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Dispersive Waves

Dispersive Waves. Dispersive Waves ( c=c(f) ). Non-Dispersive Waves. C 1 = /k 1. C 2 = /k 2. w 2. w 1. >. w. P( x,t ) = Acos ( k 1 x- w 1 t). + Acos ( k 2 x- w 2 t). k. Dispersive Waves. cos ( 1/2 {k 1 - k 2 )x – ½ { w 1 - w 2 )t ). Long l. Short l.

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Dispersive Waves

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  1. Dispersive Waves

  2. Dispersive Waves (c=c(f))

  3. Non-Dispersive Waves C1= /k1 C2= /k2 w 2 w 1 > w P(x,t) = Acos(k1x-w1t) + Acos(k2x-w2t) k Dispersive Waves cos( 1/2{k1-k2)x – ½ {w 1-w 2)t ) Long l Short l = 2Acos( 1/2{k1+k2)x – ½ {w 1+w 2)t ) {w 1+w 2)/(k1+k2) = Vavg dw/dk~{w 1-w 2)/(k1-k2) = Vgroup Long l Short l x

  4. Dispersive Waves Which one is group?

  5. dw/dk = vgroup w Dispersive Waves w/k = vphase k If both phase and group having same sign then they go in same directions; otherwise go in opposite directions Long l Short l x

  6. Dispersive Waves http://www.thefullwiki.org/Rayleigh_wave

  7. Dispersive Waves Model S40RTS is our more recently developed shear velocity model of the mantle. The model represesnts isotropic shear velocity perturbations to the (anisotropic) PREM oodel that provide the best fit to 20 million Rayleigh wave dispersion [Van Heijst & Woodhouse, 1999], 500,000 shear-wave Traveltime [Ritsema & Van Heijst, 2001], and 1100 normal-mode Splitting function.measurements. The construction of S40RTS is discussed in Ritsema, J., A. Deuss, H.J. van Heijst & J.H. Woodhouse, Geophys. J. Int., 2010.

  8. Dispersive Waves Model S40RTS is our more recently developed shear velocity model of the mantle. The model represesnts isotropic shear velocity perturbations to the (anisotropic) PREM oodel that provide the best fit to 20 million Rayleigh wave dispersion [Van Heijst & Woodhouse, 1999], 500,000 shear-wave Traveltime [Ritsema & Van Heijst, 2001], and 1100 normal-mode Splitting function.measurements. The construction of S40RTS is discussed in Ritsema, J., A. Deuss, H.J. van Heijst & J.H. Woodhouse, Geophys. J. Int., 2010.

  9. Dispersive Waves

  10. K F-K transform Measured Dispersion Curve K K/w w w = Cphase K Fund. Mode Phase velocity Fund. Mode Group velocity 1/w

  11. Rayleigh Dipsersion Curve

  12. Rayleigh Dipsersion Curve Predicted Dipsersion Curve

  13. Rayleigh Dipsersion Curve Predicted Dipsersion Curve

  14. Rayleigh Dipsersion Curve Predicted Dipsersion Curve

  15. Optical Dispersive Waves(Intrinsic Dispersion)

  16. (Interference Dispersion) Water Dispersive Waves

  17. NDE EM Dispersive Waves

  18. NDE EM Dispersive Waves http://www.ndt.net/article/wcndt00/papers/idn508/idn508.htm

  19. Cloak EM Dispersive Waves By installing a transmission-line network through the structural parts of this strongly scattering object, the radiation waves, guided by the network, are made to go through the object like water finding its course (see Video 2).The transmission-line network thus makes the mesh-like object transparent ("invisible"): in other words, it does not scatter light in any direction in practice," explains the researcher. http://www.tkk.fi/en/current_affairs/news/view/vaitos-esineet_nakymattomiksi_siirtojohtoverkoilla/ In his dissertation, PekkaAlitalo, M.Sc. (Eng.), developed a new method to make objects invisible. The method is based on transmission-line networks, which can transfer electromagnetic radiation through, for example, a metallic mesh-like object. There has been no previous research from this viewpoint on transparency, i.e. electromagnetic cloaking, as it is called. The idea can be applied to many different practical problems; when compared with other scientifically proved "invisibility techniques", this technique works in a comparatively wide frequency range and can be used with transmission-line structures that are easy to manufacture. PekkaAlitalo will publicly defend his doctoral thesis at Helsinki University of Technology (TKK) on 7th August 2009.

  20. dw/dk = vgroup w/k = vphase Group (dw/dk) & Phase Velocity (w/k) might not be Parallel

  21. interference attenuation ikx-ax e Apparent vs Intrinsic Dispersion ax = [Re(a)+Im(a)]x Hilbert Transform pair Short l x

  22. Homogeneous, non-attenuative media: Cphase = Cgroup; Group and Phase velocity vectors ||. • 1. Cphase = w/k; Cgroup=dw/dk. • 2. • 3. Layered Media: Guided waves, surface waves exhibit apparent dispersion (interference effects):Cphase=Cgroup Summary cos(Cgroupx-wt) cos(Cphasex-wt) • Measured dispersion curves Cphase(w) VS(x,z) • 5. Attenuation coefficient a complex a for • satisfaction of causality. Intrinsic dispersion.

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