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3.1 Linear waves

We begin by assuming that we have a steady, wave-free, non-magnetic background state. For discussions of background states see for example Cox (1980Jump To The Next Citation Point) or Unno et al. (1989Jump To The Next Citation Point). In general, the background will satisfy the force balance
ρ0v0 ⋅ ∇v0 = − ∇p0 + ρ0g0, (2 )
where ρ 0, p 0, v 0, and g 0 are the density, pressure, velocity, and gravitational acceleration in the background state. The energy equation and equation of state must also be satisfied in the background state (see, e.g., Cox, 1980Unno et al., 1989, for details).

We describe the wave motions that occur on top of the background state by the displacement ξ(r, t), which is the displacement of a fluid parcel that would have been at location r at time t had there been no wave motion. The continuity equation for the wave motion is then

δρ + ρ0∇ ⋅ ξ = 0, (3 )
where δρ is the Lagrangian density perturbation. Notice that Equation (3View Equation) holds even when the background flow v0 is non zero. The momentum equation can be written as (Lynden-Bell and Ostriker, 1967Jump To The Next Citation Point)
ℒ ξ = 0, (4 )
with
d20ξ- ℒ ξ = − ρ0dt2 + ∇ [γp0∇ ⋅ ξ + ξ ⋅ ∇p0 ] − (∇ ⋅ ξ) ∇p0 − ξ ⋅ ∇ (∇p0 ). (5 )
The symbol d0∕dt = ∂t + v0 ⋅ ∇ is the material derivative in the background flow. To obtain Equation (4View Equation) Lynden-Bell and Ostriker (1967) assumed that the Lagrangian pressure and density changes associated with the wave are related by
δp-= γδρ-. (6 ) p0 ρ0
For the case of adiabatic motion, γ is the first adiabatic exponent. Equation (4View Equation) is the equation of motion for small amplitude waves that we will refer to throughout this review.
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