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Krzysztof /Kris Murawski
UMCS Lublin
Frequency shift and amplitude alteration of waves in random fields
Outline:
1. Doppler effect
2. Motivation
3. Modelling of random waves
4. Summary
DopplerDoppler e effffeect ct
Acoustic waves in a homogeneous mediumAcoustic 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 + Ve k
Acoustic waves in an inhomogeneous mediumAcoustic 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
Global solar oscillations
P-Mode SpectrumP-Mode Spectrum
Solar granulation
Euler equationsEuler equations
t + (V) = 0
[Vt + (V)V] = -p + g
pt + (pV) = (1-) p V
Sound waves in simple random fieldsA space-dependent random flow
One-dimensional (/y=/z=0) equilibrium:
e= 0 = const.
ue = ur(x)
pe = p0 = const.
A weak random field
ur(x) = 0
The perturbation technique dispersion relation
2 2 – c– css22kk22 = 4k = 4k 2 2 --
E(E(-k) d-k) d / [ / [2 2 - - ccss
2222]]
For instance, Gaussian spectrum
E(k) = (2 lx / exp(-k2lx2)
Approximate solution
Expansion = c0k + 2 2 +
2 lx/c0 = - 2/1/2 k2lx2D(2klx)
- i k2lx2[1-exp(-4k2lx2)]
D() = exp(-2) 0 exp(t2) dtDawson's integral
Dispersion relation
Re(2) Im (2)
Re(2 ) < 0 frequency reduction
Red shift
Im(2 ) < 0 amplitude attenuation
Typical realization of a Random Gaussian field
Mędrek i Murawski (2002)
Random waves – numerical simulations
(Murawski & Mędrek 2002)
Numerical (asterisks, diamonds)
vs.
analytical (dashed lines) data
Sound waves in random fields
= Re r - 0, a = Im r - 0
< 0 (> 0) a red (blue) shift
a < 0 ( a > 0) attenuation (amplification)
r(x) r(t) ur(x) ur(t) pr(x) pr(t)
>0 >0 <0 >0 <0 <0
a <0 >0 <0 >0 <0 >0
Sound waves in complex fields
An example: r(x,t)
Dispersion relation
2 2 - K- K2 2 = =
22 ----((22 E( E(-K,-K,--)) d)) d d d / (/ (22--22))
K = klx
= lx/cs
Wave noise
E(K,E(K,) = ) = 22// E(K) E(K) --rr(K)) (K))
Spectrum
Dispersionless noise
rr(K) = c(K) = crr K K
r(x,t) = r(x-crt,t=0)
2 2 = K/(2= K/(23/23/2) [c) [crr22/(c/(crr
22-1) K D(2/c-1) K D(2/c+ + K)]K)]
+ i K+ i K22/(4/(4 [1/c [1/c--+|c+|c-- / c / c++|1/c|1/c++ exp(-4Kexp(-4K22/c/c++
22)])]
Dispersion relation:
cc = c = crr 1 1
Re 2 Im 2
cr = -2 cr = 2
Re(2)
Im(2)
K=2
An analogy with Landau damping in plasma physics
Re(2)
Im(2)
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