Fale akustyczne w ośrodku jednorodnym Stan równowagi: = e =const., p = p e =const, V e =0 Fale o małej amplitudzie: P tt – c s 2 p xx = 0 c s 2 = p e / e Związek dyspersyjny 2 =c s 2 k 2 Efekt Dopplera (V 0) = c s k + V e k
Fale akustyczne w ośrodku niejednorodnym Stan równowagi: = e (x), p = p e =const, V=0 Fale o małej amplitudzie: P tt – c s 2 (x)p xx = 0 Rozpraszanie – warunek Bragga k i k s = k h i s = h
The Sun l Dedicated satellite measurements are allowing many of the Sun’s properties to be understood. è Most of these are restricted to measurements of the solar surface.
A hypothesis or theory is clear, decisive, and positive but it is believed by none but the man who created it. Experimental findings, on the other hand are messy, inexact things which are believed by everyone except the man who did the work. Harlow Shapley
Równania Eulera ,t + ( V) = 0 [V,t + (V )V] =- p + g p,t + (pV) =(1- )p V
p 1 = p 2, z= (x,t) ( / t + V )( -z) = 0, z = Warunki brzegowe na międzypowierzchni z= (x,t) : z= (x,t)
Sound waves in simple random fields An example: A space-dependent random flow One-dimensional ( / y= / z=0) equilibrium: = 0 = const u e = u r (x) p e = p 0 = const
A weak random field: u r (x) = 0 The perturbation technique (e.g., Murawski & Roberts 1993) 2 -c 2 k 2 = 4k 2 - E( -k) d / [ 2 -c 0 2 2 ] For instance, Gaussian spectrum E(k) = ( 2 l x / exp(-k 2 l x 2 )
Approximate solution Expansion = c 0 k + 2 2 + 2 l x /c 0 = -2/ 1/2 k 2 l x 2 D(2kl x ) - ik 2 l x 2 [1-exp(-4k 2 l x 2 )] where D( )=exp(- 2 ) 0 exp(t 2 ) dt is Dawson's integral (Press et al. 1992).
Real (left) and imaginary (right) parts of 2 for V r (x).
Random Gaussian mass density for its typical realization.
Numerical (asterisks, diamonds) and analytical (dashed lines) data (Murawski & Mędrek 2002)
Various random fields Sound waves in random fields: =Re r - 0, a = Im r - 0. 0) denotes a red (blue) shift. a 0) corresponds to 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
Sound waves in complex fields An example: a space- and time-dependent random mass density field The dispersion relation for r (x,t): 2 - K 2 = 2 - - ( 2 E( -K, - )) d d / ( 2 - 2 ) K = kl x, = l x /c 0
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)
2 = K/(2 3/2 ) [c r 2 /(c r 2 -1) KD(2/c + K)] + i K 2 /(4 [1/c - +|c - / c + |1/c + exp(-4K 2 /c + 2 )] Dispersion relation: c = c r 1
Real (solid lines) and imaginary (dashed lines) parts of 2 versus K for c r =-2 (left) and c r =2 (right).
Real (solid line) and imaginary (dashed line) parts of 2 for K=2. An analogy with Landau damping in plasma physics.
Conclusions Random fields change frequencies and alter amplitudes of waves The random p-modes problem is analogous to the random trapped waves problem Numerical verification (Nocera et al. 2001, Murawski et al. 2001) A number of problems remain to be solved both analytically and numerically
Modelling improvement Gabriel (1976), Phil. Trans. A281, 339 Dowdy et al. (1986) Solar Phys., 105, 35