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