Biomedical Engineering Reference
In-Depth Information
Limit III
The next significant limit is the case of the maximum damping coefficient.
This arises when the frequencies of the transverse and longitudinal modes at
the surface are equal. The frequency and damping coefficient are then given,
respectively, as
ω
o
=
1-
10
16
σ
k
3
ρ
-9
/
17
ψ
(24)
and
=
√
2
2
ση
1+
32
.
1
/
4
2
k
7
ρ
4
11
ψ
-5
/
α
(25)
3
The limit of maximum damping coefficient at intermediate elasticities was
first recognized by Dorrenstein [49] and corresponds to the maximum res-
onant coupling of the two modes at the surface, and the correction to the
original Dorrenstein equation is prescribed by Esker. Lucassen-Reynders and
Lucassen [55] provide some insight into the nature of this limit, which is illus-
trated in Fig. 5. The maximum in the damping coefficient signifies a change
in the phase between the longitudinal and transverse modes from 90
◦
at the
pure liquid, Limit I, to 180
◦
at the maximum damping coefficient, Limit III.
Hence, the motion of the surface particles (monolayer) is at a 45
◦
angle to the
propagation direction of the capillary waves.
Fig. 5
Wave motion at maximum damping and infinite dilational elasticity.
A
Motion at
the maximum damping coefficient where optimal resonant mode coupling implies that
a surface fluid element moves at a 45
◦
angle to the direction of wave propagation.
B
Wa v e
Motion at infinite dilational elasticity, where the same element is only able to move in the
transverse direction
