1 / 24

Krzysztof /Kris Murawski UMCS Lublin

Frequency shift and amplitude alteration of waves in random fields. Krzysztof /Kris Murawski UMCS Lublin. Outline:. Doppler effect Motivation Modelling of random waves Summary. Doppler e f f e ct. Acoustic waves in a homogeneous medium. Still equilibrium

adie
Télécharger la présentation

Krzysztof /Kris Murawski UMCS Lublin

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Frequency shift and amplitude alteration of waves in random fields Krzysztof /Kris Murawski UMCS Lublin

  2. Outline: • Doppler effect • Motivation • Modelling of random waves • Summary

  3. Doppler effect

  4. Acoustic waves in a homogeneous medium Still equilibrium e = const., pe = const, Ve = 0 Small amplitude waves Ptt – cs2 pxx = 0 cs2 =  pe/e Dispersion relation 2 = cs2k2 Flowing equilibrium (Ve 0) - Doppler effect  =  cs k + Vek

  5. Acoustic waves in an inhomogeneous medium Equilibrium e(x), pe = const, Ve = 0 Small amplitude waves Ptt – cs2(x) pxx = 0 Scattering – Bragg condition Ki  ks =  kh i s = h

  6. Global solar oscillations

  7. P-Mode Spectrum

  8. Solar granulation

  9. Euler equations • t + (V) = 0 • [Vt + (V)V] = -p + g • pt + (pV) = (1-) p V

  10. Sound waves in simple random fields A space-dependent random flow One-dimensional (/y=/z=0) equilibrium: e= 0 = const. ue = ur(x) pe = p0 = const.

  11. A weak random field ur(x)  = 0 The perturbation technique  dispersion relation 2 – cs2k2 = 4k 2 -  E(-k) d / [2 - cs22] For instance, Gaussian spectrum E(k) = (2 lx /) exp(-k2lx2)

  12. Approximate solution = c0k + 22 +  Expansion Dispersion relation 2 lx/c0 = - 2/1/2 k2lx2D(2klx) - i k2lx2[1-exp(-4k2lx2)] Dawson's integral D() = exp(-2) 0 exp(t2) dt

  13. Re(2) Im (2) Re(2) < 0  frequency reduction Red shift Im(2) < 0  amplitude attenuation

  14. Typical realization of a Random Gaussian field

  15. Random waves – numerical simulations Mędrek i Murawski (2002)

  16. Numerical (asterisks, diamonds) vs. analytical (dashed lines) data (Murawski & Mędrek 2002)

  17. Sound waves in random fields  = Re r - 0,  a = Im r - 0  < 0 ( > 0)  a red (blue) shift  a < 0 ( a > 0)  attenuation (amplification)

  18. Sound waves in complex fields An example:r(x,t) Dispersion relation 2 - K2 = 2-- (2 E(-K,-)) d d/ (2-2) K = klx  =  lx/cs

  19. Wave noise Spectrum E(K,) = 2/ E(K) (-r(K)) Dispersionless noise r(K) = cr K r(x,t) = r(x-crt,t=0)

  20. Dispersion relation: 2 = K/(23/2) [cr2/(cr2-1) K D(2/c+ K)] + i K2/(4) [1/c-+|c- / c+|1/c+ exp(-4K2/c+2)] c = cr 1

  21. Re 2 Im 2

  22. cr=-2 cr=2 Re(2) Im(2)

  23. Re(2) Im(2) K=2 An analogy with Landau damping in plasma physics

  24. Conclusions • Random fields alter frequencies and amplitudes of waves • Numerical verification of analytical results (Nocera et al. 2001, Murawski et al. 2001) • A number of problems remain to be solved both analytically • and numerically

More Related