3.3. PERTURBATION THEORY 29
The Energy conservation equation can also be written in Lagrangian form, similar to
the NavierStokes formulation of momentum conservation. From Eq. (3.11b) we use the
continuity equation as before and resubstitute X,
(@t + (v r)) "+@t
��1
2v2
= r
v+kthrT
��(vr)
��1
2v2
��
��1
2v2
(rv): (3.12a)
Using the continuity equation (3.9a) to rewrite r v = ��1
(@t + v r), and letting the
dierential operators work on the product 1
2v2, yields
@t + (v r)
(" + 1
2v2) = r
v + kthrT
: (3.12b)
When divided by the density , this equation states that the material derivative @t+(vr)
of the energy per mass " + 1
2v2 equals the energy loss per mass due to dissipation and
thermal conduction.
This concludes the treatment of the governing equations for the conservation of mass,
momentum, and energy in a compressible Newtonian uid.
3.3 Perturbation theory
The coupled non-linear equations for mass, momentum, and energy conservation can be
solved directly using advanced non-linear solvers, and this is indeed necessary in situations
with large amplitudes of the acoustic elds, such as shock waves and noise from jet
engines. However, for situations where the acoustic eld is a small perturbation to the
equilibrium state, i.e. the amplitude of the acoustic density oscillations is small compared
to the equilibrium density, the coupled nonlinear governing equations can be expanded
into sets of coupled linear equations. In this thesis and the papers presented, we use the
notation of implicit perturbation parameter, and there is no background ow v0 = 0. The
perturbation expansion to second order becomes
= 0 + 1 + 2 ; 0 j1j j2j ; (3.13a)
p = p0 + p1 + p2 ; p0 jp1j jp2j ; (3.13b)
T = T0 + T1 + T2 ; T0 jT1j jT2j ; (3.13c)
v = 0 + v1 + v2 ; cs jv1j jv2j ; (3.13d)
where the subscripts indicate the perturbation order. 0 and T0 are simply the equilibrium
density and temperature of the uid, both in the case of a gas and a liquid. p0 is the
equilibrium pressure in the case of a gas, however, it is not the ambient equilibrium pressure
in the case of a liquid. When treating liquids, as we do in the studies presented in the
papers, p0 should be thought of as an energy density related to the intermolecular bonds in
the liquid. p0 is not discussed in the literature, since only the gradient and time-derivative
of the pressure is present in the governing equations, when the energy conservation equation
has been reduced to the form Eq. (3.11d). The separation of the nonlinear governing
equations into sets of linear equations is achieved by inserting the perturbation expansion
Eq. (3.13) into the governing equations (3.9b), (3.10a), and (3.11d), and separating the