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

3. Outline: 1.Doppler effect 2.Motivation 3.Modelling of random waves 4.Summary

4. Doppler effect

5. Acoustic waves in a homogeneous medium Still equilibrium  e = const., p e = const, V e = 0 Small amplitude waves P tt – c s 2 p xx = 0 c s 2 =  p e /  e Dispersion relation  2 = c s 2 k 2 Flowing equilibrium (Ve  0) - Doppler effect  =  c s k + V e k

6. Acoustic waves in an inhomogeneous medium Equilibrium  e (x), p e = const, V e = 0 Small amplitude waves P tt – c s 2 (x) p xx = 0 Scattering – Bragg condition K i  k s =  k h  i   s =   h

7. Global solar oscillations

8. P-Mode Spectrum

9. Solar granulation

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

11. Sound waves in simple random fields A space-dependent random flow One-dimensional (  /  y=  /  z=0) equilibrium:  e  =  0 = const. u e = u r (x) p e = p 0 = const.

12. A weak random field  u r (x)  = 0 The perturbation technique  dispersion relation  2 – c s 2 k 2 = 4k  2  -    E(  -k) d  / [  2 - c s 2  2 ] For instance, Gaussian spectrum E(k) = (  2 l x /   exp(-k 2 l x 2 )

13. Approximate solution Expansion  = c 0 k +  2  2 +   2 l x /c 0 = - 2/  1/2 k 2 l x 2 D(2kl x ) - i k 2 l x 2 [1-exp(-4k 2 l x 2 )] D(  ) = exp(-  2 )  0  exp(t 2 ) dt Dawson's integral Dispersion relation

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

15. Typical realization of a Random Gaussian field

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

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

18. Sound waves in random fields  = Re  r -  0,  a = Im  r -  0  0)  a red (blue) shift  a 0)  attenuation (amplification)  r (x)  r (t) u r (x)u r (t)p r (x)p r (t)  >0 <0>0<0  a <0>0<0>0<0>0

19. Sound waves in complex fields An example:  r (x,t) Dispersion relation  2 - K 2 =  2  -    -   (  2 E(  -K,  -  )) d  d  / (  2 -  2 ) K = kl x  =  l x /c s

20. Wave noise E(K,  ) =  2 /  E(K)  -  r (K)) Spectrum Dispersionless noise  r (K) = c r K  r (x,t) =  r (x-c r t,t=0)

21.  2 = K/(2  3/2 ) [c r 2 /(c r 2 -1) K D(2/c + K)] + i K 2 /(4  [1/c - +|c - / c + |1/c + exp(-4K 2 /c + 2 )] + i K 2 /(4  [1/c - +|c - / c + |1/c + exp(-4K 2 /c + 2 )] Dispersion relation: c  = c r  1

22. Re  2 Im  2

23. c r = -2 cr = 2cr = 2 Re(  2 ) Im(  2 )

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

25. 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