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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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Frequency shift and amplitude alteration of waves in random fields Krzysztof /Kris Murawski UMCS Lublin
Outline: • Doppler effect • Motivation • Modelling of random waves • Summary
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
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
Euler equations • t + (V) = 0 • [Vt + (V)V] = -p + g • pt + (pV) = (1-) p V
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.
A weak random field ur(x) = 0 The perturbation technique dispersion relation 2 – cs2k2 = 4k 2 - E(-k) d / [2 - cs22] For instance, Gaussian spectrum E(k) = (2 lx /) exp(-k2lx2)
Approximate solution = c0k + 22 + 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
Re(2) Im (2) Re(2) < 0 frequency reduction Red shift Im(2) < 0 amplitude attenuation
Random waves – numerical simulations Mędrek i Murawski (2002)
Numerical (asterisks, diamonds) vs. analytical (dashed lines) data (Murawski & Mędrek 2002)
Sound waves in random fields = Re r - 0, a = Im r - 0 < 0 ( > 0) a red (blue) shift a < 0 ( a > 0) attenuation (amplification)
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
Wave noise Spectrum E(K,) = 2/ E(K) (-r(K)) Dispersionless noise r(K) = cr K r(x,t) = r(x-crt,t=0)
Dispersion relation: 2 = K/(23/2) [cr2/(cr2-1) K D(2/c+ K)] + i K2/(4) [1/c-+|c- / c+|1/c+ exp(-4K2/c+2)] c = cr 1
Re 2 Im 2
cr=-2 cr=2 Re(2) Im(2)
Re(2) Im(2) K=2 An analogy with Landau damping in plasma physics
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