Geometric approximation

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)

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