60 CHAPTER 6. STREAMING STUDIES
(a) (b)
−Fmount
Fdrag
Fext
Fmount
Fg
(c) (d)
Fixed hard walls
Viscous
fluid
Oscillating wall
w
h
water
water
1.955 1.960 1.965 1.970 1.975 1.980
100
75
50
25
0
f MHz
Eac #J m−3$
2 = 2
2 = 42
fres = 1.9670 MHz
Q = 416
=E = 66 t0
fres = 1.9647 MHz
Q = 205
=E = 33 t0
"f
fres
water
water
water
water
0 200 400 600 800 1000
1.2
1
0.8
0.6
0.4
0.2
0
Eac normalized
t/t0
Num. 2
Ana. 2
Num. 42
Ana. 42
f = fres
f = fres + 1
2
"f
Figure 6.5: Resonance curves and energy build-up in an acoustic cavity, and a comparison to the
analytical solution for a sinusoidally-driven damped harmonic oscillator. (a) Sketch of a pendulum, which
for small amplitudes can be considered a damped harmonic oscillator. The forces acting on the pendulum
are the gravitational force Fg, the drag force from the surrounding uid Fdrag, the force from the mounting
Fmount balancing the radial component of Fg, and the external oscillating force Fext. Here Fext is depicted
as a body force on the pendulum mass, but it could as well be a torque on the rotation axis. (b) Sketch of the
numerical model of the channel cross section, with arrows indicating the in-phase oscillations of the left and
right boundaries. (c) Resonance curves showing the steady acoustic energy density Eac versus the actuation
frequency f for the numerical model of the water-lled channel, with viscosity = water (brown) and
articially increased viscosity = 4water (purple). The inset exhibits the tted resonance frequency fres,
the quality factor Q = fres=f, where f is the full width at half maximum, and the resonance relaxation
1
time E = Q=(2fres). The maximum energy scales by , while the cavity quality factor Q scales by
2 .
(d) Build-up of the time-dependent energy Eac(t) versus time t for constant actuation amplitude. Graphs
for = water, numerical (brown) and analytical (green), and for = 4water, numerical (purple) and
analytical (pink), all four cases for actuation at resonance f = fres (long dashes) and for actuation o
resonance f = fres + f
2 (short dashes). The gure is an extended version of similar gures presented in
Ref. 30 Appendix E.