Introduction to Acoustics

262 Part B Physical and Nonlinear Acoustics
Part B 8.4
quantities, namely the variation of sound velocity
with pressure and temperature. Introducing the
definition of the sound velocity the following formulation
chain holds: (∂2 p/∂ϱ2 )s,ϱ=ϱ0 = (∂c2 /∂ϱ)s,ϱ=ϱ0 =
2c0(∂c/∂ϱ)s,ϱ=ϱ0 = 2c0(∂c/∂p)s,p=p0 (∂p/∂ϱ)s,ϱ=ϱ0 =
2c3(∂c/∂p)s,p=p0 0 . Insertion of this result into (8.9)
yields
B = ϱ 2 �
∂2 p
0
∂ϱ2 �
= 2ϱ
s,ϱ=ϱ0
2 0c3 � �
∂c
0
(8.14)
∂p s,p=p0
and into (8.13) results in
� �
B ∂c
= 2ϱ0c0
. (8.15)
A ∂p s,p=p0
Here B/A is essentially given by the variation of
sound velocity c with pressure p at constant entropy
s. This quantity can be measured with sound waves
when the pressure is varied sufficiently rapidly but still
smoothly (no shocks) to maintain isentropic conditions.
Equation (8.15) can be transformed further [8.44,53]
using standard thermodynamic manipulations and definitions.
Starting from c = c(p, t, s = const.) it follows
that dc = (∂c/∂p)T,p=p0 dp + (∂c/∂p)p,T=T0 dT or
� � � �
∂c
∂c
=
∂p s,p=p0
∂p T,p=p0 � � � �
∂c ∂T
+
.
∂T p,T=T0
∂p s,p=p0
(8.16)
From the general thermodynamic relation Tds =
cpdT − (αT /ϱ)Tdp = 0 for the isentropic case the relation
�∂T �
=
∂p
T0αT
follows, where
. (8.17)
ϱ0cp
αT = 1
� �
∂V
=−
V ∂T p,T=T0
1
� �
∂ϱ
ϱ0 ∂T p,T=T0
(8.18)
is the isobaric volume coefficient of the thermal expansion
and cp is the specific heat at constant pressure of
the liquid. Insertion of (8.17) into(8.16) together with
(8.15) yields
� �
B ∂c
= 2ϱ0c0
A ∂p T,p=p0
+ 2 c0T0αT
� �
∂c
. (8.19)
ϱ0cp ∂T p,T=T0
This form of B/A divides its value into an isothermal
(first) and isobaric (second) part. It has been found that
the isothermal part dominates.
For liquids, B/A mostly varies between 2 and 12.
For water under normal conditions it is about 5. Gases
(with B/A smaller than one, see before) are much less
nonlinear than liquids. Water with gas bubbles, however,
mayhaveaverylargevalueofB/A, strongly depending
on the volume fraction, bubble sizes and frequency of
the sound wave. Extremely high values on the order of
1000 to 10 000 have been reported [8.54, 55]. Tables
with values of B/A for a large number of materials can
be found in [8.43, 44].
As water is the most common fluid two tables
(Table 8.1, 2) ofB/A for water as a function of temperature
and pressure are given (see also [8.56]). Two
more tables (Table 8.3, 4) list B/A values for organic liquids
and for liquid metals and gases, both at atmospheric
pressure.
s,p=p0
8.4 The Coefficient of Nonlinearity β
In a sound wave the propagation velocity dx/dt of
a quantity (pressure, density) as observed by an outside
stationary observer changes along the wave. As
shown by Riemann [8.57] andEarnshaw [8.58], for
a forward-traveling plane wave it is given by
dx
= c + u , (8.20)
dt
c being the sound velocity in the medium without the
particle velocity u introduced by the sound wave. The
sound velocity c is given by (8.1), c2 = (∂p/∂ϱ)s, and
contains the nonlinear properties of the medium. It is
customary to incorporate this nonlinearity in (8.20) in
a second-order approximation as [8.59, 60]
dx
dt = c0 + βu , (8.21)
introducing a coefficient of nonlinearity β.Herec0is the
sound velocity in the limit of vanishing sound pressure
amplitude. The coefficient of nonlinearity β is related to
the parameter of nonlinearity B/A as derived from the
Taylor expansion of the isentropic equation of state [8.7]
via
β = 1 + B
. (8.22)
2A