AST242 LECTURE NOTES PART 5 Contents 1. Waves and ...

AST242 LECTURE NOTES PART 5 13We consider perturbations of velocity, density, temperature and pressure in theform(78)ρ(x, t) = ρ 0 + ρ 1 e i(k·x−ωt)p(x, t) = p 0 + p 1 e i(k·x−ωt)T (x, t) = T 0 + T 1 e i(k·x−ωt)u(x, t) = u 1 e i(k·x−ωt)We assume the gas without perturbations is uniform and at rest (u 0 = 0, and p 0 , ρ 0 , T 0independent of x and time).Let consider perturbations to the heating and cooling functions(79) Q(ρ, T ) = Q + (ρ 0 , T 0 ) − Q − (ρ 0 , T 0 ) + Q ρ ρ 1 + Q T T 1so that(80)Q ρ = ∂(Q+ − Q − )∂ρQ T = ∂(Q+ − Q − )∂T∣∣ρ0 ,T 0∣∣ρ0 ,T 0To first order in our perturbations, the continuity equation, Euler’s equation, theequation of state and our equation for conservation of energy become(81)(82)(83)(84)Note γp 0ρ 0−iωρ 1 + ik · u 1 ρ 0 = 0−iωu 1 = −ik p 1p 1= ρ 1+ T 1p 0 ρ 0 T 0− iω (p 1 − γp )0ρ 1 = −λk 2 T 1 + Q T T 1 + Q ρ ρ 1γ − 1 ρ 0= c 2 s where c s is the sound speed. We can dot k with equation (82) to findρ 0(85) ωk · u 1 = k 2 p 1ρ 0and this can be used to eliminate the velocity from the continuity equation(86) ω 2 ρ 1 = k 2 p 1