The Wave and Helmholtz Equations

CMSC828D notes (adapted from material written with Nail Gumerov). Work in progress 1 Acoustic Waves 1.1 Barotropic Fluids The usual assumptions for acoustic problems are that acoustic waves are perturbations of the medium (fluid) density ρ (r,t), pressure p (r,t), and mass velocity, v (r,t), where t is time. It is also assumed that the medium is inviscid, and that perturbations are small, so that ρ = ρ0 + ρ 0, p = p0 + p0, ρ0 ¿ ρ0, p0 ¿ p0, |v0| ¿ c ∼ r p0 ρ0 . (1) Here the perturbations are near an initial spatially uniform state (ρ0, p0) of the fluid at rest (v0 = 0) and are denoted by primes. The latter equation states that the mass velocity of the fluid is much smaller than the speed of sound c in that medium. In this case the linearized continuity (mass conservation) and momentum conservation equations can be written as ∂ρ0 ∂t +∇ · (ρ0v 0) = 0, ρ0 ∂v0 ∂t +∇p0 = 0, (2) where ∇ = ix ∂ ∂x + iy ∂ ∂y + iz ∂ ∂z , (3) is the invariant “nabla” operator, represented by formula (3) in Cartesian coordinates, where (ix, iy, iz) are the Cartesian basis vectors. Differentiating the former equation with respect to t and excluding from the obtained expression ∂v0/∂t due to the latter equation, we obtain ∂2ρ0 ∂t2 = ∇2p0. (4) Note now that system (2) is not closed since the number of variables (three components of velocity, pressure, and density) is larger than the number of equations. The relation needed to close the system is equation of state, which relates perturbations of the pressure and density. The simplest form of this relation is provided by barotropic fluids, where the pressure is a function of density only: p = p(ρ). (5) We can expand this in series near the unperturbed state p = p(ρ0) + dp dρ ̄̄̄̄ ρ=ρ0 (ρ− ρ0) +O 3 (ρ− ρ0) 2 ́ . (6)