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5 Acoustics in Biology and Medicine
resulting wave, 1 we apply Newton’s second Law to the mass δm between the two
nearby surface segments:
(p(x) − p(x + δx))A = δm
d 2 ξ
dt 2
(5.1)
with A being the transverse area of the segments on which the pressures act.
Dividing by the area A and the initial thickness between the surface segments, δx,
then Eq. (5.1) reads
−
∂p
∂x
= ρ
∂ 2 ξ
∂t 2 .
(5.2)
where ρ is the density of the material at the position x.
The pressure variations in the material arises from changes in the material
density. For small relative displacements, the pressure will be the ambient pressure
p 0 when there is no wave plus the pressure variation because of density variation as
the wave passes. As a Taylor series, we have
p = p o +
dp
dρ
o
(ρ − ρ o ) +
1
2
d 2 p
dρ 2
o
(ρ − ρ o )
2
+ · · · .
(5.3)
Following Beyer, 2 this is often written as
p = p o + A (ρ − ρ o )/ρ o +
1
2
B(ρ − ρ o )
2 /ρ
2
o + · · · .
(5.4)
The dimensionless ratio B/A characterizes the degree of non-linearity the material
shows in response to the passing wave. In Table 5.1, values of B/A are given for
some common materials.
Table 5.1 Beyer’s
parameters B/A for
common materials
Material
B/A
Temp. ( ◦ C)
Air
0.4
20
Distilled water
5.0
20
Salt water
5.4
20
Ethanol
10.5
20
Hemoglobin (50%)
7.6
30
Liver
6.5
30
Fat
9.9
30
1 This will be a ‘longitudinal wave, i.e. a wave whose displacement occurs along the same line as
the wave moves. A wave with displacements perpendicular to the direction of movement of the
wave is called a ‘transverse wave’.
2 R.T. Beyer, Parameters of Non-linearity in Fluids, J Acoust Soc Am 32, 719–721 (1960).