2.4. THE ACOUSTIC BOUNDARY LAYER AND ACOUSTIC STREAMING 21
propagation parallel to a wall, at which the velocity eld has to fulll a no-slip boundary
condition,
v = 0; at wall. (2.58)
The amplitude of the velocity wave thus have to undergo a transition, in the direction
perpendicular to it propagation, from its amplitude va in the bulk to zero at the wall. This
transition region is called the acoustic boundary layer.
The starting point of the boundary treatment is a standing acoustic wave in the bulk
of the uid, far from the wall, given by an expression much similar to the orthogonally
oscillating plates solution Eq. (2.31),
vBulk
y1 (y; t) = Re
n
iva cos(k0y)e��i!t
o
; (2.59)
Bulk
1 (y; t) = Re
n
��
va
cs
0 sin(k0y)e��i!t
o
; (2.60)
where va is the amplitude of the velocity oscillations far from the wall. We note that the
complex i in the expression for vBulk
y1 indicates a phase shift between the velocity and
density oscillations. This out of phase oscillation of v1 and 1 was the reason why the
time-averaged product
1v1
��j1jjv1j was a factor of �� smaller than expected from
dimensional analysis in the orthogonally oscillating plates example Section 2.3.2.
Now we introduce a planar wall at z = 0, and to accommodate the no-slip condition
vy1(z = 0) = 0, we make the following ansatz for vy1(y; z; t),
vy1(y; z; t) = Re
n
iva cos(k0y)f(z)e��i!t
o
; (2.61a)
f(z) ! 1; for z ! 1; (2.61b)
f(z) ! 0; for z ! 0: (2.61c)
The task in now to determine the function f(z).
The governing equation in the frequency domain for v1 and 1, originally stated in
Eq. (2.10), are
��i!0v = ��c2s
r1 + r2v1 +
1
3 + b
r(r v1) ; (2.62a)
��i!1 = ��0r v1: (2.62b)
Substituting r v1 in Eq. (2.62a) with Eq. (2.62b) yields,
2
!0r2v1 + i2v1 ��
2!
k00
(1 �� i��0)r1 = 0; ��0 =
!
c2s
1
3
+
b
: (2.63)
The prefactor to the Laplace term is of great importance, as it leads to the denition of
the viscous acoustic boundary layer thickness s given by
s =
s
2
!0
: (2.64)