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sess11 - 11 Acoustic waves and shocks Reading Shu Vol 2 Ch...

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11. Acoustic waves and shocks Reading: Shu, Vol. 2, Ch. 15 11.1 Acoustic waves of low amplitude Let us consider an adiabatic (or isentropic s =const.) fluid in equilibrium at rest, in which the pressure will vary in coordination with the density P ρ γ γ = c p c V = ratio of specific heats (11 . 1) The pressure gradient therefore is ~ P = ∂P ∂ρ s ~ ρ = c 2 s ~ ρ (11 . 2) with the adiabatic speed of sound c s = c s, 0 ρ ρ 0 ! ( γ - 1) / 2 (11 . 3) where ρ 0 and c s, 0 = ∂P ∂ρ s,ρ 0 are the uniform density and sound speed. Let us for simplicity consider only one spatial dimension. Noting that the adiabatic relation 11.1 replaces the energy equation we write the remaining hydrodynamical equations as ∂ρ ∂t + u ∂ρ ∂x + ρ ∂u ∂x = 0 (11 . 4) ∂u ∂t + u ∂u ∂x = - 1 ρ ∂P ∂x = - c 2 s ρ ∂ρ ∂x (11 . 5) If we assume that the fluid quantities are only minimally perturbed, ρ = ρ 0 + ρ 1 u = u 1 P = P 0 + P 1 (11 . 6) we can write down perturbation equations containing only the terms that are linear in the perturbations. ∂ρ 1 ∂t + ρ 0 ∂u 1 ∂x = 0 (11 . 7) ∂u 1 ∂t = - c 2 s, 0 ρ ∂ρ 1 ∂x (11 . 8) and combining these two equations 2 ρ 1 ∂t 2 = c 2 s, 0 2 ρ ∂x 2 ρ 1 = f ( x - c s, 0 t ) + g ( x + c s, 0 t ) (11 . 9) 1
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The perturbations propagate with the sound speed an do not change their shape! This is equivalent to saying that in the low-amplitude limit sound waves do not show dispersion, ω is not a function of k and the phase velocity of sound waves is constant. Does that still hold for sound waves of finite amplitude? Equation 11.3 tells us that can no longer set the adiabatic sound speed constant, so some dispersion would occur. This is the result of the non-linear nature of the hydrodynamical equations. On should note that wave damping, e.g. by viscous friction, adds other complications to the dispersion relation. 11.2 Acoustic waves with finite amplitudes We now have to study Eq.11.4 and 11.5, where we would like to replace the differentials in ρ by those in c s . Using Eq.11.3 c s = c s, 0 ρ ρ 0 ! ( γ - 1) / 2 dc s = c s ρ (11 . 10) we obtain ∂t 2 γ - 1 c s ! + u ∂x 2 γ - 1 c s ! + c s ∂u ∂x = 0 (11 . 11) ∂u ∂t + u ∂u ∂x + c s ∂x 2 γ - 1 c s ! = 0 (11 . 12) Adding and subtracting these two coupled equations yields two individual equations " ∂t + ( u + c s ) ∂x # u + 2 γ - 1 c s ! = 0 (11 . 13) " ∂t + ( u - c s ) ∂x # u - 2 γ - 1 c s ! = 0 (11 . 14) The solutions are R u + 2 γ - 1 c s = const . on characteristic dx dt = u + c s (11 . 15) L u - 2 γ - 1 c s = const . on characteristic dx dt = u - c s (11 . 16) Let us assume there was on one wave moving to the right (in positive x-direction), so the L -solution is a strict constant. The volume in front of the wave is therefore unperturbed and homogeneous. In particular, the L -solution is given by the unperturbed velocity and sound speed L u - 2 γ - 1 c s = - 2 γ - 1 c s, 0 (11 . 17) 2
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L and R are independent solutions and thus (11.17) must be valid also in the region occupied
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