the functions $ u _ {s} $, and plays a fundamental role in all constructions of geometric approximations.

References

[1] F.G. Friedlander, "Sound pulses" , Cambridge Univ. Press (1958) MR0097233 Zbl 0079.41001
[2] V.M. Babich, V.S. Buldyrev, "Asymptotic methods in the diffraction of short waves" , Moscow (1972) (In Russian) (Translation forthcoming: Springer)

Comments

References

[a1] M. Kline, I.W. Kay, "Electromagnetic theory and geometrical optics" , Interscience (1965) MR0180094 Zbl 0123.23602
[a2] L.B. Felsen, N. Marcuvitz, "Radiation and scattering of waves" , Prentice-Hall (1973) pp. Sect. 1.7 MR0471626
How to Cite This Entry:
Geometric approximation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Geometric_approximation&oldid=11706
This article was adapted from an original article by V.M. Babich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article