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 (1980

) or Unno et al. (1989

). In general, the background will satisfy the force balance

$r0v0 . \~/ v0 = - \~/ p0 + r0g0, (2)$

where $r 0$ , $p 0$ , $v 0$ , 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, 1980 ; Unno et al., 1989 , for details).

We describe the wave motions that occur on top of the background state by the displacement $q(r, t)$ , 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

$dr + r0 \~/ .q = 0, (3)$

where $dr$ is the Lagrangian density perturbation. Notice that Equation (3

) holds even when the background flow $v0$ is non zero. The momentum equation can be written as (Lynden-Bell and Ostriker, 1967

)

with

$d20q- Lq = - r0dt2 + \~/ [gp0 \~/ .q + q . \~/ p0] - ( \~/ .q) \~/ p0 - q . \~/ ( \~/ p0) . (5)$

The symbol $d0/dt = @t + v0 . \~/$ is the material derivative in the background flow. To obtain Equation (4

) Lynden-Bell and Ostriker (1967 ) assumed that the Lagrangian pressure and density changes associated with the wave are related by

For the case of adiabatic motion, $g$ is the first adiabatic exponent. Equation (4

) is the equation of motion for small amplitude waves that we will refer to throughout this review.