3
C. First-order time-domain equations
The homogeneous, isotropic, quiescent thermodynamic
equilibrium state is taken to be the zeroth-order state in
the acoustic perturbation expansion. Following standard
perturbation theory, all fields g are written in the form
g = g0 + g1, for which g0 is the value of the zerothorder
state, and g1 is the acoustic perturbation which
by definition has to be much smaller than g0. For the
velocity, the value of the zeroth-order state is v0 = 0, and
thus v = v1. The zeroth-order terms solve the governing
equations in the zeroth-order state and thus drop out of
the equations. Keeping only first-order terms, we obtain
the following first-order equations.
The first-order continuity equation (3b) becomes
s@tp1 = −∇· v1, (5)
and likewise, the momentum equation (4c) becomes
0@tv1 = ∇·
1 − p11
, (6a)
where 1 is given by
1 = 0
∇v1 + (∇v1)T
+
b
0 −
2
3
0
(∇· v1) 1. (6b)
Equations (5) and (6) determine together with a set of
boundary conditions the time evolution of the first-order
acoustic fields p1 and v1.
D. Second-order time-domain equations
Moving on to second-order perturbation theory, we
write the fields as g = g0 + g1 + g2, with g1 and g2 depending
on both time and space. For simplicity and in
contrast to Ref. 11, we do not include perturbations in
and b. This will cause the magnitude of the streaming to
be slightly off, as does the adiabatic approximation, however
the qualitative behavior is not expected to change.
The second-order time-domain continuity equation (3b)
becomes
s@tp2 = −∇· v2 − sv1 ·∇p1, (7)
and the momentum equation (4c) takes the form
0@tv2 = −1@tv1 +∇·
2 −p2 1
−0(v1 ·∇)v1, (8a)
where 2 is given by
2 = 0
∇v2 + (∇v2)T
+
b
0 −
2
3
0
(∇· v2) 1. (8b)
Using Eq. (1) in the form 1 = 0sp1 and the first-order
momentum equation (6a), we rewrite Eq. (8a) to
0@tv2 =∇·
2 − p2 1 − sp11 + 1
2sp 2
1 1
+ s∇p1 · 1 − 0(v1 ·∇)v1. (8c)
This particular form of the second-order momentum
equation is chosen to minimize numerical errors as described
in Section IIIA.
E. Periodic frequency-domain equations
When solving for the periodic state at t → ∞, it is
advantageous to formulate the first-order equations in
the frequency domain. The harmonic
first-
order fields
are all written as g1(r, t) = Re
gfd
e−i!t
gfd
1 (r), where 1
is the complex field amplitude in the frequency domain.
The first-order frequency-domain equations are derived
from Eqs. (5) and (6a) by the substitution @t → −i!,
1 − i!spfd
∇· vfd
1 = 0, (9)
∇·
fd
1 − pfd
1 1
+ i!0vfd
1 = 0. (10)
The steady time-averaged streaming flow is obtained
from the time-averaged
second-order frequency-domain
equations, where
gfd
2
denotes time averaging over one
oscillation period of the periodic second-order field. The
time-average of products of two harmonic first-order
fields gfd
1 and ˜gfd
1 is given by
gfd
1 ˜gfd
1
= 1
2 Re
��
gfd
1
∗˜gfd
1
,
as in Ref. 11, where the asterisk denotes complex conjugation.
In the periodic state, the fields may consist
of harmonic terms and a steady term, and thus all full
time-derivatives average to zero
@tgfd
2
= 0. The timeaveraged
second-order frequency-domain equations are
derived from Eqs. (7) and (4a),
∇·
vfd
2
+ s
vfd
1 ·∇pfd
1
= 0, (11)
∇·
fd
2
−
pfd
2
1 − 0
vfd
1 vfd
1
= 0. (12)
F. Acoustic energy and cavity Q-factor
The total acoustic energy of the system in the time
domain Eac(t) and in the frequency domain
Efd
ac(∞)
is
given by
Eac(t) =
Z
V
1
2
sp 2
1 +
1
2
0v 2
1
dV, (13a)
Efd
ac(∞)
=
Z
V
1
2
s
pfd
1 pfd
1
+
1
2
0
vfd
1 · vfd
1
dV.
(13b)
Moreover, the time derivative of Eac(t) is
@tEac =
Z
V
@t
1
2
sp 2
1 +
1
2
0v 2
1
dV
=
Z
V
sp1@tp1 + 0v1 · @tv1
dV
=
Z
V
∇·
h
v1 · (1−p11)
i
−∇v1 :1
dV, (14a)
where we have used Eqs. (5) and (6a). Applying Gauss’s
theorem on the first term in Eq. (14a), we arrive at
@tEac =
Z
A
h
v1 · (1 − p11)
i
· n dA −
Z
V ∇v1 : 1 dV
= Ppump − Pdis, (14b)