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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

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Krzysztof /Kris Murawski UMCS Lublin

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  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

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