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{{Wikify|date=January 2012}}
{{Wikify|date=September 2011}}


{{Refimprove|Date|date=August 2011}}
In mathematics, the '''Blasius function''', introduced by {{harvs|txt|last=Blasius|authorlink=Paul Richard Heinrich Blasius|year=1908}}, is the solution to the third-order [[differential equation]]
{{Orphan|Date|date=August 2011}}


The '''T-matrix method''' is a computatational technique of [[light scattering]] by nonspherical particles originally formulated by [http://www.giss.nasa.gov/staff/mmishchenko/t_matrix.html P. C. Waterman] (1928-2012) in 1965.<ref>M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, T-matrix computations of light scattering by nonspherical particles: A review, J. Quant. Spectrosc. Radiat. Transfer, 55, 535-575 (1996).</ref> The technique is also known as null field method and extended boundary technique method (EBCM). In the method, matrix elements are obtained by matching boundary conditions for solutions of [[Maxwell equations]].
:<math>2f_{xxx} + f\,f_{xx}=0</math>


== Definition of the T-Matrix ==
with the boundary conditions ''f''(0)&nbsp;=&nbsp;''f''<sub>''x''</sub>(0)&nbsp;=&nbsp;0, ''f''<sub>x</sub>(∞)&nbsp;=&nbsp;1.


The incident and scattered electric field are expanded into spherical vector wave functions (SVWF), which are also encountered in [[Mie scattering]]. They are the fundamental solutions of the vector [[Helmholtz equation]] and
==See also==
can be generated from the scalar fundamental solutions in spherical coordinates, the spherical [[Bessel functions]] of the first kind and the spherical Hankel Functions. Accordingly, there are two linearly independent sets of solutions
denoted as <math>\mathbf{M}^1,\mathbf{N}^1</math> and <math>\mathbf{M}^3,\mathbf{N}^3</math>, respectively. They are also called regular and propagating SVWFs, respectively. With this, we can write the incident field as


<math>\mathbf{E}_{inc}= \sum_{n=1}^\infty \sum_{m=-n}^n a_{mn} \mathbf{M}^1_{mn}+ b_{mn} \mathbf{N}^1_{mn}.</math>
*[[Blasius boundary layer]]


The scattered field is expanded into radiating SVWFs:
==References==


<math>\mathbf{E}_{scat}= \sum_{n=1}^\infty \sum_{m=-n}^n f_{mn} \mathbf{M}^3_{mn}+ g_{mn} \mathbf{N}^3_{mn}.</math>
*{{Citation | last1=Blasius | first1=H. | title=Grenzschichten in Flüssigkeiten mit kleiner Reibung | language=German | year=1908 | journal=Zeitschrift für Mathematik und Physik | volume=56 | pages=1–37}}
*{{Citation | last1=Boyd | first1=John P. | title=The Blasius function in the complex plane | url=http://projecteuclid.org/getRecord?id=euclid.em/1047262359 | mr=1737233 | year=1999 | journal=Experimental Mathematics | issn=1058-6458 | volume=8 | issue=4 | pages=381–394}}


The T-Matrix relates the expansion coefficients of the incident field to those of the scattered field.
[[Category:Special functions]]
 
<math>\begin{pmatrix} a_{mn}\\ b_{mn}\end{pmatrix} = T \begin{pmatrix} f_{mn} \\ g_{mn} \end{pmatrix}</math>
 
The T-Matrix is determined by the scatterer shape and material and for given indcient field allows to calculate the scattered
field.
 
== Calculation of the T-Matrix ==
 
The standard way to actually calculate the T-Matrix method is the Null-Field Method, that relies on the Stratton-Chu equations <ref>J.A. Stratton & L.J. Chu: Diffraction Theory of Electromagnetic Waves Phys. Rev., American Physical Society, 56 (1939)</ref>. They basically state that the electromagnetic fields outside a given volume can be expressed as integrals over the surface enclosing the volume involving only the tangential components of the fields on the surface. If the observation point is located inside this volume, the integrals vanish.
 
By making use of the boundary conditions for the tangential field components on the scatterer surface
<math>\mathbf{n} \times (\mathbf{E}_{scat} + \mathbf{E}_{inc}) = \mathbf{E}_{int}</math> and <math>\mathbf{n} \times (\mathbf{H}_{scat} + \mathbf{H}_{inc}) = \mathbf{H}_{int}</math>, where <math>\mathbf{n}</math> is the normal vector to the scatterer surface, one can derive an integral representation of the scattered field in terms of the tangential components of the internal fields on the scatterer surface. A similar representation can be derived for the incident field.
 
By expanding the internal field in terms of SVWFs and exploiting their orthogonality on spherical surfaces, one arrives at an expression for the T-Matrix. Numerical codes for the evaluation of the T-Matrix can be found online [http://www.scattport.org/index.php/light-scattering-software/t-matrix-codes/list].
 
== References ==
{{reflist|2}}
 
{{DEFAULTSORT:T-matrix method}}
[[Category:Computational physics]]
[[Category:Electromagnetism]]
[[Category:Electrodynamics]]
[[Category:Scattering, absorption and radiative transfer (optics)]]

Revision as of 14:33, 18 August 2014

Glazier Anton Broadbent from Fort St. John, has pastimes which includes entertaining, property developers in singapore new condo launches and aircraft spotting. Last month very recently made a trip to Selous Game Reserve.

In mathematics, the Blasius function, introduced by Template:Harvs, is the solution to the third-order differential equation

2fxxx+ffxx=0

with the boundary conditions f(0) = fx(0) = 0, fx(∞) = 1.

See also

References

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